diff options
| -rw-r--r-- | Changelog | 25 | ||||
| -rw-r--r-- | NOTICE | 2 | ||||
| -rw-r--r-- | README | 11 | ||||
| -rw-r--r-- | doc/Makefile | 2 | ||||
| -rw-r--r-- | doc/bibliography/bibliography.tex | 7 | ||||
| -rw-r--r-- | doc/libs/code.sty | 2 | ||||
| -rw-r--r-- | doc/libs/dlmf.sty | 2 | ||||
| -rw-r--r-- | doc/libs/ian.cls | 53 | ||||
| -rw-r--r-- | doc/libs/iantheo.sty | 2 | ||||
| -rw-r--r-- | doc/libs/largearray.sty | 2 | ||||
| -rw-r--r-- | doc/libs/point.sty | 12 | ||||
| -rw-r--r-- | doc/simplesolv-doc.tex | 1110 | ||||
| -rw-r--r-- | src/anyeq.jl | 1483 | ||||
| -rw-r--r-- | src/chebyshev.jl | 323 | ||||
| -rw-r--r-- | src/easyeq.jl | 1036 | ||||
| -rw-r--r-- | src/integration.jl | 50 | ||||
| -rw-r--r-- | src/interpolation.jl | 36 | ||||
| -rw-r--r-- | src/main.jl | 168 | ||||
| -rw-r--r-- | src/multithread.jl | 16 | ||||
| -rw-r--r-- | src/optimization.jl | 94 | ||||
| -rw-r--r-- | src/potentials.jl | 58 | ||||
| -rw-r--r-- | src/print.jl | 2 | ||||
| -rw-r--r-- | src/simpleq-Kv.jl | 68 | ||||
| -rw-r--r-- | src/simpleq-hardcore.jl | 486 | ||||
| -rw-r--r-- | src/simpleq-iteration.jl | 49 | ||||
| -rw-r--r-- | src/tools.jl | 22 |
26 files changed, 4295 insertions, 826 deletions
diff --git a/Changelog b/Changelog new file mode 100644 index 0000000..9e363cf --- /dev/null +++ b/Changelog @@ -0,0 +1,25 @@ +0.4: + + * feature: compute the 2-point correlation function in easyeq. + + * feature: compute the Fourier transform of the 2-point correlation function + in anyeq and easyeq. + + * feature: compute the local maximum of the 2-point correlation function and + its Fourier transform. + + * feature: compute the compressibility for anyeq. + + * feature: allow for linear spacing of rho's. + + * feature: print the scattering length. + + * change: ux and uk now return real numbers. + + * fix: error in the computation of the momentum distribution: wrong + definition of delta functions. + + * fix: various minor bugs. + + * optimization: assign explicit types to variables. + @@ -1,4 +1,4 @@ simplesolv: Licensed under the Apache 2.0 License (see LICENSE for details) -Copyright Ian Jauslin 2021 +Copyright Ian Jauslin 2021-2023 @@ -29,7 +29,9 @@ published in the following papers: * E.A. Carlen, I. Jauslin, E.H. Lieb - Analysis of a simple equation for the ground state energy of the Bose gas II: Monotonicity, Convexity and - Condensate Fraction, to appear in the SIAM Journal on Mathematical Analysis + Condensate Fraction, SIAM Journal on Mathematical Analysis, volume 53, + number 5, pages 5322-5360, 2021 + https://doi.org/10.1137/20M13768 https://arxiv.org/abs/2010.13882 http://ian.jauslin.org/publications/20cjl @@ -40,6 +42,13 @@ published in the following papers: https://arxiv.org/abs/2011.10869 http://ian.jauslin.org/publications/20chjl + * I. Jauslin - The Simplified approach to the Bose gas without translation + invariance, 2023 + http://ian.jauslin.org/publications/23j + + * I. Jauslin - Evidence of a liquid phase in repulsive Bosonic systems using + the Simplified approach to the Bose gas, 2023 + http://ian.jauslin.org/publications/23j_b Dependencies: diff --git a/doc/Makefile b/doc/Makefile index 4b58b3c..d0c4c64 100644 --- a/doc/Makefile +++ b/doc/Makefile @@ -1,4 +1,4 @@ -## Copyright 2021 Ian Jauslin +## Copyright 2021-2023 Ian Jauslin ## ## Licensed under the Apache License, Version 2.0 (the "License"); ## you may not use this file except in compliance with the License. diff --git a/doc/bibliography/bibliography.tex b/doc/bibliography/bibliography.tex index 53454da..e8893ca 100644 --- a/doc/bibliography/bibliography.tex +++ b/doc/bibliography/bibliography.tex @@ -4,10 +4,13 @@ doi:{\tt\color{blue}\href{http://dx.doi.org/10.1103/PhysRevA.103.053309}{10.1103 \bibitem[CJL20]{CJL20}E.A. Carlen, I. Jauslin, E.H. Lieb - {\it Analysis of a simple equation for the ground state energy of the Bose gas}, Pure and Applied Analysis, volume~\-2, issue~\-3, pages~\-659-684, 2020,\par\penalty10000 doi:{\tt\color{blue}\href{http://dx.doi.org/10.2140/paa.2020.2.659}{10.2140/paa.2020.2.659}}, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/1912.04987}{1912.04987}}.\par\medskip -\bibitem[CJL20b]{CJL20b}E.A. Carlen, I. Jauslin, E.H. Lieb - {\it Analysis of a simple equation for the ground state of the Bose gas II: Monotonicity, Convexity and Condensate Fraction}, 2020, to appear in the SIAM journal of Mathematical Analysis,\par\penalty10000 -arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/2010.13882}{2010.13882}}.\par\medskip +\bibitem[CJL21]{CJL21}E.A. Carlen, I. Jauslin, E.H. Lieb - {\it Analysis of a Simple Equation for the Ground State of the Bose Gas II: Monotonicity, Convexity, and Condensate Fraction}, SIAM Journal on Mathematical Analysis, volume~\-53, number~\-5, pages~\-5322-5360, 2021,\par\penalty10000 +doi:{\tt\color{blue}\href{http://dx.doi.org/10.1137/20M1376820}{10.1137/20M1376820}}, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/2010.13882}{2010.13882}}.\par\medskip \bibitem[DLMF]{DLMF1.1.3}F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders, H.S. Cohl, M.A. McClain (editors) - {\it NIST Digital Library of Mathematical Functions}, Release~\-1.1.3 of~\-2021-09-15, 2021.\par\medskip +\bibitem[Ja23]{Ja23}I. Jauslin - {\it The Simplified approach to the Bose gas without translation invariance}, 2023\par\penalty10000 +{\tt\color{blue}\href{http://ian.jauslin.org/publications/23j/}{http://ian.jauslin.org/publications/23j/}}.\par\medskip + \bibitem[Ta87]{Ta87}Y. Taguchi - {\it Fourier coefficients of periodic functions of Gevrey classes and ultradistributions}, Yokohama Mathematical Journal, volume~\-35, pages~\-51-60, 1987.\par\medskip diff --git a/doc/libs/code.sty b/doc/libs/code.sty index d4f4070..e067dfe 100644 --- a/doc/libs/code.sty +++ b/doc/libs/code.sty @@ -1,4 +1,4 @@ -%% Copyright 2021 Ian Jauslin +%% Copyright 2021-2023 Ian Jauslin %% %% Licensed under the Apache License, Version 2.0 (the "License"); %% you may not use this file except in compliance with the License. diff --git a/doc/libs/dlmf.sty b/doc/libs/dlmf.sty index f435e96..f66157b 100644 --- a/doc/libs/dlmf.sty +++ b/doc/libs/dlmf.sty @@ -1,4 +1,4 @@ -%% Copyright 2021 Ian Jauslin +%% Copyright 2021-2023 Ian Jauslin %% %% Licensed under the Apache License, Version 2.0 (the "License"); %% you may not use this file except in compliance with the License. diff --git a/doc/libs/ian.cls b/doc/libs/ian.cls index f29e6bd..762f679 100644 --- a/doc/libs/ian.cls +++ b/doc/libs/ian.cls @@ -219,6 +219,7 @@ % counters \stepcounter{subsectioncount} \setcounter{subsubsectioncount}{0} + \setcounter{subsubsubsectioncount}{0} \ifsubsectionsineq \setcounter{seqcount}0 \fi @@ -268,6 +269,7 @@ \def\subsubsection#1{ % counters \stepcounter{subsubsectioncount} + \setcounter{subsubsubsectioncount}{0} % space before subsubsection (if not first) \ifnum\thesubsubsectioncount>1 @@ -302,6 +304,49 @@ \medskip\penalty10000 } +%% subsubsubsection +% counter +\newcounter{subsubsubsectioncount} +% space before subsubsubsection +\newlength\subsubsubsecskip +\setlength\subsubsubsecskip{20pt} +\def\subsubsubsection#1{ + % counters + \stepcounter{subsubsubsectioncount} + + % space before subsubsubsection (if not first) + \ifnum\thesubsubsubsectioncount>1 + \vskip\subsubsubsecskip + \penalty-500 + \fi + + % hyperref anchor + \hrefanchor + % define tag (for \label) + \xdef\tag{\sectionprefix\thesectioncount.\thesubsectioncount.\thesubsubsectioncount.\thesubsubsubsectioncount} + + % get widths + \def\@secnum{{\bf\hskip1.5cm\sectionprefix\thesectioncount.\thesubsectioncount.\thesubsubsectioncount.\thesubsubsubsectioncount.\hskip5pt}} + \settowidth\secnumwidth{\@secnum} + \setlength\sectitlewidth\textwidth + \addtolength\sectitlewidth{-\secnumwidth} + % print name + \parbox{\textwidth}{ + \@secnum + \parbox[t]{\sectitlewidth}{\bf #1}} + + % write to table of contents + \iftoc + % save lncount in aux variable which is written to toc + \immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@subsubsubsec.\thesectioncount.\thesubsectioncount.\thesubsubsectioncount.\thesubsubsubsectioncount\endcsname{\thelncount}} + \write\tocoutput{\noexpand\tocsubsubsubsection{#1}{\thepage}} + \fi + + % space + \par\penalty10000 + \medskip\penalty10000 +} + %% itemize \newlength\itemizeskip % left margin for items @@ -638,6 +683,7 @@ \stepcounter{tocsectioncount} \setcounter{tocsubsectioncount}{0} \setcounter{tocsubsubsectioncount}{0} + \setcounter{tocsubsubsubsectioncount}{0} % write \smallskip\hyperlink{ln.\csname toc@sec.\thetocsectioncount\endcsname}{{\bf \tocsectionprefix\thetocsectioncount}.\hskip5pt {\color{blue}#1}\leaderfill#2}\par } @@ -645,6 +691,7 @@ \def\tocsubsection #1#2{ \stepcounter{tocsubsectioncount} \setcounter{tocsubsubsectioncount}{0} + \setcounter{tocsubsubsubsectioncount}{0} % write {\hskip10pt\hyperlink{ln.\csname toc@subsec.\thetocsectioncount.\thetocsubsectioncount\endcsname}{{\bf \thetocsectioncount.\thetocsubsectioncount}.\hskip5pt {\color{blue}\small #1}\leaderfill#2}}\par } @@ -654,6 +701,12 @@ % write {\hskip20pt\hyperlink{ln.\csname toc@subsubsec.\thetocsectioncount.\thetocsubsectioncount.\thetocsubsubsectioncount\endcsname}{{\bf \thetocsectioncount.\thetocsubsectioncount.\thetocsubsubsectioncount}.\hskip5pt {\color{blue}\small #1}\leaderfill#2}}\par } +\newcounter{tocsubsubsubsectioncount} +\def\tocsubsubsubsection #1#2{ + \stepcounter{tocsubsubsubsectioncount} + % write + {\hskip30pt\hyperlink{ln.\csname toc@subsubsubsec.\thetocsectioncount.\thetocsubsectioncount.\thetocsubsubsectioncount.\thetocsubsubsubsectioncount\endcsname}{{\bf \thetocsectioncount.\thetocsubsectioncount.\thetocsubsubsectioncount.\thetocsubsubsubsectioncount}.\hskip5pt {\color{blue}\small #1}\leaderfill#2}}\par +} \def\tocappendices{ \medskip \setcounter{tocsectioncount}0 diff --git a/doc/libs/iantheo.sty b/doc/libs/iantheo.sty index 1945a5f..31842a4 100644 --- a/doc/libs/iantheo.sty +++ b/doc/libs/iantheo.sty @@ -1,4 +1,4 @@ -%% Copyright 2021 Ian Jauslin +%% Copyright 2021-2023 Ian Jauslin %% %% Licensed under the Apache License, Version 2.0 (the "License"); %% you may not use this file except in compliance with the License. diff --git a/doc/libs/largearray.sty b/doc/libs/largearray.sty index cf9075f..3a8762e 100644 --- a/doc/libs/largearray.sty +++ b/doc/libs/largearray.sty @@ -1,4 +1,4 @@ -%% Copyright 2021 Ian Jauslin +%% Copyright 2021-2023 Ian Jauslin %% %% Licensed under the Apache License, Version 2.0 (the "License"); %% you may not use this file except in compliance with the License. diff --git a/doc/libs/point.sty b/doc/libs/point.sty index a396d1c..796e0d2 100644 --- a/doc/libs/point.sty +++ b/doc/libs/point.sty @@ -1,4 +1,4 @@ -%% Copyright 2021 Ian Jauslin +%% Copyright 2021-2023 Ian Jauslin %% %% Licensed under the Apache License, Version 2.0 (the "License"); %% you may not use this file except in compliance with the License. @@ -33,12 +33,15 @@ \newif\ifresetatsubsubsection \DeclareOption{reset_at_subsubsection}{\resetatsubsubsectiontrue} \DeclareOption{no_reset_at_subsubsection}{\resetatsubsubsectionfalse} +\newif\ifresetatsubsubsubsection +\DeclareOption{reset_at_subsubsubsection}{\resetatsubsubsubsectiontrue} +\DeclareOption{no_reset_at_subsubsubsection}{\resetatsubsubsubsectionfalse} \newif\ifresetattheo \DeclareOption{reset_at_theo}{\resetattheotrue} \DeclareOption{no_reset_at_theo}{\resetattheofalse} \def\point@defaultoptions{ - \ExecuteOptions{reset_at_section, reset_at_subsection, reset_at_subsubsection, no_reset_at_theo} + \ExecuteOptions{reset_at_section, reset_at_subsection, reset_at_subsubsection, reset_at_subsubsubsection, no_reset_at_theo} \ProcessOptions %% reset at every new section @@ -56,6 +59,11 @@ \let\point@oldsubsubsection\subsubsection \gdef\subsubsection{\resetpointcounter\point@oldsubsubsection} \fi + %% reset at every new subsubsubsection + \ifresetatsubsubsubsection + \let\point@oldsubsubsubsection\subsubsubsection + \gdef\subsubsubsection{\resetpointcounter\point@oldsubsubsubsection} + \fi %% reset at every new theorem \ifresetattheo diff --git a/doc/simplesolv-doc.tex b/doc/simplesolv-doc.tex index 722282a..23c7247 100644 --- a/doc/simplesolv-doc.tex +++ b/doc/simplesolv-doc.tex @@ -1,4 +1,4 @@ -%% Copyright 2021 Ian Jauslin +%% Copyright 2021-2023 Ian Jauslin %% %% Licensed under the Apache License, Version 2.0 (the "License"); %% you may not use this file except in compliance with the License. @@ -27,7 +27,7 @@ \hfil{\bf\Large {\tt simplesolv}\par \medskip -\hfil \large v0.3 +\hfil \large v0.4 } \vfill @@ -41,7 +41,7 @@ \pagestyle{plain} \indent -{\tt simplesolv} is a tool to compute the solution of the equations of the ``Simplified approach'' to the repulsive Bose gas introduced in\-~\cite{CJL20,CJL20b,CHe21}. +{\tt simplesolv} is a tool to compute the solution of the equations of the ``Simplified approach'' to the repulsive Bose gas introduced in\-~\cite{CJL20,CJL21,CHe21}. This approach provides an approximation to various observables of the ground state of the Hamiltonian \begin{equation} H_N=-\frac12\sum_{i=1}^N\Delta_i+\sum_{1\leqslant i\leqslant j\leqslant N}v(|x_i-x_j|) @@ -85,6 +85,10 @@ The available methods are (see section\-~\ref{sec:methods} for further details) \item \refname{sec:simpleq-iteration}{{\tt simpleq-iteration}} for the Simple equation with a soft potential using the iteration defined in\-~\cite{CJL20}. \end{itemize} Each method is described in detail below, along with the list of commands ({\tt command}) and parameters ({\tt params}) compatible with them. +In addition, the scattering length can be displayed using the {\tt scattering\_length} command: +\begin{code} + julia \$SIMPLESOLV/src/main.jl scattering\_length +\end{code} \bigskip \indent @@ -165,10 +169,10 @@ Unless otherwise noted, this method takes the following parameters (specified vi {\tt order} ({\tt Int64}, default: 100): order used for all Gauss quadratures (denoted by $N$ below). \item\makelink{param:easyeq_minlrho_init}{} - {\tt minlrho\_init} ({\tt Float64}, default: {\tt rho}): to initialize the Newton algorithm, we first compute the solution for a smaller $\rho$, {\tt minlrho} is the minimal value for $\log_{10}\rho$ to start this initialization process. + {\tt minlrho\_init} ({\tt Float64}, default: $-6$): to initialize the Newton algorithm, we first compute the solution for a smaller $\rho$, {\tt minlrho} is the minimal value for $\log_{10}\rho$ to start this initialization process. \item\makelink{param:easyeq_nlrho_init}{} - {\tt nlrho\_init} ({\tt Int64}, default: 1): number of steps in the initialization process described above. Set to 1 to disable the incremental initialization process. + {\tt nlrho\_init} ({\tt Int64}, default: 0): number of steps in the initialization process described above. Set to 0 to disable the incremental initialization process. \item\makelink{param:easyeq_bK}{} {\tt bK}, {\tt bL} ({\tt Float64}, default: 1, 1): the values of $\beta_K$ and $\beta_L$. @@ -188,7 +192,7 @@ The available {\tt commands} are the following. \item {\tt energy\_rho}\makelink{command:easyeq_energy_rho}{}: compute the energy $e$ as a function of $\rho$. The Newton algorithm is initialized with the hardcore scattering solution (\ref{easyeq_init}) for the lowest $\rho$, and with the previously computed $\rho$ for the larger densities.\par - \underline{Disabled parameters}: \refname{param:easyeq_rho}{{\tt rho}}, \refname{param:easyeq_minlrho_init}{{\tt minlrho\_init}} and \refname{param:easyeq_nlrho_init}{{\tt nlrho\_init}}.\par + \underline{Disabled parameters}: \refname{param:easyeq_rho}{{\tt rho}}.\par \underline{Extra parameters}: \begin{itemize} \item\makelink{param:easyeq_minlrho}{} @@ -200,8 +204,18 @@ The available {\tt commands} are the following. \item\makelink{param:easyeq_nlrho}{} {\tt nlrho} ({\tt Int64}, default: $100$): number of values for $\rho$ (spaced logarithmically). + \item\makelink{param:easyeq_minrho}{} + {\tt minrho} ({\tt Float64}, default: $10^{-6}$): minimal value for $\rho$. + + \item\makelink{param:easyeq_maxrho}{} + {\tt maxrho} ({\tt Float64}, default: $10^{2}$): maximal value for $\rho$. + + \item\makelink{param:easyeq_nrho}{} + {\tt nrho} ({\tt Int64}, default: $0$): number of values for $\rho$ (spaced linearly). If {\tt nrho} is $\neq0$, then the linear spacing will be used, and \refname{param:easyeq_minlrho}{{\tt minlrho}}, \refname{param:easyeq_maxlrho}{{\tt maxlrho}}, \refname{param:easyeq_nlrho}{{\tt nlrho}} will be ignored. Otherwise, the logarithmic spacing will be used and \refname{param:easyeq_minrho}{{\tt minrho}}, \refname{param:easyeq_maxrho}{{\tt maxrho}} will be ignored. + \item\makelink{param:easyeq_rhos}{} - {\tt rhos} ({\tt Array\{Float64\}}, default: $(10^{{\tt minlrho}+\frac{{\tt maxlrho}-{\tt minlrho}}{{\tt nlrho}}n})_n$: list of values for $\rho$, specified as a `{\tt,}' separated list. + {\tt rhos} ({\tt Array\{Float64\}}): list of values for $\rho$, specified as a `{\tt,}' separated list. + This parameter takes precedence over \refname{param:easyeq_minlrho}{{\tt minlrho}}, \refname{param:easyeq_maxlrho}{{\tt maxlrho}}, \refname{param:easyeq_nlrho}{{\tt nlrho}}, \refname{param:easyeq_minrho}{{\tt minrho}}, \refname{param:easyeq_maxrho}{{\tt maxrho}}, \refname{param:easyeq_nrho}{{\tt nrho}}. \end{itemize} \underline{Output} (one line for each value of $\rho$): [$\rho$] [$e$] [Newton error $\epsilon$]. @@ -241,15 +255,100 @@ The available {\tt commands} are the following. \underline{Extra parameters}: Same as \refname{command:easyeq_ux}{{\tt ux}}.\par \underline{Output} (one line for each value of $x$): [$|x|$] [$2u(|x|)-\rho u\ast u(|x|)$] + \item\makelink{command:easyeq_2pt}{} + {\tt 2pt}: compute the spherically averaged two-point correlation function $C_2(|x|)$ at a given $\rho$.\par + \underline{Extra parameters}: same as \refname{command:easyeq_ux}{{\tt ux}}, plus\par + \begin{itemize} + \item\makelink{param:easyeq_window_L}{} + {\tt window\_L} ({\tt Float64}, default: $10^3$): size of the Hann window used to numerically invert the Fourier transform in the computation of the two-point correlation function, see\-~(\ref{hann}). + \end{itemize} + \underline{Output} (one line for each value of $|x|$): [$|x|$] [$C_2(|x|)$] + + \item\makelink{command:easyeq_2pt_max}{} + {\tt 2pt\_max}: compute the maximum of the spherically averaged two-point correlation function $C_2(|x|)$ at a given $\rho$.\par + \underline{Extra parameters}: \refname{param:easyeq_window_L}{{\tt window\_L}} plus + \begin{itemize} + \item\makelink{param:easyeq_dx}{} + {\tt dx} ({\tt Float64}, default: $10^{-7}$): step used to numerically approximate derivatives. + + \item\makelink{param:easyeq_x0}{} + {\tt x0} ({\tt Float64}, default: $1$): initial guess for the maximum is $\rho^{-1/3}\mathrm{\tt x0}$. + + \item\makelink{param:easyeq_maxstep}{} + {\tt maxstep} ({\tt Float64}, default: $\infty$): maximal size of single step in maximization algorithm. + + \item\makelink{param:easyeq_tolerance_max}{} + {\tt tolerance\_max} ({\tt Float64}, default: \refname{param:easyeq_tolerance}{{\tt tolerance}}): same as \refname{param:easyeq_tolerance}{{\tt tolerance}}, used for the Newton algorithm underlying the maximization algorithm. + \end{itemize} + \underline{Output}: [$|x_{\mathrm{max}}|$] [$C_2(|x_{\mathrm{max}}|)$] + + \item\makelink{command:easyeq_2pt_max_rho}{} + {\tt 2pt\_max\_rho}: compute the maximum of the spherically averaged two-point correlation function $C_2(|x|)$ for a range of $\rho$.\par + \underline{Extra parameters}: same as \refname{command:easyeq_2pt_max}{{\tt easyeq\_2pt\_max}} plus those of \refname{command:easyeq_energy_rho}{{\tt energy\_rho}}.\par + \underline{Output} (one line for each value of $\rho$): [$\rho$] [$|x_{\mathrm{max}}|$] [$C_2(|x_{\mathrm{max}}|)$]\par + \underline{Multithread support}: yes, different values of $\rho$ split up among workers. + + \item\makelink{command:easyeq_2pt_fourier}{} + {\tt 2pt\_fourier}: compute the spherically averaged Fourier transform of the two-point correlation function $\hat C_2(|k|)$ at a given $\rho$.\par + \underline{Extra parameters}: + \begin{itemize} + \item\makelink{param:easyeq_kmin}{} + {\tt kmin} ({\tt Float64}, default: 0): minimum of the range of $|k|$ to be printed. + + \item\makelink{param:easyeq_kmax}{} + {\tt kmax} ({\tt Float64}, default: 10): maximum of the range of $|k|$ to be printed. + + \item\makelink{param:easyeq_nk}{} + {\tt nk} ({\tt Int64}, default: 100): number of $|k|$'s to be printed. + + \item\makelink{param:easyeq_2pt_fourier_window_L}{} + {\tt window\_L} ({\tt Float64}, default: 1000): what is actually computed is $\hat C_2$ convolved with a Gaussian of variance $1/\sqrt L$ with $L=\mathrm{\tt window\_L}$, see\-~(\ref{easyeq_gaussian}). + \end{itemize} + \underline{Output} (one line for each value of $|k|$): [$|k|$] [$\hat C_2(|k|)$].\par + \underline{Multithread support}: yes, different values of $k$ are split up among workers. + + \item\makelink{command:easyeq_2pt_fourier_max}{} + {\tt 2pt\_fourier\_max}: compute the maximum of the spherically averaged Fourier transformed two-point correlation function $\hat C_2(|k|)$.\par + \underline{Extra parameters}: \refname{param:easyeq_2pt_fourier_window_L}{{\tt window\_L}}, \refname{param:easyeq_maxstep}{{\tt maxstep}} plus + \begin{itemize} + \item\makelink{param:easyeq_dk}{} + {\tt dk} ({\tt Float64}, default: $10^{-7}$): step used to numerically approximate derivatives. + + \item\makelink{param:easyeq_k0}{} + {\tt k0} ({\tt Float64}, default: $1$): initial guess for the maximum is $\rho^{1/3}\mathrm{\tt k0}$. + \end{itemize} + \underline{Output}: [$|k_{\mathrm{max}}|$] [$\hat C_2(|k_{\mathrm{max}}|)$]\par + + \item\makelink{command:easyeq_2pt_fourier_max_rho}{} + {\tt 2pt\_fourier\_max\_rho}: compute the maximum of the spherically averaged Fourier transformed two-point correlation function $\hat C_2(|k|)$ for a range of $\rho$.\par + \underline{Extra parameters}: same as those of \refname{command:easyeq_2pt_fourier_max}{{\tt 2pt\_fourier\_max}} plus those of \refname{command:easyeq_energy_rho}{{\tt energy\_rho}}.\par + \underline{Output} (one line for each value of $\rho$): [$\rho$] [$|k_{\mathrm{max}}|$] [$\hat C_2(|k_{\mathrm{max}}|)$]\par + \underline{Multithread support}: yes, different values of $\rho$ split up among workers. + + \item\makelink{command:easyeq_momentum_distribution}{} + {\tt momentum\_distribution}: compute the momentum distribution $\mathcal M(|k|)$ at a given $\rho$. The momentum distribution is computed for $k=k_{l,j}$ (see\-~(\ref{klj})).\par + \underline{Extra parameters}: + \begin{itemize} + \item\makelink{param:easyeq_momentum_kmin}{} + {\tt kmin} ({\tt Float64}, default: 0): minimum of the range of $|k|$ to be printed. + + \item\makelink{param:easyeq_momentum_kmax}{} + {\tt kmax} ({\tt Float64}, default: 10): maximum of the range of $|k|$ to be printed. + + \item\makelink{param:easyeq_momentum_window_L}{} + {\tt window\_L} ({\tt Float64}, default: 1000): what is actually computed is $\mathfrak M_k$ convolved with a Gaussian of variance $1/\sqrt L$ where $L=\sqrt{\mathrm{\tt window\_L}}/k^2$, see\-~(\ref{easyeq_gaussian_momt}). + \end{itemize} + \underline{Output} (one line for each value of $|k|$): [$|k|$] [$\mathcal M(|k|)$] + \end{itemize} \subsubsection{Description} -\point{\bf Fourier space formulation.} +\subsubsubsection{Fourier space formulation} The computation is carried out in Fourier space. We take the convention that the Fourier transform of a function $f(|x|)$ is \begin{equation} \hat f(|k|)=\int_{\mathbb R^3} dx\ e^{ikx}f(|x|) - =\frac{4\pi}{|k|}\int_0^\infty dr\ \frac{\sin(|k|r)}rf(r) + =\frac{4\pi}{|k|}\int_0^\infty dr\ r\sin(|k|r)f(r) . \end{equation} In Fourier space, (\ref{easyeq}) becomes @@ -268,6 +367,7 @@ with ,\quad \hat f\hat\ast\hat g(|k|):=\int_{\mathbb R^3}\frac{dp}{(2\pi)^3}\ \hat f(|k-p|)\hat g(|p|) . + \label{hatS_easyeq} \end{equation} We write this as a quadratic equation for $\hat u$, and solve it, keeping the solution that decays as $|k|\to\infty$: \begin{equation} @@ -287,6 +387,15 @@ with \Phi(x):=\frac{2(1-\sqrt{1-x})}x . \end{equation} +We can write this as a root finding problem: +\begin{equation} + \Omega(u):= + \rho\hat u(|k|)- + \frac{\hat S(|k|)}{2A_K(|k|)(\xi(|k|)+1)}\Phi\left({\textstyle\frac{A_L(|k|)}{A_K^2(|k|)}\frac{\hat S(|k|)}{(\xi(|k|)+1)^2}}\right) + =0 + . + \label{Omega_easyeq} +\end{equation} Furthermore, using bipolar coordinates (see lemma\-~\ref{lemma:bipolar}), we write $\hat S$ as \begin{equation} \hat S(|k|)=\hat v(|k|)-\frac1{8\pi^3}\int_0^\infty dt\ t\hat u(t)\Eta(|k|,t) @@ -305,7 +414,7 @@ By a simple change of variables, \end{equation} \bigskip -\point{\bf Evaluating integrals.} +\subsubsubsection{Evaluating integrals} To compute these integrals numerically, we will use Gauss-Legendre quadratures: \begin{equation} \int_0^1 dt\ f(t)\approx\frac12\sum_{i=1}^N w_if\left({\textstyle\frac{r_i+1}2}\right) @@ -353,6 +462,7 @@ with \mathbb S_i:=\hat v_i-\frac1{16\pi^3\rho}\sum_{j=1}^N w_j\frac{(1-y_j)\mathbb U_j\Eta(k_i,k_j)}{y_j^3} ,\quad \mathbb E:=\hat v(0)-\frac1{16\pi^3\rho}\sum_{j=1}^N w_j\frac{(1-y_j)\mathbb U_j\Eta(0,k_j)}{y_j^3} + \label{bbSE_easyeq} \end{equation} \begin{equation} \mathbb X_i:=\frac{k_i^2}{2\rho\mathbb A_{K,i}} @@ -361,12 +471,12 @@ with This is a discrete equation for the vector $(\mathbb U_i)_{i=1}^N$. \bigskip -\point{\bf Main algorithm to compute $\mathbb U$.}\par\penalty10000 +\subsubsubsection{Main algorithm to compute $\mathbb U$}\par\penalty10000 \medskip\penalty10000 -\subpoint +\point We rewrite\-~(\ref{bbU_easyeq}) as a root finding problem: \begin{equation} - \Xi_i(\mathbb U):=\mathbb U_i-\frac{\mathbb T_i}{2(\mathbb X_i+1)}\Phi\left({\textstyle B_i\frac{\mathbb T_i}{(\mathbb X_i+1)^2}}\right) + \Xi_i(\mathbb U):=\mathbb U_i-\frac{\mathbb T_i}{2(\mathbb X_i+1)}\Phi\left({\textstyle\mathbb B_i\frac{\mathbb T_i}{(\mathbb X_i+1)^2}}\right) =0 \label{root_easyeq} \end{equation} @@ -381,7 +491,7 @@ where $D\Xi$ is the Jacobian of $\Xi$: \end{equation} \bigskip -\subpoint +\point For small values of $\rho$, we initialize the algorithm with the hardcore scattering solution \begin{equation} \hat u_0(k)=\frac{4\pi a_0}{k^2} @@ -397,7 +507,7 @@ This is a good approximation for small $\rho$. For larger $\rho$, we choose $\mathbb U^{(0)}$ as the solution of {\tt easyeq} for a slightly smaller $\rho$, and proceed inductively (using the parameters \refname{param:easyeq_minlrho_init}{{\tt minlrho\_init}} and \refname{param:easyeq_nlrho_init}{{\tt nlrho\_init}}). \bigskip -\subpoint +\point We are left with computing the Jacobian of $\Xi$: \begin{equation} \begin{largearray} @@ -436,7 +546,7 @@ with \end{equation} \bigskip -\subpoint +\point We iterate the Newton algorithm until the Newton relative error $\epsilon$ becomes smaller than the \refname{param:easyeq_tolerance}{{\tt tolerance}} parameter. The Newton error is defined as \begin{equation} @@ -466,8 +576,8 @@ and the solution $u$ in real space is obtained by inverting the Fourier transfor To compute $2u-\rho u\ast u$, we replace $\hat u$ with $2\hat u-\rho\hat u^2$ in the previous equation. \bigskip -\point{\bf Condensate fraction.} -Finally, to compute the uncondensed fraction, we solve the modified {\tt easyeq} (see\-~\cite{CJL20b}) +\subsubsubsection{Condensate fraction} +Finally, to compute the uncondensed fraction, we solve the modified {\tt easyeq} (see\-~\cite{CJL21}) \begin{equation} (-\Delta+2\mu)u_\mu=v(1-u_\mu)-2\rho K+\rho^2L \label{easyeq_mu} @@ -518,6 +628,351 @@ We then compute, using\-~(\ref{jacobian_easyeq}), . \end{equation} +\subsubsubsection{Correlation function (spherical average)} +The two-point correlation function is +\begin{equation} + c_2(x):= + 2\rho\frac{\delta e}{\delta v(x)} +\end{equation} +and its spherical average is +\begin{equation} + C_2(|x|):=\frac1{4\pi|x|^2}\int dy\ \delta(|x|-|y|)c_2(y) + . +\end{equation} +In Fourier space, +\begin{equation} + c_2(x)= + 2\rho + \int dk\ e^{ikx}\frac{\delta e}{\delta\hat v(k)} +\end{equation} +so +\begin{equation} + C_2(|x|)= + 2\rho\int dk\ \left(\frac1{4\pi|x|^2}\int dy\ \delta(|x|-|y|)e^{iky}\right)\frac{\delta e}{\delta\hat v(k)} + =2\rho\int dk\ \frac{\sin(|k||x|)}{|k||x|}\frac{\delta e}{\delta\hat v(k)} + . + \label{easyeq_C2fourier} +\end{equation} +\bigskip + +\point +We can compute this quantity by considering a modified {\tt easyeq} in Fourier space, by formally replacing $\hat v$ with +\begin{equation} + \hat v+\lambda g(|k|) + ,\quad + g(|k|):=\frac{\sin(|k||x|)}{|k||x|} + . + \label{2pt_addv} +\end{equation} +Indeed, if $e_\lambda$ denotes the energy of this modified equation, +\begin{equation} + \partial_\lambda e_\lambda|_{\lambda=0} + =\int dk\ \frac{\delta e}{\delta\hat v(k)}\partial_\lambda({\textstyle \hat v(k)+\lambda g(|k|)}) + =\int dk\ g(|k|)\frac{\delta e}{\delta\hat v(k)} + . +\end{equation} +So, denoting the solution of the modified equation by $u_\lambda$, +\begin{equation} + C_2(x)=2\rho\partial_\lambda e_\lambda|_{\lambda=0} + =\rho^2g(0)-\rho^2\int\frac{dk}{(2\pi)^3}\ (g(k)\hat u(k)+\hat v(k)\partial_\lambda\hat u_\lambda(k)|_{\lambda=0}) + . +\end{equation} +We compute $\partial_\lambda u_\lambda|_{\lambda=0}$ in the same way as the uncondensed fraction: we define $\Xi(\mathbb U,\lambda)$ by formally adding $\lambda g(|k|)$ to $\hat v$, solve $\Xi(\mathbb U,\lambda)=0$, and differentiate: +\begin{equation} + \partial_\lambda\mathbb U|_{\lambda=0}=-(D\Xi)^{-1}\partial_\lambda\Xi|_{\lambda=0} + . +\end{equation} +\bigskip + +\point +We compute $\partial_\lambda\Xi|_{\lambda=0}$: +\begin{equation} + \begin{largearray} + \partial_\lambda\Xi_i + = + -\frac1{2(\mathbb X_i+1)}\left( + \partial_\lambda\mathbb T_i-\frac{\mathbb T_i\partial_\lambda\mathbb X_i}{\mathbb X_i+1} + \right)\Phi\left({\textstyle\mathbb B_i\frac{\mathbb T_i}{(\mathbb X_i+1)^2}}\right) + \\[0.5cm]\hfill + -\frac{\mathbb T_i}{2(\mathbb X_i+1)^3}\left( + \mathbb B_i\partial_\lambda\mathbb T_i+\mathbb T_i\partial_\lambda\mathbb B_i-2\frac{\mathbb B_i\mathbb T_i\partial_\lambda\mathbb X_i}{\mathbb X_i+1} + \right)\partial\Phi\left({\textstyle\mathbb B_i\frac{\mathbb T_i}{(\mathbb X_i+1)^2}}\right) + \end{largearray} + \label{dXi_2pt_easyeq} +\end{equation} +with +\begin{equation} + \partial_\lambda\mathbb B_i= + (\beta_L(1-\beta_K)-\beta_K(1-\beta_L)) + \frac{\mathbb E\partial_\lambda\mathbb S_i-\mathbb S_i\partial_\lambda\mathbb E}{(\beta_K\mathbb S_i+(1-\beta_K)\mathbb E)^2} +\end{equation} +\begin{equation} + \partial_\lambda\mathbb T_i= + (1-\beta_K)\frac{\mathbb E\partial_\lambda\mathbb S_i-\mathbb S_i\partial_\lambda\mathbb E}{(\beta_K\mathbb S_i+(1-\beta_K)\mathbb E)^2} + ,\quad + \partial_\lambda\mathbb A_{K,i}=\beta_K\partial_\lambda\mathbb S_i+(1-\beta_K)\partial_\lambda\mathbb E +\end{equation} +\begin{equation} + \partial_\lambda\mathbb S_i:=g(k_i) + -\frac1{16\pi^3\rho}\sum_{j=1}^N w_j\frac{(1-y_j)\mathbb U_j\partial_\lambda\Eta(k_i,k_j)}{y_j^3} +\end{equation} +\begin{equation} + \partial_\lambda \mathbb E:=g(0)-\frac1{16\pi^3\rho}\sum_{j=1}^N w_j\frac{(1-y_j)\mathbb U_j\partial_\lambda\Eta(0,k_j)}{y_j^3} +\end{equation} +\begin{equation} + \partial_\lambda\mathbb X_i:=-\frac{k_i^2}{2\rho\mathbb A_{K,i}^2}\partial_\lambda\mathbb A_{K,i} +\end{equation} +where $\partial\lambda\Eta$ is computed similarly to $\Eta$\-~(\ref{Eta}) but with $\hat v$ replaced by $g$: +\begin{equation} + \partial_\lambda\Eta(y,t)= + 4\pi\left(\mathds 1_{y>t}\frac{t}y+\mathds 1_{y\leqslant t}\right) + \int_0^1 ds\ ((y+t)s+|y-t|(1-s))g((y+t)s+|y-t|(1-s)) + . + \label{GG} +\end{equation} +\bigskip + +\point +In order to invert the Fourier transform in\-~(\ref{easyeq_C2fourier}) numerically, we will use a Hann window (see appendix\-~\ref{appendix:hann}) +\begin{equation} + H_L(k):=\mathds 1_{|k|<\frac L2}\cos^2({\textstyle\frac{\pi|k|}{L}}) + . + \label{hann} +\end{equation} +The parameter $L$ is set using \refname{param:easyeq_window_L}{{\tt window\_L}}. +The computation is changed only in that $g$ is changed to $H_L(k)\frac{\sin(|k||x|)}{|k||x|}$. +\bigskip + +\point +To compute the maximum of $C_2$, we use a modified Newton algorithm. +The initial guess for the maximum is $|x_0|=\rho^{-\frac13}$\refname{param:easyeq_x0}{{\tt x0}}. +The modified Newton algorithm is an iteration: +\begin{equation} + x_{n+1}=x_n+\frac{\partial C_2(|x_n|)}{|\partial^2C_2(|x_n|)|} + \label{easyeq_newton_2pt} +\end{equation} +in which the derivatives are approximated using finite differences: +\begin{equation} + \partial C_2(x)\approx \frac{C_2(|x|+dx)-C_2(|x|)}{dx} + ,\quad + \partial^2 C_2(x)\approx \frac{C_2(|x|+dx)+C_2(|x|-dx)-2C_2(|x|)}{dx^2} + . + \label{easyeq_dx_2pt} +\end{equation} +This is a modification of the usual Newton iteration $x_n+\partial C_2/\partial^2C_2$ which is designed to follow the direction of the gradient, and thus to move toward a local maximum. +In addition, if $|\partial C_2|/|\partial^2 C_2|$ is larger than \refname{param:easyeq_maxstep}{{\tt maxstep}}, then the step is replaced with $\pm$\refname{param:easyeq_maxstep}{{\tt maxstep}}. +This prevents the algorithm from stepping over a maximum and land on another, further away. +This is useful if one has a good idea of where the global maximum is, and does not want to get trapped in a smaller local maximum. +\bigskip + +\indent +The algorithm is run for a maximum of \refname{param:easyeq_maxiter}{{\tt maxiter}} iterations, or until $|x_{n+1}-x_n|$ is smaller than \refname{param:easyeq_tolerance}{{\tt tolerance}}. +If the maximal number of iterations is reached, or if the solution found is not a local maximum, then the algorithm fails, and returns $+\infty$. +The point thus computed is therefore a local maximum, but it is not guaranteed to be the global maximum. + +\subsubsubsection{Fourier transform of two-point correlation (spherical average)} +The Fourier transform of the two-point correlation function is +\begin{equation} + \hat c_2(q):= + 2\rho\frac{\delta e}{\delta v(q)} +\end{equation} +and its spherical average is +\begin{equation} + \hat C_2(|q|):=\frac1{4\pi|q|^2}\int dk\ \delta(|q|-|k|)c_2(k) + = + \frac\rho{2\pi|q|^2}\int dk\ \delta(|q|-|k|)\frac{\delta e}{\delta\hat v(k)} + . +\end{equation} +\bigskip + +\point +To compute $\frac{\delta e}{\delta\hat v(q)}$, one idea would be to proceed in the same way as for the two-point correlation function, by replacing $\hat v$ with +\begin{equation} + \hat v+\lambda g(|k|) + ,\quad + g(|k|):=\frac1{4\pi|q|^2}\delta(|q|-|k|) +\end{equation} +where $\delta$ is the Dirac-delta function distribution (compare this with\-~(\ref{2pt_addv})). +However, the $\delta$ function causes all sorts of problems with the quadratures. +\bigskip + +\point +Instead, we approximate $\hat C_2$ by convolving it with a normalized Gaussian: let +\begin{equation} + \Gamma_L(|k|):=\left(\frac{L}{2\pi}\right)^{\frac32}e^{-\frac L2k^2} + \label{easyeq_gaussian} +\end{equation} +\begin{equation} + \hat{\mathfrak C}_2(|q|) + := + \int dp\ \hat C_2(|q-p|)\Gamma_L(|p|) + =\int dk\ \int dp\ \frac\rho{2\pi|q-p|^2}\delta(|q-p|-|k|)\frac{\delta e}{\delta\hat v(k)}\Gamma_L(|p|) +\end{equation} +which by lemma\-~\ref{lemma:bipolar} is +\begin{equation} + \hat{\mathfrak C}_2(|q|) + =\int dk\ + \frac\rho{|q|} + \frac{\delta e}{\delta\hat v(k)} + \int_0^\infty dt\ \int_{||q|-t|}^{|q|+t}ds\ s\frac{\delta(t-|k|)}t\Gamma_L(s) +\end{equation} +that is +\begin{equation} + \hat{\mathfrak C}_2(|q|) + =\int dk\ + \frac{\delta e}{\delta\hat v(k)} + \frac{\rho}{|q||k|} + \int_{||q|-|k||}^{|q|+|k|}ds\ s\Gamma_L(s) +\end{equation} +which is the directional derivative of $e$ with respect to $\hat v$ in the direction of $2\rho g$ with +\begin{equation} + g(|k|):=\frac1{2|q||k|}\int_{||q|-|k||}^{|q|+|k|}ds\ s\Gamma_L(s) + = + \frac1{2|k|rL}(\Gamma_L(|k|-r)-\Gamma_L(|k|+r)) + . + \label{easyeq_2pt_fourier_g} +\end{equation} +Note that +\begin{equation} + g(0):=\Gamma_L(|q|) + . +\end{equation} +To compute this derivative, we replace $\hat v$ with +\begin{equation} + \hat v+\lambda g(|k|) +\end{equation} +so, denoting the solution of the modified equation by $u_\lambda$, for $q\neq 0$, +\begin{equation} + \hat{\mathfrak C}_2(|q|)=2\rho\partial_\lambda e_\lambda|_{\lambda=0} + =\rho^2\left( + -\int\frac{dk}{(2\pi)^3}\ g(|k|)\hat u(|k|) + -\int\frac{dk}{(2\pi)^3}\ \hat v(|k|)\partial_\lambda\hat u_\lambda(|k|)|_{\lambda=0} + \right) + . +\end{equation} +To compute $\partial_\lambda\hat u_\lambda|_{\lambda=0}$, we differentiate $\Xi(\mathbb U,\lambda)=0$: +\begin{equation} + \partial_\lambda\mathbb U|_{\lambda=0}=-(D\Xi)^{-1}\partial_\lambda\Xi|_{\lambda=0} + . +\end{equation} +The computation of $\partial_\lambda\Xi|_{\lambda=0}$ is identical to\-~(\ref{dXi_2pt_easyeq}), but with the $g$ defined in\-~(\ref{easyeq_2pt_fourier_g}). +\bigskip + +\point +To compute the maximum of $\hat C_2$, we proceed as for $C_2$, see\-~(\ref{easyeq_newton_2pt})-(\ref{easyeq_dx_2pt}). +The only difference is that the algorithm is initialized with $|k_0|=\rho^{\frac13}$\refname{param:easyeq_k0}{{\tt k0}}. + +\subsubsubsection{Momentum distribution} +To compute the momentum distribution (see\-~\cite{CHe21}), we add a parameter $\lambda$ to {\tt easyeq}: +\begin{equation} + -\Delta u_\lambda(|x|) + = + (1-u_\lambda(|x|))v(|x|)-2\rho K(|x|)+\rho^2 L(|x|) + -2\lambda \hat u_0(q)\cos(q\cdot x) +\end{equation} +($\hat u_0\equiv\hat u_\lambda|_{\lambda=0}$). +The momentum distribution is then +\begin{equation} + \mathcal M(q)=\partial_\lambda e|_{\lambda=0} + =-\frac\rho2\int\frac{dk}{(2\pi)^3}\ \hat v(k)\partial_\lambda\hat u_\lambda(k)|_{\lambda=0} + . +\end{equation} +\bigskip + +\point +Note that the Fourier transform of $2\lambda\hat u_0(q)\cos(q\cdot x)$ is +\begin{equation} + -(2\pi)^3\lambda\hat u_0(q)(\delta(q+k)+\delta(q-k)) + . + \label{fouriermomentum} +\end{equation} +The presence of delta functions does not play well with the quadratures. +To get around this, we instead compute a regularization of $\mathcal M(q)$ by convolving it with a peaked spherically symmetric function. +Let $\Gamma_L$ denote the Gaussian with variance $1/\sqrt L$ +\begin{equation} + \Gamma_L(|k|):=\left(\frac{L}{2\pi}\right)^{\frac32}e^{-\frac L2k^2} + . + \label{easyeq_gaussian_momt} +\end{equation} +In fact, we will scale $L$ with $k$, and set $L$ to +\begin{equation} + L=\sqrt{\mathrm{\refname{param:easyeq_momentum_window_L}{{\tt window\_L}}}}/k^2 + . +\end{equation} +To compute +\begin{equation} + \mathfrak M(q):=\mathcal M\ast\Gamma_L(q) +\end{equation} +we solve the equation +\begin{equation} + -\Delta u_\lambda(|x|) + = + (1-u_\lambda(|x|))v(|x|)-2\rho K(|x|)+\rho^2 L(|x|) + -2\lambda\int dk\ \hat u_0(k)\cos(k\cdot x)\Gamma_L(q-k) + . +\end{equation} +Note that the Fourier transform of +\begin{equation} + -2\lambda\int dk\ \hat u_0(k)\cos(k\cdot x)\Gamma_L(q-k) +\end{equation} +is +\begin{equation} + -(2\pi)^3\lambda\hat u_0(q)(\Gamma_L(k+q)+\Gamma_L(k-q)) + . +\end{equation} +Since the ground state is unique, $\mathcal M$ is spherically symmetric. +The term $\Gamma_L(k\pm q)$ is not, so we take its spherical average (which will not change the final result): by lemma\-~\ref{lemma:bipolar}, +\begin{equation} + -\frac1{4\pi r^2}\int dq\ \delta(|q|-r)(2\pi)^3\lambda\hat u_0(q)(\Gamma_L(k+q)+\Gamma_L(k-q)) + = + -\frac{(2\pi)^3}{|k|r}\lambda\hat u_0(r)\int_{||k|-r|}^{|k|+r} ds\ s\Gamma_L(s) + . +\end{equation} +In this setup, the approximation of the delta function is thus +\begin{equation} + \tilde\delta(|k|,r):= + \frac1{2|k|r}\int_{||k|-r|}^{|k|+r} ds\ s\Gamma_L(s) + = + \frac{1}{2|k|rL}(\Gamma_L(|k|-r)-\Gamma_L(|k|+r)) + . +\end{equation} +\bigskip + +\point +To compute the momentum distribution at $q$, we define $\Xi(\mathbb U,\lambda)$ by replacing $\mathbb T$ with +\begin{equation} + \mathbb T_{i}= + \frac1{\mathbb A_{K,i}} + \left( + \mathbb S_i + -2(2\pi)^3\lambda \hat u(|q|)\tilde\delta(k_i,|q|) + \right) + . +\end{equation} +Then we solve $\Xi(\mathbb U,\lambda)=0$, and differentiate: +\begin{equation} + \partial_\lambda\mathbb U|_{\lambda=0}=-(D\Xi)^{-1}\partial_\lambda\Xi|_{\lambda=0} + . +\end{equation} +Finally, +\begin{equation} + \partial_\lambda\Xi_{i}|_{\lambda=0} + = + -\partial_\lambda\mathbb T_{i}|_{\lambda=0}\left( + \frac1{2(\mathbb X_{i}+1)} + \Phi\left(\mathbb B_i{\textstyle\frac{\mathbb T_{i}}{(\mathbb X_{i}+1)^2}}\right) + +\frac{\mathbb B_i\mathbb T_{i}}{2(\mathbb X_{i}+1)^3} + \partial\Phi\left(\mathbb B_i{\textstyle\frac{\mathbb T_{i}}{(\mathbb X_{l,j}+1)^2}}\right) + \right) +\end{equation} +with +\begin{equation} + \partial_\lambda\mathbb T_{i}|_{\lambda=0}= + -\frac{2(2\pi)^3}{\mathbb A_{K,i}} \hat u(|q|)\tilde\delta(k_{i},|q|) + . +\end{equation} +\bigskip + \subsection{\tt anyeq}\label{sec:anyeq} \indent This method is used to solve any of the equations in the Simplified approach. @@ -598,11 +1053,11 @@ Unless otherwise noted, this method takes the following parameters (specified vi \item\makelink{param:anyeq_J}{} {\tt J} ({\tt Int64}, default: 10): number of splines (denoted by $J$ below). - \item\makelink{param:anyeq_minlrho_init}{} - {\tt minlrho\_init} ({\tt Float64}, default: \refname{param:anyeq_rho}{{\tt rho}}): we initialize the Newton algorithm using the solution of \refname{eq:easyeq_medeq}{{\tt medeq}}, computed using the methods in \refname{sec:easyeq}{{\tt easyeq}}. This option is passed to the underlying \refname{sec:easyeq}{{\tt easyeq}} routine. - \item\makelink{param:anyeq_nlrho_init}{} - {\tt nlrho\_init} ({\tt Int64}, default: 1): this option is passed to the underlying \refname{sec:easyeq}{{\tt easyeq}} routine to initialize the Newton algorithm. + {\tt nlrho\_init} ({\tt Int64}, default: 0): we initialize the Newton algorithm using the solution of \refname{eq:easyeq_medeq}{{\tt medeq}}, computed using the methods in \refname{sec:easyeq}{{\tt easyeq}}. If {\tt nlrho\_init} is $\neq 0$, then the solution of \refname{eq:easyeq_medeq}{{\tt medeq}} is first computed for {\tt nlrho} smaller values of $\rho$ starting from $10^{\mathrm{\refname{param:anyeq_minlrho_init}{{\tt minlrho\_init}}}}$. This is useful when $\rho$ is too large for the solution of \refname{eq:easyeq_medeq}{{\tt medeq}} to be computed directly. + + \item\makelink{param:anyeq_minlrho_init}{} + {\tt minlrho\_init} ({\tt Float64}, default: $-6$): see \refname{param:anyeq_nlrho_init}{{\tt nlrho\_init}}. \item\makelink{param:anyeq_aK}{} {\tt aK}, {\tt bK}, {\tt gK}, {\tt aL1}, {\tt bL1}, {\tt aL2}, {\tt bL2}, {\tt gL2}, {\tt aL3}, {\tt bL3}, {\tt gL3} ({\tt Float64}, default: 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0): the values of $\alpha_K$, $\beta_K$, $\gamma_K$, $\alpha_{L,1}$, $\beta_{L,1}$, $\alpha_{L,2}$, $\beta_{L,2}$, $\gamma_{L,2}$, $\alpha_{L,3}$, $\beta_{L,3}$, $\gamma_{L,3}$. @@ -623,7 +1078,7 @@ The available {\tt commands} are the following. \item\makelink{command:anyeq_energy_rho}{} {\tt energy\_rho}: compute the energy $e$ as a function of $\rho$. The Newton algorithm is initialized with the solution of \refname{eq:easyeq_medeq}{{\tt medeq}}.\par - \underline{Disabled parameters}: \refname{param:anyeq_rho}{{\tt rho}}, \refname{param:anyeq_minlrho_init}{{\tt minlrho\_init}} and \refname{param:anyeq_nlrho_init}{{\tt nlrho\_init}}.\par + \underline{Disabled parameters}: \refname{param:anyeq_rho}{{\tt rho}}.\par \underline{Extra parameters}: \begin{itemize} \item\makelink{param:anyeq_minlrho}{} @@ -635,8 +1090,18 @@ The available {\tt commands} are the following. \item\makelink{param:anyeq_nlrho}{} {\tt nlrho} ({\tt Int64}, default: $100$): number of values for $\rho$ (spaced logarithmically). + \item\makelink{param:anyeq_minrho}{} + {\tt minrho} ({\tt Float64}, default: $10^{-6}$): minimal value for $\rho$. + + \item\makelink{param:anyeq_maxrho}{} + {\tt maxrho} ({\tt Float64}, default: $10^{2}$): maximal value for $\rho$. + + \item\makelink{param:anyeq_nrho}{} + {\tt nrho} ({\tt Int64}, default: $0$): number of values for $\rho$ (spaced linearly). If {\tt nrho} is $\neq0$, then the linear spacing will be used, and \refname{param:anyeq_minlrho}{{\tt minlrho}}, \refname{param:anyeq_maxlrho}{{\tt maxlrho}}, \refname{param:anyeq_nlrho}{{\tt nlrho}} will be ignored. Otherwise, the logarithmic spacing will be used and \refname{param:anyeq_minrho}{{\tt minrho}}, \refname{param:anyeq_maxrho}{{\tt maxrho}} will be ignored. + \item\makelink{param:anyeq_rhos}{} - {\tt rhos} ({\tt Array\{Float64\}}, default: $(10^{{\tt minlrho}+\frac{{\tt maxlrho}-{\tt minlrho}}{{\tt nlrho}}n})_n$: list of values for $\rho$, specified as a `{\tt,}' separated list. + {\tt rhos} ({\tt Array\{Float64\}}): list of values for $\rho$, specified as a `{\tt,}' separated list. + This parameter takes precedence over \refname{param:anyeq_minlrho}{{\tt minlrho}}, \refname{param:anyeq_maxlrho}{{\tt maxlrho}}, \refname{param:anyeq_nlrho}{{\tt nlrho}}, \refname{param:anyeq_minrho}{{\tt minrho}}, \refname{param:anyeq_maxrho}{{\tt maxrho}}, \refname{param:anyeq_nrho}{{\tt nrho}}. \end{itemize} \underline{Output} (one line for each value of $\rho$): [$\rho$] [$e$] [Newton error $\epsilon$].\par \underline{Multithread support}: yes, different values of $\rho$ split up among workers. @@ -686,14 +1151,10 @@ The available {\tt commands} are the following. {\tt nx} ({\tt Int64}, default: 100): number of points to print (linearly spaced). \end{itemize} - \underline{Output} (one line for each value of $x$): [$|x|$] [$\mathcal Re(u(|x|))$] [$\mathcal Im(u(|x|))$] - - \item\makelink{command:anyeq_momentum_distribution}{} - {\tt momentum\_distribution}: compute the momentum distribution $\mathfrak M(|k|)$ at a given $\rho$.\par - \underline{Output} (one line for each value of $|k|$): [$|k|$] [$\mathfrak M(|k|)$] + \underline{Output} (one line for each value of $x$): [$|x|$] [$u(|x|)$] - \item\makelink{command_anyeq_2pt}{} - {\tt 2pt}: compute the two-point correlation function $C_2(|x|)$ at a given $\rho$.\par + \item\makelink{command:anyeq_2pt}{} + {\tt 2pt}: compute the spherically averaged two-point correlation function $C_2(|x|)$ at a given $\rho$.\par \underline{Extra parameters}: same as \refname{command:anyeq_ux}{{\tt ux}}, plus\par \begin{itemize} \item\makelink{param:anyeq_window_L}{} @@ -701,16 +1162,99 @@ The available {\tt commands} are the following. \end{itemize} \underline{Output} (one line for each value of $|x|$): [$|x|$] [$C_2(|x|)$] + \item\makelink{command:anyeq_2pt_max}{} + {\tt 2pt\_max}: compute the maximum of the spherically averaged two-point correlation function $C_2(|x|)$ at a given $\rho$.\par + \underline{Extra parameters}: \refname{param:anyeq_window_L}{{\tt window\_L}} plus + \begin{itemize} + \item\makelink{param:anyeq_dx}{} + {\tt dx} ({\tt Float64}, default: $10^{-7}$): step used to numerically approximate derivatives. + + \item\makelink{param:anyeq_x0}{} + {\tt x0} ({\tt Float64}, default: $1$): initial guess for the maximum is $\rho^{-1/3}\mathrm{\tt x0}$. + + \item\makelink{param:anyeq_maxstep}{} + {\tt maxstep} ({\tt Float64}, default: $\infty$): maximal size of single step in maximization algorithm. + + \item\makelink{param:easyeq_tolerance_max}{} + {\tt tolerance\_max} ({\tt Float64}, default: \refname{param:easyeq_tolerance}{{\tt tolerance}}): same as \refname{param:easyeq_tolerance}{{\tt tolerance}}, used for the Newton algorithm underlying the maximization algorithm. + \end{itemize} + \underline{Output}: [$|x_{\mathrm{max}}|$] [$C_2(|x_{\mathrm{max}}|)$] + + \item\makelink{command:anyeq_2pt_max_rho}{} + {\tt 2pt\_max\_rho}: compute the maximum of the spherically averaged two-point correlation function $C_2(|x|)$ for a range of $\rho$.\par + \underline{Extra parameters}: same as \refname{command:anyeq_2pt_max}{{\tt anyeq\_2pt\_max}} plus those of \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par + \underline{Output} (one line for each value of $\rho$): [$\rho$] [$|x_{\mathrm{max}}|$] [$C_2(|x_{\mathrm{max}}|)$]\par + \underline{Multithread support}: yes, different values of $\rho$ split up among workers. + + \item\makelink{command:anyeq_2pt_fourier}{} + {\tt 2pt\_fourier}: compute the spherically averaged Fourier transform of the two-point correlation function $\hat C_2(|k|)$ at a given $\rho$.\par + \underline{Extra parameters}: + \begin{itemize} + \item\makelink{param:anyeq_kmin}{} + {\tt kmin} ({\tt Float64}, default: 0): minimum of the range of $|k|$ to be printed. + + \item\makelink{param:anyeq_kmax}{} + {\tt kmax} ({\tt Float64}, default: 10): maximum of the range of $|k|$ to be printed. + + \item\makelink{param:anyeq_nk}{} + {\tt nk} ({\tt Int64}, default: 100): number of $|k|$'s to be printed. + + \item\makelink{param:anyeq_2pt_fourier_window_L}{} + {\tt window\_L} ({\tt Float64}, default: 1000): what is actually computed is $\hat C_2$ convolved with a Gaussian of variance $1/\sqrt L$ where $L=\sqrt{\mathrm{\tt window\_L}}$, see\-~(\ref{anyeq_gaussian}). + \end{itemize} + \underline{Output} (one line for each value of $|k|$): [$|k|$] [$\hat C_2(|k|)$].\par + \underline{Multithread support}: yes, different values of $k$ are split up among workers. + + \item\makelink{command:anyeq_2pt_fourier_max}{} + {\tt 2pt\_fourier\_max}: compute the maximum of the spherically averaged Fourier transformed two-point correlation function $\hat C_2(|k|)$.\par + \underline{Extra parameters}: \refname{param:anyeq_2pt_fourier_window_L}{{\tt window\_L}}, \refname{param:anyeq_maxstep}{{\tt maxstep}} plus + \begin{itemize} + \item\makelink{param:anyeq_dk}{} + {\tt dk} ({\tt Float64}, default: $10^{-7}$): step used to numerically approximate derivatives. + + \item\makelink{param:anyeq_k0}{} + {\tt k0} ({\tt Float64}, default: $1$): initial guess for the maximum is $\rho^{1/3}\mathrm{\tt k0}$. + \end{itemize} + \underline{Output}: [$|k_{\mathrm{max}}|$] [$\hat C_2(|k_{\mathrm{max}}|)$]\par + + \item\makelink{command:anyeq_2pt_fourier_max_rho}{} + {\tt 2pt\_fourier\_max\_rho}: compute the maximum of the spherically averaged Fourier transformed two-point correlation function $\hat C_2(|k|)$ for a range of $\rho$.\par + \underline{Extra parameters}: same as those of \refname{command:anyeq_2pt_fourier_max}{{\tt 2pt\_fourier\_max}} plus those of \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par + \underline{Output} (one line for each value of $\rho$): [$\rho$] [$|k_{\mathrm{max}}|$] [$\hat C_2(|k_{\mathrm{max}}|)$]\par + \underline{Multithread support}: yes, different values of $\rho$ split up among workers. + + \item\makelink{command:anyeq_momentum_distribution}{} + {\tt momentum\_distribution}: compute the momentum distribution $\mathcal M(|k|)$ at a given $\rho$. The momentum distribution is computed for $k=k_{l,j}$ (see\-~(\ref{klj})).\par + \underline{Extra parameters}: + \begin{itemize} + \item\makelink{param:anyeq_momentum_kmin}{} + {\tt kmin} ({\tt Float64}, default: 0): minimum of the range of $|k|$ to be printed. + + \item\makelink{param:anyeq_momentum_kmax}{} + {\tt kmax} ({\tt Float64}, default: 10): maximum of the range of $|k|$ to be printed. + + \item\makelink{param:anyeq_momentum_window_L}{} + {\tt window\_L} ({\tt Float64}, default: 1000): what is actually computed is $\mathfrak M_k$ convolved with a Gaussian of variance $1/\sqrt L$ where $L=\sqrt{\mathrm{\tt window\_L}}/k^2$, see\-~(\ref{anyeq_gaussian_momt}). + \end{itemize} + \underline{Output} (one line for each value of $|k|$): [$|k|$] [$\mathcal M(|k|)$] + + \item\makelink{command:anyeq_compressibility_rho}{} + {\tt compressibility\_rho}: compute the compressibility $\chi$ as a function of $\rho$. + The Newton algorithm is initialized with the solution of \refname{eq:easyeq_medeq}{{\tt medeq}}. + \underline{Disabled parameters}: same as \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par + \underline{Extra parameters}: same as \refname{command:anyeq_energy_rho}{{\tt energy\_rho}}.\par + \underline{Output} (one line for each value of $\rho$): [$\rho$] [$\chi$]. + \item \makelink{command:anyeq_save_Abar}{} {\tt save\_Abar}: compute the matrix $\bar A$. This matrix is used to compute observables for \refname{eq:anyeq_compleq}{{\tt compleq}}. This command is useful to output the value of $\bar A$ to a file once and for all, and use this file to run commands without recomputing $\bar A$.\par - \underline{Disabled parameters}: \refname{param:anyeq_rho}{{\tt rho}}, \refname{param:anyeq_tolerance}{{\tt tolerance}}, \refname{param:anyeq_maxiter}{{\tt maxiter}}, \refname{param:anyeq_minlrho_init}{{\tt minlrho\_init}}, \refname{param:anyeq_nlrho}{{\tt nlrho}}.\par + \underline{Disabled parameters}: \refname{param:anyeq_rho}{{\tt rho}}, \refname{param:anyeq_tolerance}{{\tt tolerance}}, \refname{param:anyeq_maxiter}{{\tt maxiter}}, \refname{param:anyeq_minlrho_init}{{\tt minlrho\_init}}, \refname{param:anyeq_nlrho_init}{{\tt nlrho\_init}}.\par \underline{Output}: [$\bar A$] (the output is not designed to be human-readable; it is obtained through nested {\tt for} loops; for details, see the code).\par \underline{Multithread support}: yes, the first indices are split up among workers, which produces $NJ$ jobs. \end{itemize} \subsubsection{Description} -\point{\bf Fourier space formulation.} +\subsubsubsection{Fourier space formulation} The computation is carried out in Fourier space. We take the convention that the Fourier transform of a function $f(|x|)$ is \begin{equation} @@ -770,11 +1314,15 @@ with \end{equation} Therefore, \begin{equation} + \Omega(u)=0 +\end{equation} +with +\begin{equation} + \Omega(u):= \rho^2\hat u(|k|)^2\sigma_{L,1}(|k|) -\left(\frac{k^2}\rho+2\sigma_K(|k|)+2f_1(|k|)\right)\rho\hat u(|k|) +\left(\hat S(|k|)+\frac12f_2(|k|)+G(|k|)\right) - = - 0 + \label{Omega} \end{equation} with \begin{equation} @@ -831,7 +1379,7 @@ with \end{equation} \bigskip -\point{\bf Evaluating integrals.} +\subsubsubsection{Evaluating integrals} To evaluate integrals numerically, we will split integration intervals over splines and use Gauss-Legendre quadratures. More specifically, to compute integrals of the form \begin{equation} @@ -861,7 +1409,7 @@ In these terms, \int \frac{dk}{(2\pi)^3}\ f(U(k),k) = \sum_{l=0}^{J-1} - \int_{\tau_l}^{\tau_{l+1}} d\tau\ \frac{(1-\tau)^2}{(1+\tau^4}g(U({\textstyle\frac{1-\tau}{1+\tau},\frac{1-\tau}{1+\tau}}) + \int_{\tau_l}^{\tau_{l+1}} d\tau\ \frac{(1-\tau)^2}{(1+\tau)^4}g(U({\textstyle\frac{1-\tau}{1+\tau},\frac{1-\tau}{1+\tau}}) . \end{equation} We then change variables to $r$: @@ -879,6 +1427,7 @@ and find \sum_{l=0}^{J-1} \frac{\tau_{l+1}-\tau_l}{16\pi}\int_{-1}^1 dr\ \cos({\textstyle\frac{\pi r}2})(1+\vartheta_l(r))^2\vartheta_l^2(r)f(U(\vartheta_l(r)),\vartheta_l(r)) . + \label{anyeq_preintegration} \end{equation} We then approximate the integral using a Gauss-Legendre quadrature of order $N$ (see appendix\-~\ref{appGL}): \begin{equation} @@ -901,14 +1450,14 @@ The choice of the change of variables\-~(\ref{chebyshevvars}) is made so that $U In this way, we can compute arbitrary integrals of functions of $U$ by just computing the values of $\mathbb U_{l,j}$. \bigskip -\point{\bf Chebyshev polynomial expansion.} +\subsubsubsection{Chebyshev polynomial expansion} In the case of \refname{sec:easyeq}{{\tt easyeq}}, we saw that using Gauss quadratures reduced the computation to evaluating $U$ at a fixed set of discrete momenta. This is not the case here, due to the presence of more complicated convolutions of $U$. Instead, we will approximate $U$ using polynomial approximations. We will now discuss this approximation for an arbitrary function $a(|k|)$, as we will need it for other functions than simply $U$. \bigskip -\subpoint +\point As in the previous paragraph, we compactify $[0,\infty)$ to $(-1,1]$ via the change of variables $r\mapsto\frac{1-r}{1+r}$, and split the interval into splines. Inside each spline, we use a Chebyshev polynomial expansion. However, we need to be careful in the first spline, which encapsulates the behavior at $|k|\to\infty$: $U$ decays to 0 as $|k|\to\infty$, and this behavior cannot be reproduced by a polynomial expansion. @@ -934,7 +1483,7 @@ in which $\mathds 1_{\tau_l<\tau\leqslant\tau_{l+1}}\in\{0,1\}$ is equal to 1 if (see appendix\-~\ref{appChebyshev} for a discussion of the Chebyshev polynomial expansion and the error of its truncations). \bigskip -\subpoint +\point In order to compute an approximation for $a$ using\-~(\ref{a_chebyshev_spline}), we will truncate the sum over $n$ to a finite value $P$ (given by the parameter \refname{param:anyeq_P}{{\tt P}}). In addition, to compute the integral in\-~(\ref{bfF}), we will use a Gauss-Legendre quadrature of order $N$ (given by the parameter \refname{param:anyeq_N}{{\tt N}}), see appendix\-~\ref{appGL}: \begin{equation} @@ -952,7 +1501,7 @@ with and $(x_j,w_j)$ are the abcissa and weights for Gauss-Legendre quadratures. \bigskip -\subpoint +\point All in all, we approximate \begin{equation} a({\textstyle\frac{1-\tau}{1+\tau}}) @@ -964,14 +1513,15 @@ Furthermore, using the Chebyshev polynomial expansion and Gauss-Legendre quadrat With this in mind, we will represent the function $U$ as a vector of dimension $NJ$ whose components are $\mathbb U_{l,j}$. \bigskip -\point{\bf Convolutions.} +\subsubsubsection{Convolutions} Using the Chebyshev polynomial approximation, we can compute convolutions as follows. \medskip -\subpoint +\point First, we rewrite the convolution as a two-dimensional integral, using bipolar coordinates (see lemma\-~\ref{lemma:bipolar}): \begin{equation} (a\hat\ast b)(|k|)=\frac1{4\pi^2|k|}\int_0^\infty dt\ ta(t)\int_{||k|-t|}^{|k|+t}ds\ sb(s) + \label{conv_approx1} \end{equation} We change variables to compactify the integrals $\tau=\frac{1-t}{1+t}$, $\sigma=\frac{1-s}{1+s}$: \begin{equation} @@ -990,30 +1540,31 @@ Therefore, using the approximation\-~(\ref{approxchebyshev}), if $\mathfrak a_{l \begin{equation} (a\hat\ast b)(|k_{l,j}|)\approx (\mathfrak a\odot \mathfrak b)_{l,j}:= - \frac1{4\pi^2|k_{l,j}|} \sum_{n,m=0}^P - \sum_{l,l'=0}^{J-1} - \mathfrak F_{l,n}^{(\nu_a)}(\mathfrak a)\mathfrak F_{l',m}^{(\nu_b)}(\mathfrak b) - A_{l,n;l',m}^{(\nu_a,\nu_b)}(|k_{l,j}|) + \sum_{l',l''=0}^{J-1} + \mathfrak F_{l',n}^{(\nu_a)}(\mathfrak a)\mathfrak F_{l'',m}^{(\nu_b)}(\mathfrak b) + A_{l',n;l'',m}^{(\nu_a,\nu_b)}(|k_{l,j}|) \label{odot} \end{equation} with \begin{equation} \begin{largearray} - A_{l,n;l',m}^{(\nu_a,\nu_b)}(|k|):= - \int_{\tau_l}^{\tau_{l+1}} d\tau\ \frac{2(1-\tau)}{(1+\tau)^{3-\nu_a}}T_n({\textstyle\frac{2\tau-(\tau_l+\tau_{l+1})}{\tau_{l+1}-\tau_l}}) + A_{l',n;l'',m}^{(\nu_a,\nu_b)}(|k|):= + \frac1{4\pi^2|k_{l,j}|} + \int_{\tau_{l'}}^{\tau_{l'+1}} d\tau\ \frac{2(1-\tau)}{(1+\tau)^{3-\nu_a}}T_n({\textstyle\frac{2\tau-(\tau_{l'}+\tau_{l'+1})}{\tau_{l'+1}-\tau_{l'}}}) \cdot\\[0.5cm]\hfill\cdot - \mathds 1_{\alpha_-(|\tilde k|,\tau)<\tau_{\zeta'+1}} - \mathds 1_{\alpha_+(|\tilde k|,\tau)>\tau_{\zeta'}} - \int_{\max(\tau_{l'},\alpha_-(|k|,\tau))}^{\min(\tau_{l'+1},\alpha_+(|k|,\tau))}d\sigma\ \frac{2(1-\sigma)}{(1+\sigma)^{3-\nu_b}}T_m({\textstyle\frac{2\sigma-(\tau_{l'}+\tau_{l'+1})}{\tau_{l'+1}-\tau_{l'}}}) + \mathds 1_{\alpha_-(|k|,\tau)<\tau_{l''+1}} + \mathds 1_{\alpha_+(|k|,\tau)>\tau_{l''}} + \int_{\max(\tau_{l''},\alpha_-(|k|,\tau))}^{\min(\tau_{l''+1},\alpha_+(|k|,\tau))}d\sigma\ \frac{2(1-\sigma)}{(1+\sigma)^{3-\nu_b}}T_m({\textstyle\frac{2\sigma-(\tau_{l''}+\tau_{l''+1})}{\tau_{l''+1}-\tau_{l''}}}) \end{largearray} + \label{conv_approx2} \end{equation} (we need the indicator functions to ensure that the bounds of the integral are correct). Note that $A$ is independent of $a$ and $b$, and can be computed once and for all at the beginning of execution for all values of $k_{l,j}$ (\ref{klj}). We then compute the integrals using Gauss-Legendre quadratures (as is proved in the next paragraph, the integrand is non singular provided $\nu_a,\nu_b\geqslant 2$). \bigskip -\subpoint +\point Note that these integrals are not singular as long as $\nu_a,\nu_b\geqslant 2$: indeed (since the only possible problems occur at $-1$, it suffices to consider the case with only one spline), $\alpha_-,\alpha_+>-1$ for $\tau>-1$, and \begin{equation} \alpha_\pm(k,\tau)=-1+(1+\tau)\pm\frac k2(1+\tau)^2+O(1+\tau)^3 @@ -1047,7 +1598,7 @@ where $c_{2}:=1$ and $c_{\nu_b}:=\nu_b-2$ if $\nu_b>2$. The integrand is, therefore, not singular as long as $\nu_a,\nu_b\geqslant 2$. \bigskip -\subpoint +\point Evaluating convolutions at $k=0$ is not immediate, as the formula for $A(0)$ involves a bit of a computation. To compute $A(0)$, we expand \begin{equation} @@ -1077,11 +1628,11 @@ We can then evaluate convolutions at $k=0$: \bigskip -\subpoint +\point Let us now compute some choices of $\mathfrak a,\mathfrak b$ more explicitly. \medskip -\subsubpoint +\subpoint Let us start with $\mathds 1_{l,n}\hat\ast a$ where $\mathds 1_{l,n}$ is the vector which has 0's everywhere except at position $(l,n)$. We have \begin{equation} @@ -1095,7 +1646,7 @@ so \end{equation} \bigskip -\subsubpoint +\subpoint We now turn to $a\hat\ast\hat v$. Let \begin{equation} @@ -1132,7 +1683,7 @@ The integral in\-~(\ref{Upsilon}) is computed using Gauss-Legendre quadratures, To maintain a high precision, we set the order of the integration to \refname{param:anyeq_J}{{\tt J}}$\times$\refname{param:anyeq_N}{{\tt N}}. \bigskip -\subsubpoint +\subpoint Finally, \begin{equation} (\mathds 1_{l',n}\odot\mathfrak v)_{l,i}= @@ -1146,7 +1697,7 @@ and \end{equation} \bigskip -\point{\bf Evaluating $\hat l_3$.} +\subsubsubsection{Evaluating $\hat l_3$} The only term in\-~(\ref{anyeq}) that does not involve just convolutions (whose computation was described above) is $\hat l_3$ (\ref{l3}). To evaluate it, we first change variables, using a generalization of bipolar coordinates (see lemma\-~\ref{lemma:bipolar2}): \begin{equation} @@ -1230,12 +1781,12 @@ Since the tensor $\bar A$ is quite large (it contains $(NJ)^5$ entries), and its Instead, one can use the \refname{command:anyeq_save_Abar}{{\tt save\_Abar}} method to compute $\bar A$ and save it to a file, which can then be recalled via the {\tt -s} option on the command line. \bigskip -\point{\bf Main algorithm to compute $U$.} +\subsubsubsection{Main algorithm to compute $U$} We are now ready to detail how $U\equiv\rho\hat u$ is computed. All of the observables we are interested in are approximated from the values $\mathbb U_{l,j}:=U(k_{l,j})$ (\ref{klj}). \bigskip -\subpoint +\point The equation for $(\mathbb U_{l,j})_{l\in\{0,\cdots,J-1\},j\in\{1,\cdots,N\}}$ is obtained by approximating\-~(\ref{anyeq_hatu}) according to the prescriptions detailed above: \begin{equation} \mathbb U_{l,j}:= @@ -1283,7 +1834,7 @@ The equation for $(\mathbb U_{l,j})_{l\in\{0,\cdots,J-1\},j\in\{1,\cdots,N\}}$ i (see\-~(\ref{odot}), (\ref{otimes}) and\-~(\ref{avg}) for the definitions of $\odot$, $\otimes$ and $\left<\cdot\right>$). \bigskip -\subpoint +\point We rewrite\-~(\ref{bbU}) as a root finding problem: \begin{equation} \Xi_{l,j}(\mathbb U):= @@ -1303,14 +1854,14 @@ where $D\Xi$ is the Jacobian of $\Xi$: \end{equation} \bigskip -\subpoint +\point We initialize the algorithm with the solution of the \refname{eq:easyeq_medeq}{Medium equation}, which is computed using the \refname{sec:easyeq}{{\tt easyeq}} method. However, \refname{sec:easyeq}{{\tt easyeq}} only computes $\hat u$ at the momenta given by the Gauss-Legendre quadratures\-~(\ref{easyeq_ki}). To obtain a value for $\hat u$ at $k_{l,j}$, we use a linear interpolation (code is provided for a polynomial interpolation, which has not performed as well). -The parameters \refname{param:anyeq_tolerance}{{\tt tolerance}}, \refname{param:anyeq_maxiter}{{\tt maxiter}}, \refname{param:anyeq_minlrho_init}{{\tt minlrho\_init}}, \refname{param:anyeq_nlrho_init}{{\tt nlrho\_init}} are passed on to \refname{sec:easyeq}{easyeq} as is, and the order of the Gauss-Legendre quadratures is set to \refname{param:anyeq_J}{{\tt J}}$\times$\refname{param:anyeq_N}{{\tt N}}. +The parameters \refname{param:anyeq_tolerance}{{\tt tolerance}}, \refname{param:anyeq_maxiter}{{\tt maxiter}} are passed on to \refname{sec:easyeq}{{\tt easyeq}} as is, and the order of the Gauss-Legendre quadratures is set to \refname{param:anyeq_J}{{\tt J}}$\times$\refname{param:anyeq_N}{{\tt N}}. \bigskip -\subpoint +\point We are left with computing the Jacobian of $\Xi$: \begin{equation} \begin{largearray} @@ -1444,7 +1995,7 @@ with \end{equation} \bigskip -\subpoint +\point We iterate the Newton algorithm until the Newton relative error $\epsilon$ becomes smaller than the \refname{param:easyeq_tolerance}{{\tt tolerance}} parameter. The Newton error is defined as \begin{equation} @@ -1471,8 +2022,8 @@ where $k_{l,j}$ was defined in\-~(\ref{klj}), and the solution $u$ in real space \end{equation} \bigskip -\point{\bf Condensate fraction.} -To compute the condensate fraction, we solve the modified {\tt anyeq} (see\-~\cite{CJL20b}): +\subsubsubsection{Condensate fraction} +To compute the condensate fraction, we solve the modified {\tt anyeq} (see\-~\cite{CJL21}): \begin{equation} (-\Delta+2\mu)u_\mu = @@ -1519,41 +2070,53 @@ We then approximate (see\-~(\ref{avg})). \bigskip -\point{\bf Correlation function.} +\subsubsubsection{Correlation function (spherical average)} The two-point correlation function is \begin{equation} - C_2(x):= + c_2(x):= 2\rho\frac{\delta e}{\delta v(x)} +\end{equation} +and its spherical average is +\begin{equation} + C_2(|x|):=\frac1{4\pi|x|^2}\int dy\ \delta(|x|-|y|)c_2(y) . \end{equation} In Fourier space, \begin{equation} - C_2(x)= - 2\rho\int dk\ e^{ikx}\frac{\delta e}{\delta\hat v(k)} + c_2(x)= + 2\rho + \int dk\ e^{ikx}\frac{\delta e}{\delta\hat v(k)} +\end{equation} +so +\begin{equation} + C_2(|x|)= + 2\rho\int dk\ \left(\frac1{4\pi|x|^2}\int dy\ \delta(|x|-|y|)e^{iky}\right)\frac{\delta e}{\delta\hat v(k)} =2\rho\int dk\ \frac{\sin(|k||x|)}{|k||x|}\frac{\delta e}{\delta\hat v(k)} . \label{C2fourier} \end{equation} \bigskip -\subpoint +\point We can compute this quantity by considering a modified {\tt anyeq} in Fourier space, by formally replacing $\hat v$ with \begin{equation} \hat v+\lambda g(|k|) ,\quad g(|k|):=\frac{\sin(|k||x|)}{|k||x|} . + \label{2pt_addv} \end{equation} Indeed, if $e_\lambda$ denotes the energy of this modified equation, \begin{equation} \partial_\lambda e_\lambda|_{\lambda=0} - =\int dk\ \frac{\delta e}{\hat v(k)}\partial_\lambda({\textstyle \hat v(k)+\lambda g(|k|)}) - =\int dk\ g(|k|)\frac{\delta e}{\hat v(k)} + =\int dk\ \frac{\delta e}{\delta\hat v(k)}\partial_\lambda({\textstyle \hat v(k)+\lambda g(|k|)}) + =\int dk\ g(|k|)\frac{\delta e}{\delta\hat v(k)} + . \end{equation} -so, denoting the solution of the modified equation by $u_\lambda$, +So, denoting the solution of the modified equation by $u_\lambda$, \begin{equation} C_2(x)=2\rho\partial_\lambda e_\lambda|_{\lambda=0} - =-\rho^2\int dx\ (g(x)u_0(x)+v(x)\partial_\lambda u_\lambda(x)|_{\lambda=0}) + =\rho^2g(0)-\rho^2\int\frac{dk}{(2\pi)^3}\ (g(k)\hat u(k)+\hat v(k)\partial_\lambda\hat u_\lambda(k)|_{\lambda=0}) . \end{equation} We compute $\partial_\lambda u_\lambda|_{\lambda=0}$ in the same way as the uncondensed fraction: we define $\Xi(\mathbb U,\lambda)$ by formally adding $\lambda g(|k|)$ to $\hat v$, solve $\Xi(\mathbb U,\lambda)=0$, and differentiate: @@ -1563,7 +2126,7 @@ We compute $\partial_\lambda u_\lambda|_{\lambda=0}$ in the same way as the unco \end{equation} \bigskip -\subpoint +\point We compute $\partial_\lambda\Xi|_{\lambda=0}$: \begin{equation} \begin{largearray} @@ -1581,6 +2144,7 @@ We compute $\partial_\lambda\Xi|_{\lambda=0}$: -\partial_\lambda\mathbb Y_{l,j} \right)\ \partial\Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right) \end{largearray} + \label{2pt_dXi} \end{equation} with \begin{equation} @@ -1610,7 +2174,7 @@ with \begin{equation} \partial_\lambda\mathbb I_{l,j}= \frac1{\rho}\gamma_{L,2}\left( - \beta_{L,2}(\mathbb U\odot(\mathbb U\partial_\lambda\mathbb S)_{l,j} + \beta_{L,2}(\mathbb U\odot(\mathbb U\partial_\lambda\mathbb S))_{l,j} + (1-\beta_{L,2})\partial_\lambda\mathbb E(\mathbb U\odot\mathbb U)_{l,j} \right) @@ -1623,7 +2187,7 @@ with \begin{equation} \begin{largearray} \partial_\lambda\mathbb G_{l,j}= - \frac2{\rho}\gamma_K\alpha_K\beta_K\left(\partial_\lambda\mathbb S\mathbb U\right)_{l,j} + \frac2{\rho}\gamma_K\alpha_K\beta_K\left(\mathbb U\odot\left(\partial_\lambda\mathbb S\mathbb U\right)\right)_{l,j} +\frac2{\rho}\gamma_K\alpha_K(1-\beta_K)\partial_\lambda\mathbb E(\mathbb U\odot\mathbb U)_{l,j} - \frac1{\rho}\alpha_{L,1}\beta_{L,1}\left(\mathbb U\odot\partial_\lambda\mathbb S\mathbb U^2\right)_{l,j} @@ -1641,7 +2205,7 @@ To evaluate $(\mathbb U\odot\mathfrak g)$ and $\left<\mathbb U\mathfrak g\right> To do so, we replace $\hat v$ with $g$ in the computation of $\Upsilon$. \bigskip -\subpoint +\point In order to invert the Fourier transform in\-~(\ref{C2fourier}) numerically, we will use a Hann window (see appendix\-~\ref{appendix:hann}) \begin{equation} H_L(k):=\mathds 1_{|k|<\frac L2}\cos^2({\textstyle\frac{\pi|k|}{L}}) @@ -1652,7 +2216,203 @@ The parameter $L$ is set using \refname{param:anyeq_window_L}{{\tt window\_L}}. The computation is changed only in that $g$ is changed to $H_L(k)\frac{\sin(|k||x|)}{|k||x|}$. \bigskip -\point {\bf Momentum distribution.} +\point +To compute the maximum of $C_2$, we use a modified Newton algorithm. +The initial guess for the maximum is $|x_0|=\rho^{-\frac13}$\refname{param:anyeq_x0}{{\tt x0}}. +The modified Newton algorithm is an iteration: +\begin{equation} + x_{n+1}=x_n+\frac{\partial C_2(|x_n|)}{|\partial^2C_2(|x_n|)|} + \label{anyeq_newton_2pt} +\end{equation} +in which the derivatives are approximated using finite differences: +\begin{equation} + \partial C_2(x)\approx \frac{C_2(|x|+dx)-C_2(|x|)}{dx} + ,\quad + \partial^2 C_2(x)\approx \frac{C_2(|x|+dx)+C_2(|x|-dx)-2C_2(|x|)}{dx^2} + . + \label{anyeq_dx_2pt} +\end{equation} +This is a modification of the usual Newton iteration $x_n+\partial C_2/\partial^2C_2$ which is designed to follow the direction of the gradient, and thus to move toward a local maximum. +In addition, if $|\partial C_2|/|\partial^2 C_2|$ is larger than \refname{param:anyeq_maxstep}{{\tt maxstep}}, then the step is replaced with $\pm$\refname{param:anyeq_maxstep}{{\tt maxstep}}. +This prevents the algorithm from stepping over a maximum and land on another, further away. +This is useful if one has a good idea of where the global maximum is, and does not want to get trapped in a smaller local maximum. +\bigskip + +\indent +The algorithm is run for a maximum of \refname{param:anyeq_maxiter}{{\tt maxiter}} iterations, or until $|x_{n+1}-x_n|$ is smaller than \refname{param:anyeq_tolerance}{{\tt tolerance}}. +If the maximal number of iterations is reached, or if the solution found is not a local maximum, then the algorithm fails, and returns $+\infty$. +The point thus computed is therefore a local maximum, but it is not guaranteed to be the global maximum. + +\subsubsubsection{Fourier transform of two-point correlation (spherical average)} +The Fourier transform of the two-point correlation function is +\begin{equation} + \hat c_2(q):= + 2\rho\frac{\delta e}{\delta v(q)} +\end{equation} +and its spherical average is +\begin{equation} + \hat C_2(|q|):=\frac1{4\pi|q|^2}\int dk\ \delta(|q|-|k|)c_2(k) + = + \frac\rho{2\pi|q|^2}\int dk\ \delta(|q|-|k|)\frac{\delta e}{\delta\hat v(k)} + . +\end{equation} +\bigskip + +\point +To compute $\frac{\delta e}{\delta\hat v(q)}$, one idea would be to proceed in the same way as for the two-point correlation function, by replacing $\hat v$ with +\begin{equation} + \hat v+\lambda g(|k|) + ,\quad + g(|k|):=\frac1{4\pi|q|^2}\delta(|q|-|k|) +\end{equation} +where $\delta$ is the Dirac-delta function distribution (compare this with\-~(\ref{2pt_addv})). +However, the $\delta$ function causes all sorts of problems with the Chebyshev polynomial exansion and the quadratures. +\bigskip + +\point +Instead, we approximate $\hat C_2$ by convolving it with a normalized Gaussian: let +\begin{equation} + \Gamma_L(|k|):=\left(\frac{L}{2\pi}\right)^{\frac32}e^{-\frac L2k^2} + \label{anyeq_gaussian} +\end{equation} +\begin{equation} + \hat{\mathfrak C}_2(|q|) + := + \int dp\ \hat C_2(|q-p|)\Gamma_L(|p|) + =\int dk\ \int dp\ \frac\rho{2\pi|q-p|^2}\delta(|q-p|-|k|)\frac{\delta e}{\delta\hat v(k)}\Gamma_L(|p|) +\end{equation} +which by lemma\-~\ref{lemma:bipolar} is +\begin{equation} + \hat{\mathfrak C}_2(|q|) + =\int dk\ + \frac\rho{|q|} + \frac{\delta e}{\delta\hat v(k)} + \int_0^\infty dt\ \int_{||q|-t|}^{|q|+t}ds\ s\frac{\delta(t-|k|)}t\Gamma_L(s) +\end{equation} +that is +\begin{equation} + \hat{\mathfrak C}_2(|q|) + =\int dk\ + \frac{\delta e}{\delta\hat v(k)} + \frac{\rho}{|q||k|} + \int_{||q|-|k||}^{|q|+|k|}ds\ s\Gamma_L(s) +\end{equation} +which is the directional derivative of $e$ with respect to $\hat v$ in the direction of $2\rho g$ with +\begin{equation} + g(|k|):=\frac1{2|q||k|}\int_{||q|-|k||}^{|q|+|k|}ds\ s\Gamma_L(s) + = + \frac1{2|k|rL}(\Gamma_L(|k|-r)-\Gamma_L(|k|+r)) + . + \label{anyeq_2pt_fourier_g} +\end{equation} +Note that +\begin{equation} + g(0):=\Gamma_L(|q|) + . +\end{equation} +To compute this derivative, we replace $\hat v$ with +\begin{equation} + \hat v+\lambda g(|k|) +\end{equation} +so, denoting the solution of the modified equation by $u_\lambda$, for $q\neq 0$, +\begin{equation} + \hat{\mathfrak C}_2(|q|)=2\rho\partial_\lambda e_\lambda|_{\lambda=0} + =\rho^2\left( + -\int\frac{dk}{(2\pi)^3}\ g(|k|)\hat u(|k|) + -\int\frac{dk}{(2\pi)^3}\ \hat v(|k|)\partial_\lambda\hat u_\lambda(|k|)|_{\lambda=0} + \right) + . +\end{equation} +To compute $\partial_\lambda\hat u_\lambda|_{\lambda=0}$, we differentiate $\Xi(\mathbb U,\lambda)=0$: +\begin{equation} + \partial_\lambda\mathbb U|_{\lambda=0}=-(D\Xi)^{-1}\partial_\lambda\Xi|_{\lambda=0} + . +\end{equation} +The computation of $\partial_\lambda\Xi|_{\lambda=0}$ is identical to\-~(\ref{2pt_dXi}), but taking +\begin{equation} + \mathfrak g_{l,j}=g(|k_{l,j}|) + . +\end{equation} +with $g$ defined in\-~(\ref{anyeq_2pt_fourier_g}). +\bigskip + +\point +To compute the maximum of $\hat C_2$, we proceed as for $C_2$, see\-~(\ref{anyeq_newton_2pt})-(\ref{anyeq_dx_2pt}). +The only difference is that the algorithm is initialized with $|k_0|=\rho^{\frac13}$\refname{param:anyeq_k0}{{\tt k0}}. + +\subsubsubsection{Correlation function of uncondensed particles (spherical average)} +To compute the correlation function among uncondensed particles, denoted by $\gamma_2(|\xi|)$, we solve the modified {\tt anyeq} (see\-~\cite{Ja23}): +\begin{equation} + -\Delta u_\mu + = + (1-u_\mu)v-2\rho K+\rho^2 L + -\frac{\rho\mu}{2\pi|\xi|^2} u(|\xi|)\delta(|\xi|-|x|) +\end{equation} +where $K$ and $L$ are defined as in\-~(\ref{anyeq_K})-(\ref{anyeq_e}) in which $u$ is replaced with $u_\mu$. +In Fourier space, +\begin{equation} + k^2 \hat u_\mu + = + \hat S-2\rho\hat K+\rho^2\hat L + -2\rho\mu u(|\xi|)\frac{\sin(|k||\xi|)}{|k||\xi|} + . +\end{equation} +The uncondensed correlation function is then +\begin{equation} + \gamma_2(|\xi|)=\partial_\mu e|_{\mu=0} + =-\frac\rho2\int dx\ v(x)\partial_\mu u_\mu(x)|_{\mu=0} + . +\end{equation} +To compute the energy in the presence of the parameter $\mu$, we proceed in the same way as for $\mu=0$, the only difference being that the term +\begin{equation} + \mu g(|k|):= + -\mu 2\rho u(|\xi|)\frac{\sin(|k||\xi|)}{|k||\xi|} +\end{equation} +should formally be added to the right side of\-~(\ref{Omega}). +In other words, we consider $\mathbb U_{j,l}=u_\mu(|k_{j,l}|)$ and define $\Xi(\mathbb U,\mu)$ in the same way as in\-~(\ref{root_anyeq}), except that $\mathbb Y_{l,j}$ should be replaced by +\begin{equation} + \mathbb Y_{l,j}= + \frac1{\mathbb L_{l,j}}\left(\mathbb S_{l,j}-\mathbb L_{l,j}+\frac12\mathbb J_{l,j}+\mathbb G_{l,j}+\mu g(k_{l,j})\right) + . +\end{equation} +We then solve +\begin{equation} + \Xi(\mathbb U_\mu,\mu)=0 +\end{equation} +By differentiating this identity with respect to $\mu$, we find $\partial_\mu u_\mu$: +\begin{equation} + \partial_\mu \mathbb U|_{\mu=0}=-(D\Xi)^{-1}\partial_\mu\Xi|_{\mu=0} +\end{equation} +and +\begin{equation} + \partial_\mu\Xi|_{\mu=0}= + -\frac{\partial_\mu\mathbb Y_{l,j}}{2(\mathbb X_{l,j}+1)} + \Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right) + -\frac{(1+\mathbb Y_{l,j})\partial_\mu\mathbb Y_{l,j}}{2(\mathbb X_{l,j}+1)^3} + \partial\Phi\left({\textstyle\frac{1+\mathbb Y_{l,j}}{(\mathbb X_{l,j}+1)^2}}\right) + ,\quad + \partial_\mu\mathbb Y_{l,j}=\frac{g(k_{l,j})}{\mathbb L_{l,j}} + . +\end{equation} +We then approximate +\begin{equation} + \gamma_2(\xi)\approx + -\frac12\left<\mathfrak v\partial_\mu\mathbb U\right> +\end{equation} +(see\-~(\ref{avg})). +\bigskip + +\indent +In order to avoid numerical oscillations due to the $\sin$ function, we will use a Hann window (see appendix\-~\ref{appendix:hann}) +\begin{equation} + H_L(k):=\mathds 1_{|k|<\frac L2}\cos^2({\textstyle\frac{\pi|k|}{L}}) + . + \label{hann} +\end{equation} +The parameter $L$ is set using \refname{param:anyeq_window_L}{{\tt window\_L}}. +The computation is changed only in that $g$ is changed to $H_L(k)g(k)$. + +\subsubsubsection{Momentum distribution} To compute the momentum distribution (see\-~\cite{CHe21}), we add a parameter $\lambda$ to {\tt anyeq}: \begin{equation} -\Delta u_\lambda(|x|) @@ -1664,7 +2424,7 @@ To compute the momentum distribution (see\-~\cite{CHe21}), we add a parameter $\ The momentum distribution is then \begin{equation} \mathcal M(q)=\partial_\lambda e|_{\lambda=0} - =-\frac\rho2\int dx\ v(x)\partial_\lambda u_\lambda(x)|_{\lambda=0} + =-\frac\rho2\int\frac{dk}{(2\pi)^3}\ \hat v(k)\partial_\lambda\hat u_\lambda(k)|_{\lambda=0} . \end{equation} Note that the Fourier transform of $2\lambda\hat u_0(q)\cos(q\cdot x)$ is @@ -1673,52 +2433,118 @@ Note that the Fourier transform of $2\lambda\hat u_0(q)\cos(q\cdot x)$ is . \label{fouriermomentum} \end{equation} -We compute $\partial_\lambda u_\lambda|_{\lambda=0}$ in the same way as the uncondensed fraction. +The presence of delta functions does not play well with the Chebyshev polynomial expansion and the quadratures. +\bigskip + +\point +We will consider two different ways of getting around this. \bigskip \subpoint -In order to do so we will discretize momentum space, see\-~(\ref{klj}), and so it is necessary to construct a discrete analog of the delta-functions in\-~(\ref{fouriermomentum}). -The starting point we take is +One idea is to compute a regularization of $\mathcal M(q)$ by convolving it with a peaked spherically symmetric function. +Let $\Gamma_L$ denote the Gaussian with variance $1/\sqrt L$: \begin{equation} - \int dk f(k)\delta(k-q) - =f(q) + \Gamma_L(|k|):=\left(\frac{L}{2\pi}\right)^{\frac32}e^{-\frac L2k^2} + . + \label{anyeq_gaussian_momt} +\end{equation} +In fact, we will scale $L$ with $k$, and set $L$ to +\begin{equation} + L=\sqrt{\mathrm{\refname{param:anyeq_momentum_window_L}{{\tt window\_L}}}}/k^2 + . +\end{equation} +To compute +\begin{equation} + \mathfrak M(q):=\mathcal M\ast\Gamma_L(q) +\end{equation} +we solve the equation +\begin{equation} + -\Delta u_\lambda(|x|) + = + (1-u_\lambda(|x|))v(|x|)-2\rho K(|x|)+\rho^2 L(|x|) + -2\lambda\int dk\ \hat u_0(k)\cos(k\cdot x)\Gamma_L(q-k) + . +\end{equation} +Note that the Fourier transform of +\begin{equation} + -2\lambda\int dk\ \hat u_0(k)\cos(k\cdot x)\Gamma_L(q-k) +\end{equation} +is +\begin{equation} + -(2\pi)^3\lambda\hat u_0(q)(\Gamma_L(k+q)+\Gamma_L(k-q)) + . +\end{equation} +Since the ground state is unique, $\mathcal M$ is spherically symmetric. +The term $\Gamma_L(k\pm q)$ is not, so we take its spherical average (which will not change the final result): by lemma\-~\ref{lemma:bipolar}, +\begin{equation} + -\frac1{4\pi r^2}\int dq\ \delta(|q|-r)(2\pi)^3\lambda\hat u_0(q)(\Gamma_L(k+q)+\Gamma_L(k-q)) + = + -\frac{(2\pi)^3}{|k|r}\lambda\hat u_0(r)\int_{||k|-r|}^{|k|+r} ds\ s\Gamma_L(s) + . +\end{equation} +In this setup, the approximation of the delta function is thus +\begin{equation} + \tilde\delta(|k|,r):= + \frac1{2|k|r}\int_{||k|-r|}^{|k|+r} ds\ s\Gamma_L(s) + = + \frac{1}{2|k|rL}(\Gamma_L(|k|-r)-\Gamma_L(|k|+r)) + . +\end{equation} +To choose this method, set \refname{param:anyeq_window_L}{{\tt window\_L}} to a finite value. +\bigskip + +\subpoint +Another approach is to contruct a discrete analog of the delta-functions in\-~(\ref{fouriermomentum}). +The starting point we take is, for $q\neq0$, +\begin{equation} + \int dk\ f(|k|)\delta(k-q) + \equiv\frac1{4\pi|q|^2}\int dk\ f(|k|)\delta(|k|-|q|) + =f(|q|) \end{equation} so, when approximating the integral according to\-~(\ref{anyeq_integration}), we find \begin{equation} - \frac{\pi^2}2\sum_{l=0}^{J-1} + \frac{\pi}{8|q|^2}\sum_{l=0}^{J-1} (\tau_{l+1}-\tau_l) \sum_{j=1}^Nw_j - \cos({\textstyle\frac{\pi x_j}2})(1+\vartheta_l(x_j))^2\vartheta_l^2(x_j)f(\vartheta_l(x_j))\tilde\delta(\vartheta_l(x_j)-k_{l',i}) - \approx - f(k_{l',i}) + \cos({\textstyle\frac{\pi x_j}2})(1+\vartheta_l(x_j))^2\vartheta_l^2(x_j)f(\vartheta_l(x_j))\tilde\delta(\vartheta_l(x_j),|q|) + = + f(|q|) \end{equation} where $\tilde\delta$ is the approximation of the delta-function. Since \begin{equation} \vartheta_l(x_j)\equiv k_{l,j} \end{equation} -(see\-~(\ref{chebyshevvars})), we find that the definition of $\tilde\delta$ must be +(see\-~(\ref{chebyshevvars})), we define the approximation of the delta function as \begin{equation} - \tilde\delta_{l,j;l',i} + \tilde\delta(k_{l,j},k_{l',i}) := \delta_{l,l'}\delta_{j,i} - \frac2{\pi^2} + \frac{8}{\pi} \left( (\tau_{l+1}-\tau_l) w_j \cos({\textstyle\frac{\pi x_j}2}) - (1+k_{l,j})^2k_{l,j}^2 + (1+k_{l,j})^2 \right)^{-1} . \end{equation} -Note that, due to the invariance under rotation, the approximation for $\delta(q+k)$ is equal to that for $\delta(q-k)$. +To choose this method, set \refname{param:anyeq_window_L}{{\tt window\_L}} to {\tt Inf}. +{\color{red}This method seems to yield some fairly poor results!} \bigskip -\subpoint -To compute the momentum distribution at $q=k_{l',i}$, we define $\Xi(\mathbb U,\lambda)$ by formally adding $-2(2\pi)^3\lambda \hat u_0(k_{l',i})\tilde\delta_{l,j;l',i}$ to $\mathbb G_{l,j}$, which corresponds to replacing $\mathbb Y$ with +\point +To compute the momentum distribution at $q$, we define $\Xi(\mathbb U,\lambda)$ by formally adding +\begin{equation} + -2(2\pi)^3\lambda \hat u_0(|q|)\tilde\delta(k_{l,j},|q|) +\end{equation} +to $\mathbb G_{l,j}$, which corresponds to replacing $\mathbb Y$ with \begin{equation} \mathbb Y_{l,j}= - \frac1{\mathbb L_{l,j}}\left(\mathbb S_{l,j}-\mathbb L_{l,j}+\frac12\mathbb J_{l,j}+\mathbb G_{l,j}-2(2\pi)^3\lambda\frac{\mathbb U_{l',i}}\rho\tilde\delta_{l,j;l',i}\right) + \frac1{\mathbb L_{l,j}}\left(\mathbb S_{l,j}-\mathbb L_{l,j}+\frac12\mathbb J_{l,j}+\mathbb G_{l,j} + -2(2\pi)^3\lambda \hat u(|q|)\tilde\delta(k_{l,j},|q|) + \right) + . \end{equation} Then we solve $\Xi(\mathbb U,\lambda)=0$, and differentiate: \begin{equation} @@ -1739,7 +2565,33 @@ Finally, with \begin{equation} \partial_\lambda\mathbb Y_{l,j}|_{\lambda=0}= - \frac{2(2\pi)^3\mathbb U_{l',i}\tilde\delta_{l,j;l',i}}{\rho\mathbb L_{l,j}} + -\frac{2(2\pi)^3}{\mathbb L_{l,j}} \hat u(|q|)\tilde\delta(k_{l,j},|q|) + . +\end{equation} +\bigskip + +\subsubsubsection{Compressibility} +The compressibility is defined as +\begin{equation} + \chi:=\frac1{\rho^2\partial_\rho^2(\rho e)} + =\frac1{\partial_{\log\rho}^2(\rho e)-\partial_{\log\rho}(\rho e)} + . +\end{equation} +We approximate these derivatives by finite differences: +\begin{equation} + \partial_{\log\rho}^2 f(\rho) + \approx + \frac{f(\rho_{j+1})+f(\rho_{j-1})-2f(\rho_j)}{(\log\rho_{j+2}-\log\rho_{j+1})(\log\rho_{j+1}-\log\rho_j)} +\end{equation} +and +\begin{equation} + \partial_{\log\rho} f(\rho) + \approx + \frac12\left( + \frac{f(\rho_{j+1})-f(\rho_j)}{\log\rho_{j+1}-\log\rho_{j}} + + + \frac{f(\rho_{j})-f(\rho_{j-1})}{\log\rho_{j}-\log\rho_{j-1}} + \right) . \end{equation} @@ -1755,7 +2607,7 @@ The method is used to compute observables for the simple equation . \label{simpleq} \end{equation} -One can show\-~\cite[Theorem\-~1.6]{CJL20b} that the condensate fraction is +One can show\-~\cite[Theorem\-~1.6]{CJL21} that the condensate fraction is \begin{equation} \eta=\frac{\rho\int v(x)\mathfrak Ku(x)\ dx}{1-\rho\int v(x)\mathfrak K(2u(x)-\rho u\ast u(x))\ dx} \label{simpleq-Kv_eta} @@ -1824,8 +2676,18 @@ The available {\tt commands} are the following. \item\makelink{param:simpleq-Kv_nlrho}{} {\tt nlrho} ({\tt Int64}, default: $100$): number of values for $\rho$ (spaced logarithmically). + \item\makelink{param:simpleq-Kv_minrho}{} + {\tt minrho} ({\tt Float64}, default: $10^{-6}$): minimal value for $\rho$. + + \item\makelink{param:simpleq-Kv_maxrho}{} + {\tt maxrho} ({\tt Float64}, default: $10^{2}$): maximal value for $\rho$. + + \item\makelink{param:simpleq-Kv_nrho}{} + {\tt nrho} ({\tt Int64}, default: $0$): number of values for $\rho$ (spaced linearly). If {\tt nrho} is $\neq0$, then the linear spacing will be used, and \refname{param:simpleq-Kv_minlrho}{{\tt minlrho}}, \refname{param:simpleq-Kv_maxlrho}{{\tt maxlrho}}, \refname{param:simpleq-Kv_nlrho}{{\tt nlrho}} will be ignored. Otherwise, the logarithmic spacing will be used and \refname{param:simpleq-Kv_minrho}{{\tt minrho}}, \refname{param:simpleq-Kv_maxrho}{{\tt maxrho}} will be ignored. + \item\makelink{param:simpleq-Kv_rhos}{} - {\tt rhos} ({\tt Array\{Float64\}}, default: $(10^{{\tt minlrho}+\frac{{\tt maxlrho}-{\tt minlrho}}{{\tt nlrho}}n})_n$: list of values for $\rho$, specified as a `{\tt,}' separated list. + {\tt rhos} ({\tt Array\{Float64\}}): list of values for $\rho$, specified as a `{\tt,}' separated list. + This parameter takes precedence over \refname{param:simpleq-Kv_minlrho}{{\tt minlrho}}, \refname{param:simpleq-Kv_maxlrho}{{\tt maxlrho}}, \refname{param:simpleq-Kv_nlrho}{{\tt nlrho}}, \refname{param:simpleq-Kv_minrho}{{\tt minrho}}, \refname{param:simpleq-Kv_maxrho}{{\tt maxrho}}, \refname{param:simpleq-Kv_nrho}{{\tt nrho}}. \end{itemize} \underline{Output} (one line for each value of $\rho$): [$\rho$] [$\eta$] @@ -1980,8 +2842,18 @@ The available {\tt commands} are the following. \item\makelink{param:simpleq-hardcore_nlrho}{} {\tt nlrho} ({\tt Int64}, default: $100$): number of values for $\rho$ (spaced logarithmically). + \item\makelink{param:simpleq-hardcore_minrho}{} + {\tt minrho} ({\tt Float64}, default: $10^{-6}$): minimal value for $\rho$. + + \item\makelink{param:simpleq-hardcore_maxrho}{} + {\tt maxrho} ({\tt Float64}, default: $10^{2}$): maximal value for $\rho$. + + \item\makelink{param:simpleq-hardcore_nrho}{} + {\tt nrho} ({\tt Int64}, default: $0$): number of values for $\rho$ (spaced linearly). If {\tt nrho} is $\neq0$, then the linear spacing will be used, and \refname{param:simpleq-hardcore_minlrho}{{\tt minlrho}}, \refname{param:simpleq-hardcore_maxlrho}{{\tt maxlrho}}, \refname{param:simpleq-hardcore_nlrho}{{\tt nlrho}} will be ignored. Otherwise, the logarithmic spacing will be used and \refname{param:simpleq-hardcore_minrho}{{\tt minrho}}, \refname{param:simpleq-hardcore_maxrho}{{\tt maxrho}} will be ignored. + \item\makelink{param:simpleq-hardcore_rhos}{} - {\tt rhos} ({\tt Array\{Float64\}}, default: $(10^{{\tt minlrho}+\frac{{\tt maxlrho}-{\tt minlrho}}{{\tt nlrho}}n})_n$: list of values for $\rho$, specified as a `{\tt,}' separated list. + {\tt rhos} ({\tt Array\{Float64\}}): list of values for $\rho$, specified as a `{\tt,}' separated list. + This parameter takes precedence over \refname{param:simpleq-hardcore_minlrho}{{\tt minlrho}}, \refname{param:simpleq-hardcore_maxlrho}{{\tt maxlrho}}, \refname{param:simpleq-hardcore_nlrho}{{\tt nlrho}}, \refname{param:simpleq-hardcore_minrho}{{\tt minrho}}, \refname{param:simpleq-hardcore_maxrho}{{\tt maxrho}}, \refname{param:simpleq-hardcore_nrho}{{\tt nrho}}. \end{itemize} \underline{Output} (one line for each value of $\rho$): [$\rho$] [$e$] [Newton error $\epsilon$].\par \underline{Multithread support}: yes, different values of $\rho$ split up among workers. @@ -2023,7 +2895,7 @@ In order to carry out the computation of the solution of\-~(\ref{linearhc}) and for $|x|>1$. \bigskip -\point{\bf Energy.} +\subsubsubsection{Energy} To compute the energy $e$ of this equation, with the extra parameter $\mu$, we consider the limit of the soft sphere potential $\lambda\mathds 1_{|x|<1}$ (see\-~(\ref{easyeq}) with $\beta_K=\beta_L=0$): \begin{equation} (-\Delta+2\mu+4e)u(x)=s_\lambda(x)+2e\rho u\ast u @@ -2056,7 +2928,7 @@ Therefore, \end{equation} \bigskip -\point{\bf Integral equation.} +\subsubsubsection{Integral equation} We turn the differential equation in\-~(\ref{hardcore_simple}) into an integral equation. Let \begin{equation} @@ -2115,7 +2987,7 @@ We change variables in the last integral: \end{equation} \bigskip -\point{\bf The auto-convolution term.} We split +\subsubsubsection{The auto-convolution term} We split \begin{equation} u(r)=\mathds 1_{r>1}u_+(r-1)+\mathds 1_{r\leqslant 1} . @@ -2253,7 +3125,7 @@ where is the same as is defined for \refname{sec:anyeq}{{\tt anyeq}} (\ref{anyeq_alpha}). \bigskip -\point{\bf Chebyshev polynomial expansion.} We use the same interpolation as we used in \refname{sec:anyeq}{{\tt anyeq}}: (\ref{a_chebyshev_spline}) +\subsubsubsection{Chebyshev polynomial expansion} We use the same interpolation as we used in \refname{sec:anyeq}{{\tt anyeq}}: (\ref{a_chebyshev_spline}) \begin{equation} \frac1{(1+\tau)^{\nu_u}}u_+({\textstyle\frac{1-\tau}{1+\tau}})=\sum_{l=0}^{J-1}\mathds 1_{\tau_l\leqslant\tau<\tau_{l+1}}\sum_{n=0}^\infty \mathbf F_{l,n}^{(\nu_u)}(u_+) T_n({\textstyle\frac{2\tau-(\tau_l+\tau_{l+1})}{\tau_{l+1}-\tau_l}}) \label{a_chebyshev_spline} @@ -2343,10 +3215,10 @@ and \end{equation} \bigskip -\point{\bf Energy.} We now compute the approximation for the energy, using\-~(\ref{hardcore_energy}). +\subsubsubsection{Energy} We now compute the approximation for the energy, using\-~(\ref{hardcore_energy}). \bigskip -\subpoint We start with $\partial u|_{|x|\searrow 1}$. +\point We start with $\partial u|_{|x|\searrow 1}$. By\-~(\ref{uinteq}), \begin{equation} \partial u|_{|x|\searrow 1} @@ -2386,7 +3258,7 @@ and . \end{equation} -\subpoint Let us now turn to $\int_{|x|<1}dx\ u\ast u(x)$. +\point Let us now turn to $\int_{|x|<1}dx\ u\ast u(x)$. We have \begin{equation} \int_{|x|<1}dx\ u\ast u(x) @@ -2421,7 +3293,7 @@ and \end{equation} \bigskip -\subpoint Thus, +\point Thus, \begin{equation} \begin{largearray} e=2\pi\rho @@ -2452,11 +3324,11 @@ and \end{equation} \bigskip -\point{\bf Newton algorithm.} +\subsubsubsection{Newton algorithm} In this paragraph, we set $\epsilon=e$, that is, $\mu=0$. \bigskip -\subpoint +\point As we did for \refname{sec:anyeq}{{\tt anyeq}}, we discretize the integral in\-~(\ref{bfF}) by using a Gauss-Legendre quadrature, and truncate the sum over Chebyshev polynomials to order \refname{param:simpleq-hardcore_P}{{\tt P}}. We then reduce the computation to a finite system of equations, whose variables are \begin{equation} @@ -2519,7 +3391,7 @@ Gauss-Legendre and Gauss-Laguerre quadratures and their errors are discussed in The orders of the quadratures are given by the variable \refname{param:simpleq-hardcore_N}{{\tt N}}. \bigskip -\subpoint +\point We then solve $\Xi=0$ using the Newton algorithm, that is, we define a sequence of $\mathbb U$'s: \begin{equation} \mathbb U^{(n+1)}=\mathbb U^{(n)}-(D\Xi(\mathbb U^{(n)}))^{-1}\Xi(\mathbb U^{(n)}) @@ -2538,7 +3410,7 @@ We initialize the algorithm with \end{equation} \bigskip -\subpoint +\point We are left with computing the Jacobian of $\Xi$: \begin{equation} \begin{largearray} @@ -2709,7 +3581,7 @@ and Finally, to get from $D$ to $\mathbb D$ and $\gamma$ to $\mathfrak g$, we approximate the integrate using Gauss-Legendre and Gauss-Laguerre quadratures (see appendix\-~\ref{appGL}), as described above. \bigskip -\subpoint +\point We iterate the Newton algorithm until the Newton relative error $\epsilon$ becomes smaller than the \refname{param:easyeq_tolerance}{{\tt tolerance}} parameter. The Newton error is defined as \begin{equation} @@ -2723,7 +3595,7 @@ The energy thus obtained is \end{equation} \bigskip -\point{\bf Condensate fraction.} +\subsubsubsection{Condensate fraction} To compute the condensate fraction, we use the parameter $\mu$ in\-~(\ref{hardcore_simple}). The uncondensed fraction is \begin{equation} @@ -3213,7 +4085,7 @@ In $x$-space \frac{12c( x^6b^6(2e-b^2) +b^4x^4(9e-7b^2) +4b^2x^2(3e-2b^2) +(5e+16b^2))}{(1+b^2x^2)^2(4+b^2x^2)^2((1+b^2x^2)^2-c)} \end{equation} The constants $a,b,c,e$ can be set using the parameters {\tt v\_a}, {\tt v\_b}, {\tt v\_c}, {\tt v\_e}, and $a,e\in\mathbb R$, $b\neq0$, $c>0$, $c\neq9$. -Note that $v\geqslant 0$ if and only if\-~\cite{CJL20b} +Note that $v\geqslant 0$ if and only if\-~\cite{CJL21} \begin{equation} b>0 ,\quad diff --git a/src/anyeq.jl b/src/anyeq.jl index 8459cb0..fee77d8 100644 --- a/src/anyeq.jl +++ b/src/anyeq.jl @@ -1,4 +1,4 @@ -## Copyright 2021 Ian Jauslin +## Copyright 2021-2023 Ian Jauslin ## ## Licensed under the Apache License, Version 2.0 (the "License"); ## you may not use this file except in compliance with the License. @@ -28,24 +28,26 @@ end # compute energy for a given rho -# use minlrho, nlrho to incrementally compute the solution to medeq (to initialize the Newton algorithm) -function anyeq_energy(rho,minlrho,nlrho,taus,P,N,J,a0,v,maxiter,tolerance,approx,savefile) +function anyeq_energy( + rho::Float64, + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Anyeq_approx, + savefile::String +) # initialize vectors - (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) - # init Abar - if savefile!="" - Abar=anyeq_read_Abar(savefile,P,N,J) - else - Abar=anyeq_Abar_multithread(k,weights,taus,T,P,N,J,approx) - end + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) # compute initial guess from medeq - rhos=Array{Float64}(undef,nlrho) - for j in 0:nlrho-1 - rhos[j+1]=(nlrho==1 ? rho : 10^(minlrho+(log10(rho)-minlrho)/(nlrho-1)*j)) - end - u0s=anyeq_init_medeq(rhos,N,J,k,a0,v,maxiter,tolerance) - u0=u0s[nlrho] + u0=anyeq_init_medeq([rho],minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) @@ -53,72 +55,60 @@ function anyeq_energy(rho,minlrho,nlrho,taus,P,N,J,a0,v,maxiter,tolerance,approx end # compute energy as a function of rho -function anyeq_energy_rho(rhos,taus,P,N,J,a0,v,maxiter,tolerance,approx,savefile) +function anyeq_energy_rho( + rhos::Array{Float64,1}, + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Anyeq_approx, + savefile::String +) # initialize vectors - (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) - - # init Abar - if savefile!="" - Abar=anyeq_read_Abar(savefile,P,N,J) - else - Abar=anyeq_Abar_multithread(k,weights,taus,T,P,N,J,approx) - end + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) # compute initial guess from medeq - u0s=anyeq_init_medeq(rhos,N,J,k,a0,v,maxiter,tolerance) - - # save result from each task - es=Array{Float64,1}(undef,length(rhos)) - err=Array{Float64,1}(undef,length(rhos)) + u0s=anyeq_init_medeq(rhos,minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) - ## spawn workers - # number of workers - nw=nworkers() - # split jobs among workers - work=Array{Array{Int64,1},1}(undef,nw) - # init empty arrays - for p in 1:nw - work[p]=zeros(0) - end - for j in 1:length(rhos) - append!(work[(j-1)%nw+1],j) - end + # spawn workers + work=spawn_workers(length(rhos)) - count=0 - # for each worker - @sync for p in 1:nw - # for each task - @async for j in work[p] - count=count+1 - if count>=nw - progress(count,length(rhos),10000) - end - # run the task - (u,es[j],err[j])=remotecall_fetch(anyeq_hatu,workers()[p],u0s[j],P,N,J,rhos[j],a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) - end - end + # compute u + (us,es,errs)=anyeq_hatu_rho_multithread(u0s,P,N,J,rhos,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx,work) for j in 1:length(rhos) - @printf("% .15e % .15e % .15e\n",rhos[j],es[j],err[j]) + @printf("% .15e % .15e % .15e\n",rhos[j],es[j],errs[j]) end end # compute energy as a function of rho # initialize with previous rho -function anyeq_energy_rho_init_prevrho(rhos,taus,P,N,J,a0,v,maxiter,tolerance,approx,savefile) +function anyeq_energy_rho_init_prevrho( + rhos::Array{Float64,1}, + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Anyeq_approx, + savefile::String +) # initialize vectors - (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) - - # init Abar - if savefile!="" - Abar=anyeq_read_Abar(savefile,P,N,J) - else - Abar=anyeq_Abar_multithread(k,weights,taus,T,P,N,J,approx) - end + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) # compute initial guess from medeq - u0s=anyeq_init_medeq([rhos[1]],N,J,k,a0,v,maxiter,tolerance) - u=u0s[1] + u=anyeq_init_medeq([rhos[1]],minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) for j in 1:length(rhos) progress(j,length(rhos),10000) @@ -134,20 +124,26 @@ function anyeq_energy_rho_init_prevrho(rhos,taus,P,N,J,a0,v,maxiter,tolerance,ap end # compute energy as a function of rho # initialize with next rho -function anyeq_energy_rho_init_nextrho(rhos,taus,P,N,J,a0,v,maxiter,tolerance,approx,savefile) +function anyeq_energy_rho_init_nextrho( + rhos::Array{Float64,1}, + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Anyeq_approx, + savefile::String +) # initialize vectors - (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) - - # init Abar - if savefile!="" - Abar=anyeq_read_Abar(savefile,P,N,J) - else - Abar=anyeq_Abar_multithread(k,weights,taus,T,P,N,J,approx) - end + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) # compute initial guess from medeq - u0s=anyeq_init_medeq([rhos[length(rhos)]],N,J,k,a0,v,maxiter,tolerance) - u=u0s[1] + u=anyeq_init_medeq([rhos[length(rhos)]],minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) for j in 1:length(rhos) progress(j,length(rhos),10000) @@ -163,23 +159,26 @@ function anyeq_energy_rho_init_nextrho(rhos,taus,P,N,J,a0,v,maxiter,tolerance,ap end # compute u(k) -# use minlrho, nlrho to incrementally compute the solution to medeq (to initialize the Newton algorithm) -function anyeq_uk(minlrho,nlrho,taus,P,N,J,rho,a0,v,maxiter,tolerance,approx,savefile) +function anyeq_uk( + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Anyeq_approx, + savefile::String +) # init vectors - (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) - # init Abar - if savefile!="" - Abar=anyeq_read_Abar(savefile,P,N,J) - else - Abar=anyeq_Abar_multithread(k,weights,taus,T,P,N,J,approx) - end + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) + # compute initial guess from medeq - rhos=Array{Float64}(undef,nlrho) - for j in 0:nlrho-1 - rhos[j+1]=(nlrho==1 ? rho : 10^(minlrho+(log10(rho)-minlrho)/(nlrho-1)*j)) - end - u0s=anyeq_init_medeq(rhos,N,J,k,a0,v,maxiter,tolerance) - u0=u0s[nlrho] + u0=anyeq_init_medeq([rho],minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) @@ -192,58 +191,60 @@ function anyeq_uk(minlrho,nlrho,taus,P,N,J,rho,a0,v,maxiter,tolerance,approx,sav end # compute u(x) -# use minlrho, nlrho to incrementally compute the solution to medeq (to initialize the Newton algorithm) -function anyeq_ux(minlrho,nlrho,taus,P,N,J,rho,a0,v,maxiter,tolerance,xmin,xmax,nx,approx,savefile) +function anyeq_ux( + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + xmin::Float64, + xmax::Float64, + nx::Int64, + approx::Anyeq_approx, + savefile::String +) # init vectors - (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) - # init Abar - if savefile!="" - Abar=anyeq_read_Abar(savefile,P,N,J) - else - Abar=anyeq_Abar_multithread(k,weights,taus,T,P,N,J,approx) - end + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) # compute initial guess from medeq - rhos=Array{Float64}(undef,nlrho) - for j in 0:nlrho-1 - rhos[j+1]=(nlrho==1 ? rho : 10^(minlrho+(log10(rho)-minlrho)/(nlrho-1)*j)) - end - u0s=anyeq_init_medeq(rhos,N,J,k,a0,v,maxiter,tolerance) - u0=u0s[nlrho] + u0=anyeq_init_medeq([rho],minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) for i in 1:nx x=xmin+(xmax-xmin)*i/nx - ux=0. - for zeta in 0:J-1 - for j in 1:N - ux+=(taus[zeta+2]-taus[zeta+1])/(16*pi*x)*weights[2][j]*cos(pi*weights[1][j]/2)*(1+k[zeta*N+j])^2*k[zeta*N+j]*u[zeta*N+j]*sin(k[zeta*N+j]*x) - end - end - @printf("% .15e % .15e % .15e\n",x,real(ux),imag(ux)) + ux=anyeq_u_x(x,u,k,N,J,taus,weights) + @printf("% .15e % .15e % .15e\n",x,ux,) end end # compute condensate fraction for a given rho -# use minlrho, nlrho to incrementally compute the solution to medeq (to initialize the Newton algorithm) -function anyeq_condensate_fraction(rho,minlrho,nlrho,taus,P,N,J,a0,v,maxiter,tolerance,approx,savefile) +function anyeq_condensate_fraction( + rho::Float64, + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Anyeq_approx, + savefile::String +) # initialize vectors - (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) - # init Abar - if savefile!="" - Abar=anyeq_read_Abar(savefile,P,N,J) - else - Abar=anyeq_Abar_multithread(k,weights,taus,T,P,N,J,approx) - end + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) # compute initial guess from medeq - rhos=Array{Float64}(undef,nlrho) - for j in 0:nlrho-1 - rhos[j+1]=(nlrho==1 ? rho : 10^(minlrho+(log10(rho)-minlrho)/(nlrho-1)*j)) - end - u0s=anyeq_init_medeq(rhos,N,J,k,a0,v,maxiter,tolerance) - u0=u0s[nlrho] + u0=anyeq_init_medeq([rho],minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) @@ -254,161 +255,571 @@ function anyeq_condensate_fraction(rho,minlrho,nlrho,taus,P,N,J,a0,v,maxiter,tol end # condensate fraction as a function of rho -function anyeq_condensate_fraction_rho(rhos,taus,P,N,J,a0,v,maxiter,tolerance,approx,savefile) - ## spawn workers - # number of workers - nw=nworkers() - # split jobs among workers - work=Array{Array{Int64,1},1}(undef,nw) - # init empty arrays - for p in 1:nw - work[p]=zeros(0) +function anyeq_condensate_fraction_rho( + rhos::Array{Float64,1}, + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Anyeq_approx, + savefile::String +) + # spawn workers + work=spawn_workers(length(rhos)) + + # initialize vectors + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) + + # compute initial guess from medeq + u0s=anyeq_init_medeq(rhos,minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) + + # compute u + (us,es,errs)=anyeq_hatu_rho_multithread(u0s,P,N,J,rhos,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx,work) + + # compute eta + etas=Array{Float64,1}(undef,length(rhos)) + count=0 + # for each worker + @sync for p in 1:length(work) + # for each task + @async for j in work[p] + count=count+1 + if count>=length(work) + progress(count,length(rhos),10000) + end + # run the task + etas[j]=remotecall_fetch(anyeq_eta,workers()[p],us[j],P,N,J,rhos[j],weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx) + end end + for j in 1:length(rhos) - append!(work[(j-1)%nw+1],j) + @printf("% .15e % .15e % .15e\n",rhos[j],etas[j],errs[j]) end +end +# compute the momentum distribution for a given rho +function anyeq_momentum_distribution( + kmin::Float64, + kmax::Float64, + rho::Float64, + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + windowL::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Anyeq_approx, + savefile::String +) # initialize vectors - (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) - # init Abar - if savefile!="" - Abar=anyeq_read_Abar(savefile,P,N,J) + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) + + # compute initial guess from medeq + u0=anyeq_init_medeq([rho],minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) + + (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) + + # compute M + if windowL==Inf + # use discrete approximation of delta function + M=anyeq_momentum_discrete_delta(kmin,kmax,u,P,N,J,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx) else - Abar=anyeq_Abar_multithread(k,weights,taus,T,P,N,J,approx) + M=anyeq_momentum_window(kmin,kmax,u,P,N,J,windowL,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx) end + # order k's in increasing order + for zeta in 0:J-1 + for j in 1:N + q=k[(J-1-zeta)*N+j] + # drop if not in k interval + if q<kmin || q>kmax + continue + end + @printf("% .15e % .15e\n",q,M[(J-1-zeta)*N+j]) + end + end +end + +# 2 point correlation function +function anyeq_2pt_correlation( + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + windowL::Float64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + xmin::Float64, + xmax::Float64, + nx::Int64, + approx::Anyeq_approx, + savefile::String +) + # init vectors + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) + # compute initial guess from medeq - u0s=anyeq_init_medeq(rhos,N,J,k,a0,v,maxiter,tolerance) + u0=anyeq_init_medeq([rho],minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) + + # compute u and some useful integrals + (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) + (S,E,II,JJ,X,Y,sL1,sK,G)=anyeq_SEIJGXY(rho*u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx) + + for i in 1:nx + x=xmin+(xmax-xmin)*i/nx + C2=anyeq_2pt(x,u,P,N,J,windowL,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx,S,E,II,JJ,X,Y,sL1,sK) + @printf("% .15e % .15e\n",x,C2) + end +end + +# maximum of 2 point correlation function +function anyeq_2pt_correlation_max( + rho::Float64, + minlrho_init::Float64, + nlrho_init::Int64, + dx::Float64, + x0::Float64, # initial guess is x0/rho^(1/3) + maxstep::Float64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + windowL::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + tolerance_max::Float64, + approx::Anyeq_approx, + savefile::String +) + # init vectors + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) + + # compute initial guess from medeq + u0=anyeq_init_medeq([rho],minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) + + # initial guess + x0=1/rho^(1/3) + + (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) + (x,f)=anyeq_2pt_max(u,P,N,J,x0/rho^(1/3),dx,maxstep,maxiter,tolerance_max,windowL,rho,weights,T,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx) + + if(x==Inf) + @printf(stderr,"max search failed for rho=%e\n",rho) + else + @printf("% .15e % .15e\n",x,f) + end +end + +# maximum of 2 point correlation function as a function of rho +function anyeq_2pt_correlation_max_rho( + rhos::Array{Float64,1}, + minlrho_init::Float64, + nlrho_init::Int64, + dx::Float64, + x0::Float64, # initial guess is x0/rho^(1/3) + maxstep::Float64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + windowL::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + tolerance_max::Float64, + approx::Anyeq_approx, + savefile::String +) + # init vectors + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) + + # compute initial guess from medeq + u0s=anyeq_init_medeq(rhos,minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) + + # save result from each task + xs=Array{Float64,1}(undef,length(rhos)) + fs=Array{Float64,1}(undef,length(rhos)) + + # spawn workers + work=spawn_workers(length(rhos)) # compute u - us=Array{Array{Float64,1}}(undef,length(rhos)) - errs=Array{Float64,1}(undef,length(rhos)) + (us,es,errs)=anyeq_hatu_rho_multithread(u0s,P,N,J,rhos,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx,work) + count=0 # for each worker - @sync for p in 1:nw + @sync for p in 1:length(work) # for each task @async for j in work[p] count=count+1 - if count>=nw - progress(count,length(rhos),10000) + if count>=length(work) + progress(count,length(rhos),10000) end # run the task - (us[j],E,errs[j])=remotecall_fetch(anyeq_hatu,workers()[p],u0s[j],P,N,J,rhos[j],a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) + (xs[j],fs[j])=remotecall_fetch(anyeq_2pt_max,workers()[p],us[j],P,N,J,x0/rhos[j]^(1/3),dx,maxstep,maxiter,tolerance_max,windowL,rhos[j],weights,T,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx) end end - # compute eta - etas=Array{Float64}(undef,length(rhos)) + for j in 1:length(rhos) + if(xs[j]==Inf) + @printf(stderr,"max search failed for rho=%e\n",rhos[j]) + else + @printf("% .15e % .15e % .15e\n",rhos[j],xs[j],fs[j]) + end + end +end + +# Correlation function in Fourier space +function anyeq_2pt_correlation_fourier( + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + windowL::Float64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + kmin::Float64, + kmax::Float64, + nk::Int64, + approx::Anyeq_approx, + savefile::String +) + # init vectors + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) + + # compute initial guess from medeq + u0=anyeq_init_medeq([rho],minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) + + # compute u and some useful integrals + (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) + (S,E,II,JJ,X,Y,sL1,sK,G)=anyeq_SEIJGXY(rho*u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx) + + # save result from each task + C2s=Array{Float64,1}(undef,nk) + + # spawn workers + work=spawn_workers(nk) + count=0 # for each worker - @sync for p in 1:nw + @sync for p in 1:length(work) # for each task @async for j in work[p] count=count+1 - if count>=nw - progress(count,length(rhos),10000) + if count>=length(work) + progress(count,nk,10000) end # run the task - etas[j]=anyeq_eta(us[j],P,N,J,rhos[j],weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx) + C2s[j]=remotecall_fetch(anyeq_2pt_fourier,workers()[p],kmin+(kmax-kmin)*j/nk,u,P,N,J,windowL,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx,S,E,II,JJ,X,Y,sL1,sK) end end - for j in 1:length(rhos) - @printf("% .15e % .15e % .15e\n",rhos[j],etas[j],errs[j]) + for j in 1:nk + @printf("% .15e % .15e\n",kmin+(kmax-kmin)*j/nk,C2s[j]) end end -# compute the momentum distribution for a given rho -# use minlrho, nlrho to incrementally compute the solution to medeq (to initialize the Newton algorithm) -function anyeq_momentum_distribution(rho,minlrho,nlrho,taus,P,N,J,a0,v,maxiter,tolerance,approx,savefile) - # initialize vectors - (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) - # init Abar - if savefile!="" - Abar=anyeq_read_Abar(savefile,P,N,J) +# maximum of Fourier transform of 2 point correlation function +function anyeq_2pt_correlation_fourier_max( + rho::Float64, + minlrho_init::Float64, + nlrho_init::Int64, + dk::Float64, + k0::Float64, # initial guess is k0*rho^(1/3) + maxstep::Float64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + windowL::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + tolerance_max::Float64, + approx::Anyeq_approx, + savefile::String +) + # init vectors + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) + + # compute initial guess from medeq + u0=anyeq_init_medeq([rho],minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) + + # compute u + (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) + (ko,f)=anyeq_2pt_fourier_max(u,P,N,J,k0*rho^(1/3),dk,maxstep,maxiter,tolerance_max,windowL,rho,weights,T,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx) + + if(ko==Inf) + @printf(stderr,"max search failed for rho=%e\n",rho) else - Abar=anyeq_Abar_multithread(k,weights,taus,T,P,N,J,approx) + @printf("% .15e % .15e\n",ko,f) end +end + +# maximum of fourier transform of 2 point correlation function as a function of rho +function anyeq_2pt_correlation_fourier_max_rho( + rhos::Array{Float64,1}, + minlrho_init::Float64, + nlrho_init::Int64, + dk::Float64, + k0::Float64, # initial guess is k0*rho^(1/3) + maxstep::Float64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + windowL::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + tolerance_max::Float64, + approx::Anyeq_approx, + savefile::String +) + # init vectors + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) # compute initial guess from medeq - rhos=Array{Float64}(undef,nlrho) - for j in 0:nlrho-1 - rhos[j+1]=(nlrho==1 ? rho : 10^(minlrho+(log10(rho)-minlrho)/(nlrho-1)*j)) - end - u0s=anyeq_init_medeq(rhos,N,J,k,a0,v,maxiter,tolerance) - u0=u0s[nlrho] + u0s=anyeq_init_medeq(rhos,minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) - (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) + # spawn workers + work=spawn_workers(length(rhos)) - # compute M - M=anyeq_momentum(u,P,N,J,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx) + # compute u + (us,es,errs)=anyeq_hatu_rho_multithread(u0s,P,N,J,rhos,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx,work) - for zeta in 0:J-1 - for j in 1:N - # order k's in increasing order - @printf("% .15e % .15e\n",k[(J-1-zeta)*N+j],M[(J-1-zeta)*N+j]) + # save result from each task + ks=Array{Float64,1}(undef,length(rhos)) + fs=Array{Float64,1}(undef,length(rhos)) + + count=0 + # for each worker + @sync for p in 1:length(work) + # for each task + @async for j in work[p] + count=count+1 + if count>=length(work) + progress(count,length(rhos),10000) + end + # run the task + (ks[j],fs[j])=remotecall_fetch(anyeq_2pt_fourier_max,workers()[p],us[j],P,N,J,k0*rhos[j]^(1/3),dk,maxstep,maxiter,tolerance_max,windowL,rhos[j],weights,T,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx) + end + end + + for j in 1:length(rhos) + if(ks[j]==Inf) + @printf(stderr,"max search failed for rho=%e\n",rhos[j]) + else + @printf("% .15e % .15e % .15e\n",rhos[j],ks[j],fs[j]) end end end -# 2 point correlation function -function anyeq_2pt_correlation(minlrho,nlrho,taus,P,N,J,windowL,rho,a0,v,maxiter,tolerance,xmin,xmax,nx,approx,savefile) +# uncondensed 2 point correlation function +function anyeq_uncondensed_2pt_correlation( + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + windowL::Float64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + xmin::Float64, + xmax::Float64, + nx::Int64, + approx::Anyeq_approx, + savefile::String +) # init vectors - (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) - # init Abar - if savefile!="" - Abar=anyeq_read_Abar(savefile,P,N,J) - else - Abar=anyeq_Abar_multithread(k,weights,taus,T,P,N,J,approx) + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) + + # compute initial guess from medeq + u0=anyeq_init_medeq([rho],minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) + + # compute u and some useful integrals + (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) + (S,E,II,JJ,X,Y,sL1,sK,G)=anyeq_SEIJGXY(rho*u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx) + + # compute u(xi) + uxis=Array{Float64,1}(undef,nx) + for i in 1:nx + uxis[i]=anyeq_u_x(xmin+(xmax-xmin)*i/nx,u,k,N,J,taus,weights) + end + + + # spawn workers + work=spawn_workers(nx) + gamma2s=Array{Float64,1}(undef,nx) + count=0 + # for each worker + @sync for p in 1:length(work) + # for each task + @async for j in work[p] + count=count+1 + if count>=length(work) + progress(count,nx,10000) + end + # run the task + gamma2s[j]=remotecall_fetch(anyeq_uncondensed_2pt,workers()[p],xmin+(xmax-xmin)*j/nx,uxis[j],u,P,N,J,windowL,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx,S,E,II,JJ,X,Y,sL1,sK) + end end + for i in 1:nx + @printf("% .15e % .15e\n",xmin+(xmax-xmin)*i/nx,gamma2s[i]) + end +end + +# compute the compressibility +function anyeq_compressibility_rho( + rhos::Array{Float64,1}, + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Anyeq_approx, + savefile::String +) + # initialize vectors + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) + # compute initial guess from medeq - rhos=Array{Float64}(undef,nlrho) - for j in 0:nlrho-1 - rhos[j+1]=(nlrho==1 ? rho : 10^(minlrho+(log10(rho)-minlrho)/(nlrho-1)*j)) + u0s=anyeq_init_medeq(rhos,minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) + + # save result from each task + es=Array{Float64,1}(undef,length(rhos)) + err=Array{Float64,1}(undef,length(rhos)) + + # spawn workers + work=spawn_workers(length(rhos)) + + # compute u + (us,es,errs)=anyeq_hatu_rho_multithread(u0s,P,N,J,rhos,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx,work) + + for j in 1:length(rhos)-2 + # compute \rho^2\partial^2(\rho e)=\partial^2_{\log\rho}(\rho e)-\partial_{\log\rho}(\rho e) as a function of rho + # \partial^2_{\log\rho}(\rho e) + p2=(rhos[j+2]*es[j+2]-2*rhos[j+1]*es[j+1]+rhos[j]*es[j])/(log(rhos[j+2])-log(rhos[j+1]))/(log(rhos[j+1])-log(rhos[j])) + # \partial_{\log\rho}(\rho e) (take average of front and back) + p11=(rhos[j+2]*es[j+2]-rhos[j+1]*es[j+1])/(log(rhos[j+2])-log(rhos[j+1])) + p12=(rhos[j+1]*es[j+1]-rhos[j]*es[j])/(log(rhos[j+1])-log(rhos[j])) + # compressibility is 1/this + @printf("% .15e % .15e\n",rhos[j+1],1/(p2-(p11+p12)/2)) end - u0s=anyeq_init_medeq(rhos,N,J,k,a0,v,maxiter,tolerance) - u0=u0s[nlrho] +end + +# Fourier transform 2 point correlation function test: compute by transforming anyeq_2pt +function anyeq_2pt_correlation_fourier_test( + minlrho_init::Float64, + nlrho_init::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + windowL::Float64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + xmax::Float64, + kmin::Float64, + kmax::Float64, + nk::Int64, + approx::Anyeq_approx, + savefile::String +) + # init vectors + (weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v,approx,savefile) + + # compute initial guess from medeq + u0=anyeq_init_medeq([rho],minlrho_init,nlrho_init,N,J,k,a0,v,maxiter,tolerance) # compute u and some useful integrals (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) (S,E,II,JJ,X,Y,sL1,sK,G)=anyeq_SEIJGXY(rho*u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx) - for i in 1:nx - x=xmin+(xmax-xmin)*i/nx - C2=anyeq_2pt(x,u,P,N,J,windowL,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx,S,E,II,JJ,X,Y,sL1,sK,G) - @printf("% .15e % .15e\n",x,C2) + # compute C2 in real space + C2=Array{Float64,1}(undef,N*J) + for i in 1:N*J + if k[i]<xmax + C2[i]=anyeq_2pt(k[i],u,P,N,J,windowL,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx,S,E,II,JJ,X,Y,sL1,sK)-rho^2 + else + C2[i]=0. + end + end + # invert Fourier transform + for i in 1:nk + k0=kmin+(kmax-kmin)*i/nk + hatC2=inverse_fourier_chebyshev(C2,k0,k,taus,weights,N,J) + @printf("% .15e % .15e\n",k0,hatC2) end end # compute Abar -function anyeq_Abar_multithread(k,weights,taus,T,P,N,J,approx) +function anyeq_Abar_multithread( + k::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + P::Int64, + N::Int64, + J::Int64, + approx::Anyeq_approx +) if approx.bL3==0. - return [] + # empty array + return Array{Array{Float64,5}}(undef,0) end out=Array{Array{Float64,5}}(undef,J*N) - ## spawn workers - # number of workers - nw=nworkers() - # split jobs among workers - work=Array{Array{Int64,1},1}(undef,nw) - # init empty arrays - for p in 1:nw - work[p]=zeros(0) - end - for j in 1:N*J - append!(work[(j-1)%nw+1],j) - end + # spawn workers + work=spawn_workers(N*J) count=0 # for each worker - @sync for p in 1:nw + @sync for p in 1:length(work) # for each task @async for j in work[p] count=count+1 - if count>=nw + if count>=length(work) progress(count,N*J,10000) end # run the task @@ -419,14 +830,23 @@ function anyeq_Abar_multithread(k,weights,taus,T,P,N,J,approx) return out end + # initialize computation -@everywhere function anyeq_init(taus,P,N,J,v) +@everywhere function anyeq_init( + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + v::Function, + approx::Anyeq_approx, + savefile::String +) # Gauss-Legendre weights weights=gausslegendre(N) # initialize vectors V,k - V=Array{Float64}(undef,J*N) - k=Array{Float64}(undef,J*N) + V=Array{Float64,1}(undef,J*N) + k=Array{Float64,1}(undef,J*N) for zeta in 0:J-1 for j in 1:N xj=weights[1][j] @@ -437,7 +857,7 @@ end end end # potential at 0 - V0=v(0) + V0=v(0.) # initialize matrix A T=chebyshev_polynomials(P) @@ -449,39 +869,62 @@ end Upsilon=Upsilonmat(k,v,weights_plus) Upsilon0=Upsilon0mat(k,v,weights_plus) - return(weights,T,k,V,V0,A,Upsilon,Upsilon0) + # init Abar + if savefile!="" + Abar=anyeq_read_Abar(savefile,P,N,J) + else + Abar=anyeq_Abar_multithread(k,weights,taus,T,P,N,J,approx) + end + + return(weights,T,k,V,V0,A,Abar,Upsilon,Upsilon0) end # compute initial guess from medeq -@everywhere function anyeq_init_medeq(rhos,N,J,k,a0,v,maxiter,tolerance) - us_medeq=Array{Array{Float64,1}}(undef,length(rhos)) - u0s=Array{Array{Float64,1}}(undef,length(rhos)) - +@everywhere function anyeq_init_medeq( + rhos::Array{Float64,1}, + minlrho_init::Float64, + nlrho_init::Int64, + N::Int64, + J::Int64, + k::Array{Float64,1}, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64 +) weights_medeq=gausslegendre(N*J) + (u0s_medeq,es,errs)=easyeq_compute_u_prevrho(rhos,minlrho_init,nlrho_init,a0,N*J,v,maxiter,tolerance,weights_medeq,Easyeq_approx(1.,1.)) - (us_medeq[1],E,err)=easyeq_hatu(easyeq_init_u(a0,J*N,weights_medeq),J*N,rhos[1],v,maxiter,tolerance,weights_medeq,Easyeq_approx(1.,1.)) - u0s[1]=easyeq_to_anyeq(us_medeq[1],weights_medeq,k,N,J) - if err>tolerance - print(stderr,"warning: computation of initial Ansatz failed for rho=",rhos[1],"\n") - end - - for j in 2:length(rhos) - (us_medeq[j],E,err)=easyeq_hatu(us_medeq[j-1],J*N,rhos[j],v,maxiter,tolerance,weights_medeq,Easyeq_approx(1.,1.)) - u0s[j]=easyeq_to_anyeq(us_medeq[j],weights_medeq,k,N,J) - - if err>tolerance + # check errs + for j in 1:length(errs) + if errs[j]>tolerance print(stderr,"warning: computation of initial Ansatz failed for rho=",rhos[j],"\n") end end - return u0s + # return a single vector if there is a single rho + if length(rhos)>1 + u0s=Array{Array{Float64,1}}(undef,length(rhos)) + for j in 1:length(u0s_medeq) + u0s[j]=easyeq_to_anyeq(u0s_medeq[j],weights_medeq,k,N,J) + end + return u0s + else + return easyeq_to_anyeq(u0s_medeq,weights_medeq,k,N,J) + end end # interpolate the solution of medeq to an input for anyeq -@everywhere function easyeq_to_anyeq(u_simple,weights,k,N,J) +@everywhere function easyeq_to_anyeq( + u_simple::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + k::Array{Float64,1}, + N::Int64, + J::Int64 +) # reorder u_simple, which is evaluated at (1-x_j)/(1+x_j) with x_j\in[-1,1] u_s=zeros(Float64,length(u_simple)) - k_s=Array{Float64}(undef,length(u_simple)) + k_s=Array{Float64,1}(undef,length(u_simple)) for j in 1:length(u_simple) xj=weights[1][j] k_s[length(u_simple)-j+1]=(1-xj)/(1+xj) @@ -501,7 +944,27 @@ end # compute u using chebyshev expansions -@everywhere function anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) +@everywhere function anyeq_hatu( + u0::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + rho::Float64, + a0::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + V::Array{Float64,1}, + V0::Float64, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + Upsilon::Array{Array{Float64,1},1}, + Upsilon0::Array{Float64,1}, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Anyeq_approx +) # init # rescale by rho (that's how u is defined) u=rho*u0 @@ -512,7 +975,7 @@ end # run Newton algorithm for i in 1:maxiter-1 (S,E,II,JJ,X,Y,sL1,sK,G)=anyeq_SEIJGXY(u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx) - new=u-inv(anyeq_DXi(u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK))*anyeq_Xi(u,X,Y) + new=u-inv(anyeq_DXi(u,rho,k,taus,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK))*anyeq_Xi(u,X,Y) error=norm(new-u)/norm(u) if(error<tolerance) @@ -528,8 +991,60 @@ end end +# compute u for various rho +function anyeq_hatu_rho_multithread( + u0s::Array{Array{Float64,1},1}, + P::Int64, + N::Int64, + J::Int64, + rhos::Array{Float64,1}, + a0::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + V::Array{Float64,1}, + V0::Float64, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + Upsilon::Array{Array{Float64,1},1}, + Upsilon0::Array{Float64,1}, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Anyeq_approx, + work::Array{Array{Int64,1},1} +) + # compute u + us=Array{Array{Float64,1}}(undef,length(rhos)) + es=Array{Float64,1}(undef,length(rhos)) + errs=Array{Float64,1}(undef,length(rhos)) + count=0 + # for each worker + @sync for p in 1:length(work) + # for each task + @async for j in work[p] + count=count+1 + if count>=length(work) + progress(count,length(rhos),10000) + end + # run the task + (us[j],es[j],errs[j])=remotecall_fetch(anyeq_hatu,workers()[p],u0s[j],P,N,J,rhos[j],a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,approx) + end + end + + return (us,es,errs) +end + + # save Abar -function anyeq_save_Abar(taus,P,N,J,v,approx) +function anyeq_save_Abar( + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + v::Function, + approx::Anyeq_approx +) # initialize vectors (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) @@ -555,7 +1070,12 @@ function anyeq_save_Abar(taus,P,N,J,v,approx) end # read Abar -function anyeq_read_Abar(savefile,P,N,J) +function anyeq_read_Abar( + savefile::String, + P::Int64, + N::Int64, + J::Int64 +) # open file ff=open(savefile,"r") # read all lines @@ -623,13 +1143,39 @@ end # Xi # takes the vector of kj's and xn's as input -@everywhere function anyeq_Xi(U,X,Y) +@everywhere function anyeq_Xi( + U::Array{Float64,1}, + X::Array{Float64,1}, + Y::Array{Float64,1} +) return U-(Y.+1)./(2*(X.+1)).*dotPhi((Y.+1)./((X.+1).^2)) end # DXi # takes the vector of kj's as input -@everywhere function anyeq_DXi(U,rho,k,taus,v,v0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK) +@everywhere function anyeq_DXi( + U::Array{Float64,1}, + rho::Float64, + k::Array{Float64,1}, + taus::Array{Float64,1}, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + Upsilon::Array{Array{Float64,1},1}, + Upsilon0::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64, + approx::Anyeq_approx, + S::Array{Float64,1}, + E::Float64, + II::Array{Float64,1}, + JJ::Array{Float64,1}, + X::Array{Float64,1}, + Y::Array{Float64,1}, + sL1::Array{Float64,1}, + sK::Array{Float64,1} +) out=Array{Float64,2}(undef,N*J,N*J) for zetapp in 0:J-1 for n in 1:N @@ -720,7 +1266,23 @@ end end # compute S,E,I,J,X and Y -@everywhere function anyeq_SEIJGXY(U,rho,k,taus,v,v0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx) +@everywhere function anyeq_SEIJGXY( + U::Array{Float64,1}, + rho::Float64, + k::Array{Float64,1}, + taus::Array{Float64,1}, + v::Array{Float64,1}, + v0::Float64, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + Upsilon::Array{Array{Float64,1},1}, + Upsilon0::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64, + approx::Anyeq_approx +) # Chebyshev expansion of U FU=chebyshev(U,taus,weights,P,N,J,2) @@ -798,79 +1360,207 @@ end return(S,E,II,JJ,X,Y,sL1,sK,G) end + +# u(x) +@everywhere function anyeq_u_x( + x::Float64, + u::Array{Float64,1}, + k::Array{Float64,1}, + N::Int64, + J::Int64, + taus::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}} +) + ux=0. + for zeta in 0:J-1 + for j in 1:N + ux+=(taus[zeta+2]-taus[zeta+1])/(16*pi*x)*weights[2][j]*cos(pi*weights[1][j]/2)*(1+k[zeta*N+j])^2*k[zeta*N+j]*u[zeta*N+j]*sin(k[zeta*N+j]*x) + end + end + return real(ux) +end + # condensate fraction -@everywhere function anyeq_eta(u,P,N,J,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx) +@everywhere function anyeq_eta( + u::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + V::Array{Float64,1}, + V0::Float64, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + Upsilon::Array{Array{Float64,1},1}, + Upsilon0::Array{Float64,1}, + approx::Anyeq_approx +) # compute dXi/dmu (S,E,II,JJ,X,Y,sL1,sK,G)=anyeq_SEIJGXY(rho*u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx) dXidmu=(Y.+1)./(rho*sL1)./(2*(X.+1).^2).*dotPhi((Y.+1)./((X.+1).^2))+(Y.+1).^2 ./((X.+1).^4)./(rho*sL1).*dotdPhi((Y.+1)./(X.+1).^2) # compute eta - du=-inv(anyeq_DXi(rho*u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK))*dXidmu + du=-inv(anyeq_DXi(rho*u,rho,k,taus,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK))*dXidmu eta=-avg_v_chebyshev(du,Upsilon0,k,taus,weights,N,J)/2 return eta end -# momentum distribution -@everywhere function anyeq_momentum(u,P,N,J,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx) - # compute dXi/dlambda (without delta functions) - (S,E,II,JJ,X,Y,sL1,sK,G)=anyeq_SEIJGXY(rho*u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx) - dXidlambda=-(2*pi)^3*2*u./sL1.*(dotPhi((Y.+1)./((X.+1).^2))./(2*(X.+1))+(Y.+1)./(2*(X.+1).^3).*dotdPhi((Y.+1)./(X.+1).^2)) - # approximation for delta function (without Kronecker deltas) - delta=Array{Float64}(undef,J*N) - for zeta in 0:J-1 - for n in 1:N - delta[zeta*N+n]=2/pi^2/((taus[zeta+2]-taus[zeta+1])*weights[2][n]*cos(pi*weights[1][n]/2)*(1+k[zeta*N+n])^2*k[zeta*N+n]^2) - end - end - - # compute dXidu - dXidu=inv(anyeq_DXi(rho*u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK)) +# correlation function +@everywhere function anyeq_2pt( + x::Float64, + u::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + windowL::Float64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + V::Array{Float64,1}, + V0::Float64, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + Upsilon::Array{Array{Float64,1},1}, + Upsilon0::Array{Float64,1}, + approx::Anyeq_approx, + S::Array{Float64,1}, + E::Float64, + II::Array{Float64,1}, + JJ::Array{Float64,1}, + X::Array{Float64,1}, + Y::Array{Float64,1}, + sL1::Array{Float64,1}, + sK::Array{Float64,1} +) + g=(r,x)->sinc(r*x)*hann(r,windowL) + du=anyeq_dudv(g, x, u, P, N, J, rho, weights, k, taus, V, V0, A, Abar, Upsilon, Upsilon0, approx, S, E, II, JJ, X, Y, sL1, sK) - M=Array{Float64}(undef,J*N) - for i in 1:J*N - # du/dlambda - du=dXidu[:,i]*dXidlambda[i]*delta[i] + C2=rho^2*(1-integrate_f_chebyshev(s->g(s,x),u,k,taus,weights,N,J)-integrate_f_chebyshev(s->1.,V.*du,k,taus,weights,N,J)) +end - # compute M - M[i]=-avg_v_chebyshev(du,Upsilon0,k,taus,weights,N,J)/2 +# uncondensed correlation function +@everywhere function anyeq_uncondensed_2pt( + xi::Float64, + uxi::Float64, + u::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + windowL::Float64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + V::Array{Float64,1}, + V0::Float64, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + Upsilon::Array{Array{Float64,1},1}, + Upsilon0::Array{Float64,1}, + approx::Anyeq_approx, + S::Array{Float64,1}, + E::Float64, + II::Array{Float64,1}, + JJ::Array{Float64,1}, + X::Array{Float64,1}, + Y::Array{Float64,1}, + sL1::Array{Float64,1}, + sK::Array{Float64,1} +) + # compute dXi/dmu + g=Array{Float64,1}(undef,length(k)) + for i in 1:length(k) + g[i]=-2*rho*uxi*sinc(k[i]*xi)*hann(k[i],windowL) end + dXidmu=-g./sL1./(2*(X.+1)).*dotPhi((Y.+1)./((X.+1).^2))-(Y.+1).*g./sL1./(2(X.+1).^3).*dotdPhi((Y.+1)./(X.+1).^2) - return M -end + # compute gamma2 + du=-inv(anyeq_DXi(rho*u,rho,k,taus,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK))*dXidmu + gamma2=-avg_v_chebyshev(du,Upsilon0,k,taus,weights,N,J)/2 + return gamma2 +end -# correlation function -@everywhere function anyeq_2pt(x,u,P,N,J,windowL,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx,S,E,II,JJ,X,Y,sL1,sK,G) +# compute the directional derivative of u with respect to v in direction g +@everywhere function anyeq_dudv( + g::Function,# should be of the form g(k,x) where x is a parameter + x::Float64, + u::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + V::Array{Float64,1}, + V0::Float64, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + Upsilon::Array{Array{Float64,1},1}, + Upsilon0::Array{Float64,1}, + approx::Anyeq_approx, + S::Array{Float64,1}, + E::Float64, + II::Array{Float64,1}, + JJ::Array{Float64,1}, + X::Array{Float64,1}, + Y::Array{Float64,1}, + sL1::Array{Float64,1}, + sK::Array{Float64,1} +) # initialize dV - dV=Array{Float64}(undef,J*N) + dV=Array{Float64,1}(undef,J*N) for i in 1:J*N - if x>0 - dV[i]=sin(k[i]*x)/(k[i]*x)*hann(k[i],windowL) - else - dV[i]=hann(k[i],windowL) - end + dV[i]=g(k[i],x) end - dV0=1. + dV0=g(0.,x) # compute dUpsilon # Upsilonmat does not use splines, so increase precision weights_plus=gausslegendre(N*J) - dUpsilon=Upsilonmat(k,r->sin(r*x)/(r*x)*hann(r,windowL),weights_plus) - dUpsilon0=Upsilon0mat(k,r->sin(r*x)/(r*x)*hann(r,windowL),weights_plus) + dUpsilon=Upsilonmat(k,s->g(s,x),weights_plus) + dUpsilon0=Upsilon0mat(k,s->g(s,x),weights_plus) - du=-inv(anyeq_DXi(rho*u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK))*anyeq_dXidv(x,rho*u,rho,k,taus,dV,dV0,A,Abar,dUpsilon,dUpsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK) + du=-inv(anyeq_DXi(rho*u,rho,k,taus,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK))*anyeq_dXidv(rho*u,rho,k,taus,dV,dV0,A,Abar,dUpsilon,dUpsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK) # rescale rho du=du/rho - C2=rho^2*(1-integrate_f_chebyshev(s->1.,u.*dV+V.*du,k,taus,weights,N,J)) - - return C2 + return du end -# derivative of Xi with respect to v in the direction sin(kx)/kx -@everywhere function anyeq_dXidv(x,U,rho,k,taus,dv,dv0,A,Abar,dUpsilon,dUpsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK) +# derivative of Xi with respect to v in the specified by dUpsilon and dUpsilon0 +@everywhere function anyeq_dXidv( + U::Array{Float64,1}, + rho::Float64, + k::Array{Float64,1}, + taus::Array{Float64,1}, + dv::Array{Float64,1}, + dv0::Float64, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + dUpsilon::Array{Array{Float64,1},1}, + dUpsilon0::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64, + approx::Anyeq_approx, + S::Array{Float64,1}, + E::Float64, + II::Array{Float64,1}, + JJ::Array{Float64,1}, + X::Array{Float64,1}, + Y::Array{Float64,1}, + sL1::Array{Float64,1}, + sK::Array{Float64,1} +) # Chebyshev expansion of U FU=chebyshev(U,taus,weights,P,N,J,2) @@ -896,7 +1586,7 @@ end dJJ+=approx.gL3*approx.bL3*double_conv_S_chebyshev(FU,FU,FU,FU,dFS,Abar) end if approx.bL3!=1. - dJJ=approx.gL3*(1-approx.bL3)*dE*(UU/rho).^2 + dJJ+=approx.gL3*(1-approx.bL3)*dE*(UU/rho).^2 end end @@ -947,3 +1637,220 @@ end out=((Y.+1).*dX./(X.+1)-dY)./(2*(X.+1)).*dotPhi((Y.+1)./((X.+1).^2))+(Y.+1)./(2*(X.+1).^3).*(2*(Y.+1)./(X.+1).*dX-dY).*dotdPhi((Y.+1)./(X.+1).^2) return out end + +# maximum of 2 point correlation function +@everywhere function anyeq_2pt_max( + u::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + x0::Float64, + dx::Float64, + maxstep::Float64, + maxiter::Int64, + tolerance::Float64, + windowL::Float64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + T::Array{Polynomial,1}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + V::Array{Float64,1}, + V0::Float64, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + Upsilon::Array{Array{Float64,1},1}, + Upsilon0::Array{Float64,1}, + approx::Anyeq_approx +) + # compute some useful integrals + (S,E,II,JJ,X,Y,sL1,sK,G)=anyeq_SEIJGXY(rho*u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx) + + (x,f)=newton_maximum(y->anyeq_2pt(y,u,P,N,J,windowL,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx,S,E,II,JJ,X,Y,sL1,sK),x0,dx,maxiter,tolerance,maxstep) + + return(x,f) +end + + +# Fourier transform of 2pt correlation function at q +@everywhere function anyeq_2pt_fourier( + q::Float64, + u::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + windowL::Float64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + V::Array{Float64,1}, + V0::Float64, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + Upsilon::Array{Array{Float64,1},1}, + Upsilon0::Array{Float64,1}, + approx::Anyeq_approx, + S::Array{Float64,1}, + E::Float64, + II::Array{Float64,1}, + JJ::Array{Float64,1}, + X::Array{Float64,1}, + Y::Array{Float64,1}, + sL1::Array{Float64,1}, + sK::Array{Float64,1} +) + # direction in which to differentiate u + weights_plus=gausslegendre(N*J) + #g=(r,x)->(r>0. ? (1.)/(2*x*r)*integrate_legendre(s->s*gaussian(s,(1.)/windowL),abs(x-r),x+r,weights_plus) : gaussian(x,(1.)/windowL)) + g=(r,x)->(r>0. ? (1.)/(2*x*r*windowL)*(gaussian(x-r,(1.)/windowL)-gaussian(x+r,(1.)/windowL)) : gaussian(x,(1.)/windowL)) + + du=anyeq_dudv(g,q,u,P,N,J,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx,S,E,II,JJ,X,Y,sL1,sK) + + C2=rho^2*(-integrate_f_chebyshev(s->g(s,q),u,k,taus,weights,N,J)-integrate_f_chebyshev(s->1.,V.*du,k,taus,weights,N,J)) + + return C2 +end + +# maximum of Fourier transform of 2 point correlation function +@everywhere function anyeq_2pt_fourier_max( + u::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + k0::Float64, + dk::Float64, + maxstep::Float64, + maxiter::Int64, + tolerance::Float64, + windowL::Float64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + T::Array{Polynomial,1}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + V::Array{Float64,1}, + V0::Float64, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + Upsilon::Array{Array{Float64,1},1}, + Upsilon0::Array{Float64,1}, + approx::Anyeq_approx +) + # compute some useful integrals + (S,E,II,JJ,X,Y,sL1,sK,G)=anyeq_SEIJGXY(rho*u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx) + + (ko,f)=newton_maximum(y->anyeq_2pt_fourier(y,u,P,N,J,windowL,rho,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,approx,S,E,II,JJ,X,Y,sL1,sK),k0,dk,maxiter,tolerance,maxstep) + + return(ko,f) +end + +# momentum distribution, computed using a Gaussian window +@everywhere function anyeq_momentum_window( + kmin::Float64, + kmax::Float64, + u::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + windowL::Float64, # L is windowL/k^2 + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + V::Array{Float64,1}, + V0::Float64, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + Upsilon::Array{Array{Float64,1},1}, + Upsilon0::Array{Float64,1}, + approx::Anyeq_approx +) + + # compute dXi/dlambda without the delta function of u(q) + (S,E,II,JJ,X,Y,sL1,sK,G)=anyeq_SEIJGXY(rho*u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx) + dXidlambda=(16*pi^3)*(dotPhi((Y.+1)./((X.+1).^2))./(2*(X.+1))+(Y.+1)./(2*(X.+1).^3).*dotdPhi((Y.+1)./(X.+1).^2))./sL1 + + # compute dXidu + dXidu=inv(anyeq_DXi(rho*u,rho,k,taus,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK)) + + M=Array{Float64,1}(undef,N*J) + for j in 1:J*N + # the Chebyshev polynomial expansion is often not good enough to compute M away from k[i] + q=k[j] + + # drop the computation if not in k interval + if q<kmin || q>kmax + continue + end + + # delta function + delta=Array{Float64,1}(undef,J*N) + L=windowL/q^2 + for i in 1:J*N + delta[i]=(1.)/(2*k[i]*q*L)*(gaussian(k[i]-q,(1.)/L)-gaussian(k[i]+q,(1.)/L)) + end + + # du/dlambda + du=-dXidu*(dXidlambda.*delta*u[j]) + # rescale u + du=du/rho + + # compute M + M[j]=-avg_v_chebyshev(du,Upsilon0,k,taus,weights,N,J)*rho/2 + end + + return M +end + +# momentum distribution, computed using a discrete approximation of the delta function +@everywhere function anyeq_momentum_discrete_delta( + kmin::Float64, + kmax::Float64, + u::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + V::Array{Float64,1}, + V0::Float64, + A::Array{Array{Float64,2},1}, + Abar::Array{Array{Float64,5},1}, + Upsilon::Array{Array{Float64,1},1}, + Upsilon0::Array{Float64,1}, + approx::Anyeq_approx +) + # compute dXi/dlambda (without delta functions) + (S,E,II,JJ,X,Y,sL1,sK,G)=anyeq_SEIJGXY(rho*u,rho,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx) + dXidlambda=(2*pi)^3*2*u./sL1.*(dotPhi((Y.+1)./((X.+1).^2))./(2*(X.+1))+(Y.+1)./(2*(X.+1).^3).*dotdPhi((Y.+1)./(X.+1).^2)) + + # approximation for delta function (without Kronecker deltas) + delta=Array{Float64,1}(undef,J*N) + for zeta in 0:J-1 + for n in 1:N + delta[zeta*N+n]=8/pi/((taus[zeta+2]-taus[zeta+1])*weights[2][n]*cos(pi*weights[1][n]/2)*(1+k[zeta*N+n])^2) + end + end + + # compute dXidu + dXidu=inv(anyeq_DXi(rho*u,rho,k,taus,A,Abar,Upsilon,Upsilon0,weights,P,N,J,approx,S,E,II,JJ,X,Y,sL1,sK)) + + M=Array{Float64,1}(undef,J*N) + for i in 1:J*N + # drop the computation if not in k interval + if k[i]<kmin || k[i]>kmax + continue + end + + # du/dlambda + du=-dXidu[:,i]*dXidlambda[i]*delta[i] + + # compute M + M[i]=-avg_v_chebyshev(du,Upsilon0,k,taus,weights,N,J)/2 + end + + return M +end diff --git a/src/chebyshev.jl b/src/chebyshev.jl index 28c8f1f..af4be40 100644 --- a/src/chebyshev.jl +++ b/src/chebyshev.jl @@ -1,4 +1,4 @@ -## Copyright 2021 Ian Jauslin +## Copyright 2021-2023 Ian Jauslin ## ## Licensed under the Apache License, Version 2.0 (the "License"); ## you may not use this file except in compliance with the License. @@ -13,7 +13,15 @@ ## limitations under the License. # Chebyshev expansion -@everywhere function chebyshev(a,taus,weights,P,N,J,nu) +@everywhere function chebyshev( + a::Array{Float64,1}, + taus::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64, + nu::Int64 +) out=zeros(Float64,J*(P+1)) for zeta in 0:J-1 for n in 0:P @@ -27,7 +35,15 @@ end # evaluate function from Chebyshev expansion -@everywhere function chebyshev_eval(Fa,x,taus,chebyshev,P,J,nu) +@everywhere function chebyshev_eval( + Fa::Array{Float64,1}, + x::Float64, + taus::Array{Float64,1}, + chebyshev::Array{Polynomial,1}, + P::Int64, + J::Int64, + nu::Int64 +) # change variable tau=(1-x)/(1+x) @@ -48,7 +64,11 @@ end # convolution # input the Chebyshev expansion of a and b, as well as the A matrix -@everywhere function conv_chebyshev(Fa,Fb,A) +@everywhere function conv_chebyshev( + Fa::Array{Float64,1}, + Fb::Array{Float64,1}, + A::Array{Array{Float64,2},1} +) out=zeros(Float64,length(A)) for i in 1:length(A) out[i]=dot(Fa,A[i]*Fb) @@ -57,12 +77,27 @@ end end # <ab> -@everywhere function avg_chebyshev(Fa,Fb,A0) +@everywhere function avg_chebyshev( + Fa::Array{Float64,1}, + Fb::Array{Float64,1}, + A0::Float64 +) return dot(Fa,A0*Fb) end # 1_n * a -@everywhere function conv_one_chebyshev(n,zetapp,Fa,A,taus,weights,P,N,J,nu1) +@everywhere function conv_one_chebyshev( + n::Int64, + zetapp::Int64, + Fa::Array{Float64,1}, + A::Array{Array{Float64,2},1}, + taus::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64, + nu1::Int64 +) out=zeros(Float64,N*J) for m in 1:N*J for l in 0:P @@ -74,7 +109,15 @@ end return out end # a * v -@everywhere function conv_v_chebyshev(a,Upsilon,k,taus,weights,N,J) +@everywhere function conv_v_chebyshev( + a::Array{Float64,1}, + Upsilon::Array{Array{Float64,1},1}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + N::Int64, + J::Int64 +) out=zeros(Float64,J*N) for i in 1:J*N for zetap in 0:J-1 @@ -86,7 +129,16 @@ end end return out end -@everywhere function conv_one_v_chebyshev(n,zetap,Upsilon,k,taus,weights,N,J) +@everywhere function conv_one_v_chebyshev( + n::Int64, + zetap::Int64, + Upsilon::Array{Array{Float64,1},1}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + N::Int64, + J::Int64 +) out=zeros(Float64,J*N) xj=weights[1][n] for i in 1:J*N @@ -96,7 +148,14 @@ end end # <av> -@everywhere function avg_v_chebyshev(a,Upsilon0,k,taus,weights,N,J) +@everywhere function avg_v_chebyshev(a, + Upsilon0::Array{Float64,1}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + N::Int64, + J::Int64 +) out=0. for zetap in 0:J-1 for j in 1:N @@ -107,13 +166,28 @@ end return out end # <1_nv> -@everywhere function avg_one_v_chebyshev(n,zetap,Upsilon0,k,taus,weights,N) +@everywhere function avg_one_v_chebyshev( + n::Int64, + zetap::Int64, + Upsilon0::Array{Float64,1}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + N::Int64 +) xj=weights[1][n] return (taus[zetap+2]-taus[zetap+1])/(32*pi)*weights[2][n]*cos(pi*xj/2)*(1+k[zetap*N+n])^2*k[zetap*N+n]*Upsilon0[zetap*N+n] end # compute \int dq dxi u1(k-xi)u2(q)u3(xi)u4(k-q)u5(xi-q) -@everywhere function double_conv_S_chebyshev(FU1,FU2,FU3,FU4,FU5,Abar) +@everywhere function double_conv_S_chebyshev( + FU1::Array{Float64,1}, + FU2::Array{Float64,1}, + FU3::Array{Float64,1}, + FU4::Array{Float64,1}, + FU5::Array{Float64,1}, + Abar::Array{Float64,5} +) out=zeros(Float64,length(Abar)) for i in 1:length(Abar) for j1 in 1:length(FU1) @@ -133,7 +207,17 @@ end # compute A -@everywhere function Amat(k,weights,taus,T,P,N,J,nua,nub) +@everywhere function Amat( + k::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + P::Int64, + N::Int64, + J::Int64, + nua::Int64, + nub::Int64 +) out=Array{Array{Float64,2},1}(undef,J*N) for i in 1:J*N out[i]=zeros(Float64,J*(P+1),J*(P+1)) @@ -152,7 +236,11 @@ end end # compute Upsilon -@everywhere function Upsilonmat(k,v,weights) +@everywhere function Upsilonmat( + k::Array{Float64,1}, + v::Function, + weights::Tuple{Array{Float64,1},Array{Float64,1}} +) out=Array{Array{Float64,1},1}(undef,length(k)) for i in 1:length(k) out[i]=Array{Float64,1}(undef,length(k)) @@ -162,7 +250,11 @@ end end return out end -@everywhere function Upsilon0mat(k,v,weights) +@everywhere function Upsilon0mat( + k::Array{Float64,1}, + v::Function, + weights::Tuple{Array{Float64,1},Array{Float64,1}} +) out=Array{Float64,1}(undef,length(k)) for j in 1:length(k) out[j]=2*k[j]*v(k[j]) @@ -171,17 +263,36 @@ end end # alpha_- -@everywhere function alpham(k,t) +@everywhere function alpham( + k::Float64, + t::Float64 +) return (1-k-(1-t)/(1+t))/(1+k+(1-t)/(1+t)) end # alpha_+ -@everywhere function alphap(k,t) +@everywhere function alphap( + k::Float64, + t::Float64 +) return (1-abs(k-(1-t)/(1+t)))/(1+abs(k-(1-t)/(1+t))) end # compute \bar A -@everywhere function barAmat(k,weights,taus,T,P,N,J,nu1,nu2,nu3,nu4,nu5) +@everywhere function barAmat( + k::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + P::Int64, + N::Int64, + J::Int64, + nu1::Int64, + nu2::Int64, + nu3::Int64, + nu4::Int64, + nu5::Int64 +) out=zeros(Float64,J*(P+1),J*(P+1),J*(P+1),J*(P+1),J*(P+1)) for zeta1 in 0:J-1 for n1 in 0:P @@ -211,27 +322,107 @@ end return out end -@everywhere function barAmat_int1(tau,k,taus,T,weights,nu1,nu2,nu3,nu4,nu5,zeta1,zeta2,zeta3,zeta4,zeta5,n1,n2,n3,n4,n5) +@everywhere function barAmat_int1(tau, + k::Float64, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + nu1::Int64, + nu2::Int64, + nu3::Int64, + nu4::Int64, + nu5::Int64, + zeta1::Int64, + zeta2::Int64, + zeta3::Int64, + zeta4::Int64, + zeta5::Int64, + n1::Int64, + n2::Int64, + n3::Int64, + n4::Int64, + n5::Int64 +) if(alpham(k,tau)<taus[zeta2+2] && alphap(k,tau)>taus[zeta2+1]) return 2*(1-tau)/(1+tau)^(3-nu1)*T[n1+1]((2*tau-(taus[zeta1+1]+taus[zeta1+2]))/(taus[zeta1+2]-taus[zeta1+1]))*integrate_legendre(sigma->barAmat_int2(tau,sigma,k,taus,T,weights,nu2,nu3,nu4,nu5,zeta2,zeta3,zeta4,zeta5,n2,n3,n4,n5),max(taus[zeta2+1],alpham(k,tau)),min(taus[zeta2+2],alphap(k,tau)),weights) else return 0. end end -@everywhere function barAmat_int2(tau,sigma,k,taus,T,weights,nu2,nu3,nu4,nu5,zeta2,zeta3,zeta4,zeta5,n2,n3,n4,n5) +@everywhere function barAmat_int2(tau, + sigma::Float64, + k::Float64, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + nu2::Int64, + nu3::Int64, + nu4::Int64, + nu5::Int64, + zeta2::Int64, + zeta3::Int64, + zeta4::Int64, + zeta5::Int64, + n2::Int64, + n3::Int64, + n4::Int64, + n5::Int64 +) return 2*(1-sigma)/(1+sigma)^(3-nu2)*T[n2+1]((2*sigma-(taus[zeta2+1]+taus[zeta2+2]))/(taus[zeta2+2]-taus[zeta2+1]))*integrate_legendre(taup->barAmat_int3(tau,sigma,taup,k,taus,T,weights,nu3,nu4,nu5,zeta3,zeta4,zeta5,n3,n4,n5),taus[zeta3+1],taus[zeta3+2],weights) end -@everywhere function barAmat_int3(tau,sigma,taup,k,taus,T,weights,nu3,nu4,nu5,zeta3,zeta4,zeta5,n3,n4,n5) +@everywhere function barAmat_int3(tau, + sigma::Float64, + taup::Float64, + k::Float64, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + nu3::Int64, + nu4::Int64, + nu5::Int64, + zeta3::Int64, + zeta4::Int64, + zeta5::Int64, + n3::Int64, + n4::Int64, + n5::Int64 +) if(alpham(k,taup)<taus[zeta4+2] && alphap(k,taup)>taus[zeta4+1]) return 2*(1-taup)/(1+taup)^(3-nu3)*T[n3+1]((2*taup-(taus[zeta3+1]+taus[zeta3+2]))/(taus[zeta3+2]-taus[zeta3+1]))*integrate_legendre(sigmap->barAmat_int4(tau,sigma,taup,sigmap,k,taus,T,weights,nu4,nu5,zeta4,zeta5,n4,n5),max(taus[zeta4+1],alpham(k,taup)),min(taus[zeta4+2],alphap(k,taup)),weights) else return 0. end end -@everywhere function barAmat_int4(tau,sigma,taup,sigmap,k,taus,T,weights,nu4,nu5,zeta4,zeta5,n4,n5) +@everywhere function barAmat_int4(tau, + sigma::Float64, + taup::Float64, + sigmap::Float64, + k::Float64, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + nu4::Int64, + nu5::Int64, + zeta4::Int64, + zeta5::Int64, + n4::Int64, + n5::Int64 +) return 2*(1-sigmap)/(1+sigmap)^(3-nu4)*T[n4+1]((2*sigma-(taus[zeta4+1]+taus[zeta4+2]))/(taus[zeta4+2]-taus[zeta4+1]))*integrate_legendre(theta->barAmat_int5(tau,sigma,taup,sigmap,theta,k,taus,T,weights,nu5,zeta5,n5),0.,2*pi,weights) end -@everywhere function barAmat_int5(tau,sigma,taup,sigmap,theta,k,taus,T,weights,nu5,zeta5,n5) +@everywhere function barAmat_int5(tau, + sigma::Float64, + taup::Float64, + sigmap::Float64, + theta::Float64, + k::Float64, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + nu5::Int64, + zeta5::Int64, + n5::Int64 +) R=barAmat_R((1-sigma)/(1+sigma),(1-tau)/(1+tau),(1-sigmap)/(1+sigmap),(1-taup)/(1+taup),theta,k) if((1-R)/(1+R)<taus[zeta5+2] && (1-R)/(1+R)>taus[zeta5+1]) return (2/(2+R))^nu5*T[n5+1]((2*(1-R)/(1+R)-(taus[zeta5+1]+taus[zeta5+2]))/(taus[zeta5+2]-taus[zeta5+1])) @@ -240,13 +431,22 @@ end end end # R(s,t,s',t,theta,k) -@everywhere function barAmat_R(s,t,sp,tp,theta,k) - return sqrt(k^2*(s^2+t^2+sp^2+tp^2)-k^4-(s^2-t^2)*(sp^2-tp^2)-sqrt((4*k^2*s^2-(k^2+s^2-t^2)^2)*(4*k^2*sp^2-(k^2+sp^2-tp^2)^2))*cos(theta))/(sqrt(2)*k) +@everywhere function barAmat_R( + s::Float64, + t::Float64, + sp::Float64, + tp::Float64, + theta::Float64, + k::Float64 +) + return sqrt(k^2*(s^2+t^2+sp^2+tp^2)-k^4-(s^2-t^2)*(sp^2-tp^2)-sqrt((4*k^2*s^2-(k^2+s^2-t^2)^2)*(4*k^2*sp^2-(k^2+sp^2-tp^2)^2))*cos(theta))/(sqrt(2.)*k) end # compute Chebyshev polynomials -@everywhere function chebyshev_polynomials(P) - T=Array{Polynomial}(undef,P+1) +@everywhere function chebyshev_polynomials( + P::Int64 +) + T=Array{Polynomial,1}(undef,P+1) T[1]=Polynomial([1]) T[2]=Polynomial([0,1]) for n in 1:P-1 @@ -258,7 +458,15 @@ end end # compute \int f*u dk/(2*pi)^3 -@everywhere function integrate_f_chebyshev(f,u,k,taus,weights,N,J) +@everywhere function integrate_f_chebyshev( + f::Function, + u::Array{Float64,1}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + N::Int64, + J::Int64 +) out=0. for zeta in 0:J-1 for i in 1:N @@ -268,7 +476,15 @@ end return out end -@everywhere function inverse_fourier_chebyshev(u,x,k,taus,weights,N,J) +@everywhere function inverse_fourier_chebyshev( + u::Array{Float64,1}, + x::Float64, + k::Array{Float64,1}, + taus::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + N::Int64, + J::Int64 +) out=0. for zeta in 0:J-1 for j in 1:N @@ -277,3 +493,54 @@ end end return out end + +# compute B (for the computation of the fourier transform of the two-point correlation) +@everywhere function Bmat( + q::Float64, + k::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + P::Int64, + N::Int64, + J::Int64, + nu::Int64 +) + out=Array{Array{Float64,1},1}(undef,J*N) + for i in 1:J*N + out[i]=zeros(Float64,J*(P+1)) + for zeta in 0:J-1 + for n in 0:P + out[i][zeta*(P+1)+n+1]=1/(8*pi^3*k[i]*q)*(betam(k[i],q)>taus[zeta+2] || betap(k[i],q)<taus[zeta+1] ? 0. : integrate_legendre(sigma->(1-sigma)/(1+sigma)^(3-nu)*T[n+1]((2*sigma-(taus[zeta+1]+taus[zeta+2]))/(taus[zeta+2]-taus[zeta+1])),max(taus[zeta+1],betam(k[i],q)),min(taus[zeta+2],betap(k[i],q)),weights)) + end + end + end + + return out +end +# beta_- +@everywhere function betam( + k::Float64, + q::Float64 +) + return (1-k-q)/(1+k+q) +end +# beta_+ +@everywhere function betap( + k::Float64, + q::Float64 +) + return (1-abs(k-q))/(1+abs(k-q)) +end + +# mathfrak S (for the computation of the fourier transform of the two-point correlation) +@everywhere function chebyshev_frakS( + Ff::Array{Float64,1}, + B::Array{Array{Float64,1},1} +) + out=zeros(Float64,length(B)) + for i in 1:length(B) + out[i]=dot(Ff,B[i]) + end + return out +end diff --git a/src/easyeq.jl b/src/easyeq.jl index 0bde3ab..2dbbd1a 100644 --- a/src/easyeq.jl +++ b/src/easyeq.jl @@ -1,4 +1,4 @@ -## Copyright 2021 Ian Jauslin +## Copyright 2021-2023 Ian Jauslin ## ## Licensed under the Apache License, Version 2.0 (the "License"); ## you may not use this file except in compliance with the License. @@ -19,47 +19,65 @@ end # compute energy -function easyeq_energy(minlrho_init,nlrho_init,order,rho,a0,v,maxiter,tolerance,approx) +function easyeq_energy( + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Easyeq_approx +) # compute gaussian quadrature weights weights=gausslegendre(order) - # compute initial guess from previous rho - (u,E,err)=easyeq_hatu(easyeq_init_u(a0,order,weights),order,(10.)^minlrho_init,v,maxiter,tolerance,weights,approx) - for j in 2:nlrho_init - rho_tmp=10^(minlrho_init+(log10(rho)-minlrho_init)*(j-1)/(nlrho_init-1)) - (u,E,err)=easyeq_hatu(u,order,rho_tmp,v,maxiter,tolerance,weights,approx) - end + # compute u + (u,E,err)= easyeq_compute_u_prevrho([rho],minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) # print energy @printf("% .15e % .15e\n",real(E),err) end # compute energy as a function of rho -function easyeq_energy_rho(rhos,order,a0,v,maxiter,tolerance,approx) +function easyeq_energy_rho( + rhos::Array{Float64,1}, + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Easyeq_approx +) # compute gaussian quadrature weights weights=gausslegendre(order) - # init u - u=easyeq_init_u(a0,order,weights) + # compute u + (us,es,errs)= easyeq_compute_u_prevrho(rhos,minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) for j in 1:length(rhos) - # compute u (init newton with previously computed u) - (u,E,err)=easyeq_hatu(u,order,rhos[j],v,maxiter,tolerance,weights,approx) - - @printf("% .15e % .15e % .15e\n",rhos[j],real(E),err) - + @printf("% .15e % .15e % .15e\n",rhos[j],real(es[j]),errs[j]) end end # compute u(k) -function easyeq_uk(minlrho_init,nlrho_init,order,rho,a0,v,maxiter,tolerance,approx) +function easyeq_uk( + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Easyeq_approx +) weights=gausslegendre(order) - # compute initial guess from previous rho - (u,E,err)=easyeq_hatu(easyeq_init_u(a0,order,weights),order,(10.)^minlrho_init,v,maxiter,tolerance,weights,approx) - for j in 2:nlrho_init - rho_tmp=10^(minlrho_init+(log10(rho)-minlrho_init)*(j-1)/(nlrho_init-1)) - (u,E,err)=easyeq_hatu(u,order,rho_tmp,v,maxiter,tolerance,weights,approx) - end + # compute u + (u,E,err)= easyeq_compute_u_prevrho_error([rho],minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) for i in 1:order k=(1-weights[1][i])/(1+weights[1][i]) @@ -68,15 +86,24 @@ function easyeq_uk(minlrho_init,nlrho_init,order,rho,a0,v,maxiter,tolerance,appr end # compute u(x) -function easyeq_ux(minlrho_init,nlrho_init,order,rho,a0,v,maxiter,tolerance,xmin,xmax,nx,approx) +function easyeq_ux( + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + xmin::Float64, + xmax::Float64, + nx::Int64, + approx::Easyeq_approx +) weights=gausslegendre(order) - # compute initial guess from previous rho - (u,E,err)=easyeq_hatu(easyeq_init_u(a0,order,weights),order,(10.)^minlrho_init,v,maxiter,tolerance,weights,approx) - for j in 2:nlrho_init - rho_tmp=10^(minlrho_init+(log10(rho)-minlrho_init)*(j-1)/(nlrho_init-1)) - (u,E,err)=easyeq_hatu(u,order,rho_tmp,v,maxiter,tolerance,weights,approx) - end + # compute u + (u,E,err)= easyeq_compute_u_prevrho_error([rho],minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) for i in 1:nx x=xmin+(xmax-xmin)*i/nx @@ -85,15 +112,24 @@ function easyeq_ux(minlrho_init,nlrho_init,order,rho,a0,v,maxiter,tolerance,xmin end # compute 2u(x)-rho u*u(x) -function easyeq_uux(minlrho_init,nlrho_init,order,rho,a0,v,maxiter,tolerance,xmin,xmax,nx,approx) +function easyeq_uux( + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + xmin::Float64, + xmax::Float64, + nx::Int64, + approx::Easyeq_approx +) weights=gausslegendre(order) - # compute initial guess from previous rho - (u,E,err)=easyeq_hatu(easyeq_init_u(a0,order,weights),order,(10.)^minlrho_init,v,maxiter,tolerance,weights,approx) - for j in 2:nlrho_init - rho_tmp=10^(minlrho_init+(log10(rho)-minlrho_init)*(j-1)/(nlrho_init-1)) - (u,E,err)=easyeq_hatu(u,order,rho_tmp,v,maxiter,tolerance,weights,approx) - end + # compute u + (u,E,err)= easyeq_compute_u_prevrho_error([rho],minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) for i in 1:nx x=xmin+(xmax-xmin)*i/nx @@ -102,16 +138,22 @@ function easyeq_uux(minlrho_init,nlrho_init,order,rho,a0,v,maxiter,tolerance,xmi end # condensate fraction -function easyeq_condensate_fraction(minlrho_init,nlrho_init,order,rho,a0,v,maxiter,tolerance,approx) +function easyeq_condensate_fraction( + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Easyeq_approx +) # compute gaussian quadrature weights weights=gausslegendre(order) - # compute initial guess from previous rho - (u,E,err)=easyeq_hatu(easyeq_init_u(a0,order,weights),order,(10.)^minlrho_init,v,maxiter,tolerance,weights,approx) - for j in 2:nlrho_init - rho_tmp=10^(minlrho_init+(log10(rho)-minlrho_init)*(j-1)/(nlrho_init-1)) - (u,E,err)=easyeq_hatu(u,order,rho_tmp,v,maxiter,tolerance,weights,approx) - end + # compute u + (u,E,err)= easyeq_compute_u_prevrho([rho],minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) # compute eta eta=easyeq_eta(u,order,rho,v,maxiter,tolerance,weights,approx) @@ -121,25 +163,350 @@ function easyeq_condensate_fraction(minlrho_init,nlrho_init,order,rho,a0,v,maxit end # condensate fraction as a function of rho -function easyeq_condensate_fraction_rho(rhos,order,a0,v,maxiter,tolerance,approx) +function easyeq_condensate_fraction_rho( + rhos::Array{Float64,1}, + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Easyeq_approx +) weights=gausslegendre(order) - # init u - u=easyeq_init_u(a0,order,weights) + # compute u + (us,es,errs)= easyeq_compute_u_prevrho(rhos,minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) for j in 1:length(rhos) - # compute u (init newton with previously computed u) - (u,E,err)=easyeq_hatu(u,order,rhos[j],v,maxiter,tolerance,weights,approx) - # compute eta - eta=easyeq_eta(u,order,rhos[j],v,maxiter,tolerance,weights,approx) + eta=easyeq_eta(us[j],order,rhos[j],v,maxiter,tolerance,weights,approx) + @printf("% .15e % .15e % .15e\n",rhos[j],eta,errs[j]) + end +end + +# 2 pt correlation function +function easyeq_2pt( + xmin::Float64, + xmax::Float64, + nx::Int64, + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + windowL::Float64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Easyeq_approx +) + # compute gaussian quadrature weights + weights=gausslegendre(order) + + # compute u + (u,E,err)= easyeq_compute_u_prevrho_error([rho],minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) + + # compute useful terms + (V,V0)=easyeq_init_v(weights,v) + (Eta,Eta0)=easyeq_init_H(weights,v) + (E,S,A,T,B,X)=easyeq_ESATBX(rho*u,V,V0,Eta,Eta0,weights,rho,approx) + + # compute C2 + for j in 1:nx + x=xmin+(xmax-xmin)/nx*j + C2=easyeq_C2(x,u,windowL,rho,weights,Eta,Eta0,approx,E,S,A,T,B,X) + @printf("% .15e % .15e\n",x,C2) + end +end + +# maximum of 2 point correlation function +function easyeq_2pt_max( + dx::Float64, + x0::Float64, # initial guess is x0/rho^(1/3) + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + windowL::Float64, + rho::Float64, + a0::Float64, + v::Function, + maxstep::Float64, + maxiter::Int64, + tolerance::Float64, + tolerance_max::Float64, + approx::Easyeq_approx +) + # compute gaussian quadrature weights + weights=gausslegendre(order) + + # compute u + (u,E,err)= easyeq_compute_u_prevrho_error([rho],minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) + + # compute useful terms + (V,V0)=easyeq_init_v(weights,v) + (Eta,Eta0)=easyeq_init_H(weights,v) + + (x,f)=easyeq_C2_max(u,x0/rho^(1/3),dx,maxstep,maxiter,tolerance_max,windowL,rho,weights,V,V0,Eta,Eta0,approx) + + if(x==Inf) + @printf(stderr,"max search failed for rho=%e\n",rho) + else + @printf("% .15e % .15e\n",x,f) + end +end + +# maximum of 2 point correlation function as a function of rho +function easyeq_2pt_max_rho( + rhos::Array{Float64,1}, + dx::Float64, + x0::Float64, # initial guess is x0/rho^(1/3) + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + windowL::Float64, + a0::Float64, + v::Function, + maxstep::Float64, + maxiter::Int64, + tolerance::Float64, + tolerance_max::Float64, + approx::Easyeq_approx +) + # compute gaussian quadrature weights + weights=gausslegendre(order) + + # compute u + (us,es,errs)= easyeq_compute_u_prevrho_error(rhos,minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) + + # compute useful terms + (V,V0)=easyeq_init_v(weights,v) + (Eta,Eta0)=easyeq_init_H(weights,v) + + # save result from each task + xs=Array{Float64,1}(undef,length(rhos)) + fs=Array{Float64,1}(undef,length(rhos)) + + # spawn workers + work=spawn_workers(length(rhos)) + + count=0 + # for each worker + @sync for p in 1:length(work) + # for each task + @async for j in work[p] + count=count+1 + if count>=length(work) + progress(count,length(rhos),10000) + end + # run the task + (xs[j],fs[j])=remotecall_fetch(easyeq_C2_max,workers()[p],us[j],x0/rhos[j]^(1/3),dx,maxstep,maxiter,tolerance_max,windowL,rhos[j],weights,V,V0,Eta,Eta0,approx) + end + end + + for j in 1:length(rhos) + if(xs[j]==Inf) + @printf(stderr,"max search failed for rho=%e\n",rhos[j]) + else + @printf("% .15e % .15e % .15e\n",rhos[j],xs[j],fs[j]) + end + end +end + + +# Fourier transform of 2 pt correlation function +function easyeq_2pt_fourier( + kmin::Float64, + kmax::Float64, + nk::Int64, + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + windowL::Float64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Easyeq_approx +) + # compute gaussian quadrature weights + weights=gausslegendre(order) + + # compute u + (u,E,err)= easyeq_compute_u_prevrho_error([rho],minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) - @printf("% .15e % .15e % .15e\n",rhos[j],eta,err) + # compute useful terms + (V,V0)=easyeq_init_v(weights,v) + (Eta,Eta0)=easyeq_init_H(weights,v) + (E,S,A,T,B,X)=easyeq_ESATBX(rho*u,V,V0,Eta,Eta0,weights,rho,approx) + + # compute C2 + for j in 1:nk + k=kmin+(kmax-kmin)/nk*j + C2=easyeq_C2_fourier(k,u,windowL,rho,weights,Eta,Eta0,approx,E,S,A,T,B,X) + @printf("% .15e % .15e\n",k,C2) end end +# maximum of Fourier transform of 2 point correlation function +function easyeq_2pt_fourier_max( + dk::Float64, + k0::Float64, # initial guess is k0*rho^(1/3) + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + windowL::Float64, + rho::Float64, + a0::Float64, + v::Function, + maxstep::Float64, + maxiter::Int64, + tolerance::Float64, + tolerance_max::Float64, + approx::Easyeq_approx +) + # compute gaussian quadrature weights + weights=gausslegendre(order) + + # compute u + (u,E,err)= easyeq_compute_u_prevrho_error([rho],minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) -# initialize u -@everywhere function easyeq_init_u(a0,order,weights) + # compute useful terms + (V,V0)=easyeq_init_v(weights,v) + (Eta,Eta0)=easyeq_init_H(weights,v) + + (k,f)=easyeq_C2_fourier_max(u,k0*rho^(1/3),dk,maxstep,maxiter,tolerance_max,windowL,rho,weights,V,V0,Eta,Eta0,approx) + + if(k==Inf) + @printf(stderr,"max search failed for rho=%e\n",rho) + else + @printf("% .15e % .15e\n",k,f) + end +end + +# maximum of Fourier transform of 2 point correlation function as a function of rho +function easyeq_2pt_fourier_max_rho( + rhos::Array{Float64,1}, + dk::Float64, + k0::Float64, # initial guess is k0*rho^(1/3) + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + windowL::Float64, + a0::Float64, + v::Function, + maxstep::Float64, + maxiter::Int64, + tolerance::Float64, + tolerance_max::Float64, + approx::Easyeq_approx +) + # compute gaussian quadrature weights + weights=gausslegendre(order) + + # compute u + (us,es,errs)= easyeq_compute_u_prevrho_error(rhos,minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) + + # compute useful terms + (V,V0)=easyeq_init_v(weights,v) + (Eta,Eta0)=easyeq_init_H(weights,v) + + # save result from each task + ks=Array{Float64,1}(undef,length(rhos)) + fs=Array{Float64,1}(undef,length(rhos)) + + # spawn workers + work=spawn_workers(length(rhos)) + + count=0 + # for each worker + @sync for p in 1:length(work) + # for each task + @async for j in work[p] + count=count+1 + if count>=length(work) + progress(count,length(rhos),10000) + end + # run the task + (ks[j],fs[j])=remotecall_fetch(easyeq_C2_fourier_max,workers()[p],us[j],k0*rhos[j]^(1/3),dk,maxstep,maxiter,tolerance_max,windowL,rhos[j],weights,V,V0,Eta,Eta0,approx) + end + end + + for j in 1:length(rhos) + if(ks[j]==Inf) + @printf(stderr,"max search failed for rho=%e\n",rhos[j]) + else + @printf("% .15e % .15e % .15e\n",rhos[j],ks[j],fs[j]) + end + end +end + +# momentum distribution +function easyeq_momentum_distribution( + kmin::Float64, + kmax::Float64, + minlrho_init::Float64, + nlrho_init::Int64, + order::Int64, + windowL::Float64, #L=windowL/k^2 + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + approx::Easyeq_approx +) + # compute gaussian quadrature weights + weights=gausslegendre(order) + + # compute u + (u,E,err)= easyeq_compute_u_prevrho_error([rho],minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) + + # compute useful terms + (V,V0)=easyeq_init_v(weights,v) + (Eta,Eta0)=easyeq_init_H(weights,v) + (E,S,A,T,B,X)=easyeq_ESATBX(rho*u,V,V0,Eta,Eta0,weights,rho,approx) + # dXi/dlambda without the delta function and u + dXidlambda=(dotPhi(B.*T./((X.+1).^2))./(2*(X.+1))+B.*T./(2*(X.+1).^3).*dotdPhi(B.*T./(X.+1).^2))./A*(16*pi^3) + + dXi=inv(easyeq_dXi(rho*u,Eta,Eta0,weights,rho,approx,E,S,A,T,B,X)) + + # compute momentum distribution + for j in 1:order + q=(1-weights[1][j])/(1+weights[1][j]) + # drop if not in k interval + if q<kmin || q>kmax + continue + end + + # delta(k_i,q) + delta=Array{Float64,1}(undef,order) + L=windowL/q^2 + for i in 1:order + k=(1-weights[1][i])/(1+weights[1][i]) + delta[i]=(1.)/(2*k*q*L)*(gaussian(k-q,(1.)/L)-gaussian(k+q,(1.)/L)) + end + + # du/dlambda + du=-dXi*(dXidlambda.*delta*u[j]) + # rescale u + du=du/rho + + # compute M + M=-integrate_legendre_sampled(y->(1-y)/y^3,Eta0.*du,0.,1.,weights)*rho/(16*pi^3) + + @printf("% .15e % .15e\n",q,M) + end +end + + +# initialize u from scattering solution +@everywhere function easyeq_init_u( + a0::Float64, + order::Int64, + weights::Tuple{Array{Float64,1},Array{Float64,1}} +) u=zeros(Float64,order) for j in 1:order # transformed k @@ -150,8 +517,100 @@ end return u end +# compute u for an array of rhos +# use scattering solution for the first one, and the previous rho for the others +@everywhere function easyeq_compute_u_rho( + rhos::Array{Float64,1}, + a0::Float64, + order::Int64, + v::Function, + maxiter::Int64, + tolerance::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + approx::Easyeq_approx +) + us=Array{Array{Float64,1}}(undef,length(rhos)) + es=Array{Float64,1}(undef,length(rhos)) + errs=Array{Float64,1}(undef,length(rhos)) + + (us[1],es[1],errs[1])=easyeq_hatu(easyeq_init_u(a0,order,weights),order,rhos[1],v,maxiter,tolerance,weights,approx) + for j in 2:length(rhos) + (us[j],es[j],errs[j])=easyeq_hatu(us[j-1],order,rhos[j],v,maxiter,tolerance,weights,approx) + end + + return (us,es,errs) +end + +# compute u for an array of rhos +# start from a smaller rho and work up to rhos[1] +@everywhere function easyeq_compute_u_prevrho( + rhos::Array{Float64,1}, + minlrho_init::Float64, + nlrho_init::Int64, + a0::Float64, + order::Int64, + v::Function, + maxiter::Int64, + tolerance::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + approx::Easyeq_approx +) + + # only work up to rhos[1] if nlrho_init>0 + if nlrho_init>0 + rhos_init=Array{Float64,1}(undef,nlrho_init) + for j in 0:nlrho_init-1 + rhos_init[j+1]=(nlrho_init==1 ? 10^minlrho_init : 10^(minlrho_init+(log10(rhos[1])-minlrho_init)/(nlrho_init-1)*j)) + end + append!(rhos_init,rhos) + # start from rhos[1] if nlrho_init=0 + else + rhos_init=rhos + end + (us,es,errs)=easyeq_compute_u_rho(rhos_init,a0,order,v,maxiter,tolerance,weights,approx) + + # return a single value if there was a single input + if length(rhos)==1 + return (us[nlrho_init+1],es[nlrho_init+1],errs[nlrho_init+1]) + else + return (us[nlrho_init+1:length(us)],es[nlrho_init+1:length(es)],errs[nlrho_init+1:length(errs)]) + end +end +# with error message if the computation failed to be accurate enough +@everywhere function easyeq_compute_u_prevrho_error( + rhos::Array{Float64,1}, + minlrho_init::Float64, + nlrho_init::Int64, + a0::Float64, + order::Int64, + v::Function, + maxiter::Int64, + tolerance::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + approx::Easyeq_approx +) + (us,es,errs)=easyeq_compute_u_prevrho(rhos,minlrho_init,nlrho_init,a0,order,v,maxiter,tolerance,weights,approx) + # check errs + for j in 1:length(errs) + if errs[j]>tolerance + print(stderr,"warning: computation of u failed for rho=",rhos[j],"\n") + end + end + return (us,es,errs) +end + + # \hat u(k) computed using Newton -@everywhere function easyeq_hatu(u0,order,rho,v,maxiter,tolerance,weights,approx) +@everywhere function easyeq_hatu( + u0::Array{Float64,1}, + order::Int64, + rho::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + approx::Easyeq_approx +) # initialize V and Eta (V,V0)=easyeq_init_v(weights,v) (Eta,Eta0)=easyeq_init_H(weights,v) @@ -162,7 +621,8 @@ end # iterate err=Inf for i in 1:maxiter-1 - new=u-inv(easyeq_dXi(u,V,V0,Eta,Eta0,weights,rho,approx))*easyeq_Xi(u,V,V0,Eta,Eta0,weights,rho,approx) + (E,S,A,T,B,X)=easyeq_ESATBX(u,V,V0,Eta,Eta0,weights,rho,approx) + new=u-inv(easyeq_dXi(u,Eta,Eta0,weights,rho,approx,E,S,A,T,B,X))*easyeq_Xi(u,order,S,A,T,B,X) err=norm(new-u)/norm(u) if(err<tolerance) @@ -176,15 +636,23 @@ end end # \Eta -@everywhere function easyeq_H(x,t,weights,v) - return (x>t ? 2*t/x : 2)* integrate_legendre(y->2*pi*((x+t)*y+abs(x-t)*(1-y))*v((x+t)*y+abs(x-t)*(1-y)),0,1,weights) +@everywhere function easyeq_H( + x::Float64, + t::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + v::Function +) + return (x>t ? 2*t/x : 2)* integrate_legendre(y->2*pi*((x+t)*y+abs(x-t)*(1-y))*v((x+t)*y+abs(x-t)*(1-y)),0.,1.,weights) end # initialize V -@everywhere function easyeq_init_v(weights,v) +@everywhere function easyeq_init_v( + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + v::Function +) order=length(weights[1]) - V=Array{Float64}(undef,order) - V0=v(0) + V=Array{Float64,1}(undef,order) + V0=v(0.) for i in 1:order k=(1-weights[1][i])/(1+weights[1][i]) V[i]=v(k) @@ -193,29 +661,58 @@ end end # initialize Eta -@everywhere function easyeq_init_H(weights,v) +@everywhere function easyeq_init_H( + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + v::Function +) order=length(weights[1]) - Eta=Array{Array{Float64}}(undef,order) - Eta0=Array{Float64}(undef,order) + Eta=Array{Array{Float64,1},1}(undef,order) + Eta0=Array{Float64,1}(undef,order) for i in 1:order k=(1-weights[1][i])/(1+weights[1][i]) - Eta[i]=Array{Float64}(undef,order) + Eta[i]=Array{Float64,1}(undef,order) for j in 1:order y=(weights[1][j]+1)/2 Eta[i][j]=easyeq_H(k,(1-y)/y,weights,v) end y=(weights[1][i]+1)/2 - Eta0[i]=easyeq_H(0,(1-y)/y,weights,v) + Eta0[i]=easyeq_H(0.,(1-y)/y,weights,v) end return(Eta,Eta0) end # Xi(u) -@everywhere function easyeq_Xi(u,V,V0,Eta,Eta0,weights,rho,approx) +@everywhere function easyeq_Xi( + u::Array{Float64,1}, + order::Int64, + S::Array{Float64,1}, + A::Array{Float64,1}, + T::Array{Float64,1}, + B::Array{Float64,1}, + X::Array{Float64,1} +) + return u.-T./(2*(X.+1)).*dotPhi(B.*T./(X.+1).^2) +end + +# compute E,S,A,T,B,X +@everywhere function easyeq_ESATBX( + u::Array{Float64,1}, + V::Array{Float64,1}, + V0::Float64, + Eta::Array{Array{Float64,1},1}, + Eta0::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + rho::Float64, + approx::Easyeq_approx +) order=length(weights[1]) # init - out=zeros(Float64,order) + S=zeros(Float64,order) + A=zeros(Float64,order) + T=zeros(Float64,order) + B=zeros(Float64,order) + X=zeros(Float64,order) # compute E before running the loop E=easyeq_en(u,V0,Eta0,rho,weights) @@ -224,188 +721,359 @@ end # k_i k=(1-weights[1][i])/(1+weights[1][i]) # S_i - S=V[i]-1/(rho*(2*pi)^3)*integrate_legendre_sampled(y->(1-y)/y^3,Eta[i].*u,0,1,weights) + S[i]=V[i]-1/(rho*(2*pi)^3)*integrate_legendre_sampled(y->(1-y)/y^3,Eta[i].*u,0.,1.,weights) # A_K,i - A=0. + A[i]=0. if approx.bK!=0. - A+=approx.bK*S + A[i]+=approx.bK*S[i] end if approx.bK!=1. - A+=(1-approx.bK)*E + A[i]+=(1-approx.bK)*E end - # T + # T_i if approx.bK==1. - T=1. + T[i]=1. else - T=S/A + T[i]=S[i]/A[i] end - # B + # B_i if approx.bK==approx.bL - B=1. + B[i]=1. else - B=(approx.bL*S+(1-approx.bL*E))/(approx.bK*S+(1-approx.bK*E)) + B[i]=(approx.bL*S[i]+(1-approx.bL*E))/(approx.bK*S[i]+(1-approx.bK*E)) end # X_i - X=k^2/(2*A*rho) - - # U_i - out[i]=u[i]-T/(2*(X+1))*Phi(B*T/(X+1)^2) + X[i]=k^2/(2*A[i]*rho) end - return out + return (E,S,A,T,B,X) end # derivative of Xi -@everywhere function easyeq_dXi(u,V,V0,Eta,Eta0,weights,rho,approx) +@everywhere function easyeq_dXi( + u::Array{Float64,1}, + Eta::Array{Array{Float64,1},1}, + Eta0::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + rho::Float64, + approx::Easyeq_approx, + E::Float64, + S::Array{Float64,1}, + A::Array{Float64,1}, + T::Array{Float64,1}, + B::Array{Float64,1}, + X::Array{Float64,1} +) order=length(weights[1]) # init out=zeros(Float64,order,order) - # compute E before the loop - E=easyeq_en(u,V0,Eta0,rho,weights) - for i in 1:order # k_i k=(1-weights[1][i])/(1+weights[1][i]) - # S_i - S=V[i]-1/(rho*(2*pi)^3)*integrate_legendre_sampled(y->(1-y)/y^3,Eta[i].*u,0,1,weights) - - # A_K,i - A=0. - if approx.bK!=0. - A+=approx.bK*S - end - if approx.bK!=1. - A+=(1-approx.bK)*E - end - - # T - if approx.bK==1. - T=1. - else - T=S/A - end - - # B - if approx.bK==approx.bL - B=1. - else - B=(approx.bL*S+(1-approx.bL*E))/(approx.bK*S+(1-approx.bK*E)) - end - - # X_i - X=k^2/(2*A*rho) for j in 1:order y=(weights[1][j]+1)/2 dS=-1/rho*(1-y)*Eta[i][j]/(2*(2*pi)^3*y^3)*weights[2][j] dE=-1/rho*(1-y)*Eta0[j]/(2*(2*pi)^3*y^3)*weights[2][j] + dU=(i==j ? 1. : 0.) - # dA - dA=0. - if approx.bK!=0. - dA+=approx.bK*dS - end - if approx.bK!=1. - dA+=(1-approx.bK)*dE - end + out[i,j]=easyeq_dXi_of_dSdEdU(k,dS,dE,dU,E,S[i],A[i],T[i],B[i],X[i],rho,approx) + end + end + + return out +end - # dT - if approx.bK==1. - dT=0. - else - dT=(1-approx.bK)*(E*dS-S*dE)/A^2 - end +# dXi given dS, dE and dU +@everywhere function easyeq_dXi_of_dSdEdU( + k::Float64, + dS::Float64, + dE::Float64, + dU::Float64, + E::Float64, + S::Float64, + A::Float64, + T::Float64, + B::Float64, + X::Float64, + rho::Float64, + approx::Easyeq_approx +) + # dA + dA=0. + if approx.bK!=0. + dA+=approx.bK*dS + end + if approx.bK!=1. + dA+=(1-approx.bK)*dE + end - # dB - if approx.bK==approx.bL - dB=0. - else - dB=(approx.bL*(1-approx.bK)-approx.bK*(1-approx.bL))*(E*dS-S*dE)/(approx.bK*S+(1-approx.bK*E))^2 - end + # dT,dB + # nothing to do if bK=bL=1 + if approx.bK!=1. || approx.bK!=approx.bL + dB=(E*dS-S*dE)/A^2 + end + if approx.bK==1. + dT=0. + else + dT=(1-approx.bK)*dB + end + if approx.bK==approx.bL + dB=0. + else + dB=(approx.bL*(1-approx.bK)-approx.bK*(1-approx.bL))*dB + end - dX=-k^2/(2*A^2*rho)*dA + dX=-k^2/(2*A^2*rho)*dA - out[i,j]=(i==j ? 1 : 0)-(dT-T*dX/(X+1))/(2*(X+1))*Phi(B*T/(X+1)^2)-T/(2*(X+1)^3)*(B*dT+T*dB-2*B*T*dX/(X+1))*dPhi(B*T/(X+1)^2) - end + return dU-(dT-T*dX/(X+1))/(2*(X+1))*Phi(B*T/(X+1)^2)-T/(2*(X+1)^3)*(B*dT+T*dB-2*B*T*dX/(X+1))*dPhi(B*T/(X+1)^2) +end + +# derivative of Xi with respect to mu +@everywhere function easyeq_dXidmu( + u::Array{Float64,1}, + order::Int64, + rho::Float64, + A::Array{Float64,1}, + T::Array{Float64,1}, + B::Array{Float64,1}, + X::Array{Float64,1} +) + # init + out=zeros(Float64,order) + for i in 1:order + out[i]=T[i]/(2*rho*A[i]*(X[i]+1)^2)*Phi(B[i]*T[i]/(X[i]+1)^2)+B[i]*T[i]^2/(rho*A[i]*(X[i]+1)^4)*dPhi(B[i]*T[i]/(X[i]+1)^2) end return out end -# derivative of Xi with respect to mu -@everywhere function easyeq_dXidmu(u,V,V0,Eta,Eta0,weights,rho,approx) +# energy +@everywhere function easyeq_en( + u::Array{Float64,1}, + V0::Float64, + Eta0::Array{Float64,1}, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}} +) + return V0-1/(rho*(2*pi)^3)*integrate_legendre_sampled(y->(1-y)/y^3,Eta0.*u,0.,1.,weights) +end + +# condensate fraction +@everywhere function easyeq_eta( + u::Array{Float64,1}, + order::Int64, + rho::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + approx::Easyeq_approx +) + (V,V0)=easyeq_init_v(weights,v) + (Eta,Eta0)=easyeq_init_H(weights,v) + + (E,S,A,T,B,X)=easyeq_ESATBX(rho*u,V,V0,Eta,Eta0,weights,rho,approx) + du=-inv(easyeq_dXi(rho*u,Eta,Eta0,weights,rho,approx,E,S,A,T,B,X))*easyeq_dXidmu(rho*u,order,rho,A,T,B,X) + + eta=-1/(2*(2*pi)^3)*integrate_legendre_sampled(y->(1-y)/y^3,Eta0.*du,0.,1.,weights) + + return eta +end + +# inverse Fourier transform +@everywhere function easyeq_u_x( + x::Float64, + u::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}} +) + order=length(weights[1]) + out=integrate_legendre_sampled(y->(1-y)/y^3*sin(x*(1-y)/y)/x/(2*pi^2),u,0.,1.,weights) + return out +end + + +# correlation function +@everywhere function easyeq_C2( + x::Float64, + u::Array{Float64,1}, + windowL::Float64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + Eta::Array{Array{Float64,1},1}, + Eta0::Array{Float64,1}, + approx::Easyeq_approx, + E::Float64, + S::Array{Float64,1}, + A::Array{Float64,1}, + T::Array{Float64,1}, + B::Array{Float64,1}, + X::Array{Float64,1} +) + g=(r,x)->(r>0. ? sin(r*x)/(r*x)*hann(r,windowL) : hann(r,windowL)) + (dEta,dEta0)=easyeq_init_H(weights,k->g(k,x)) + du=easyeq_dudv(g,x,u,rho,weights,Eta,Eta0,dEta,dEta0,approx,E,S,A,T,B,X) + + return rho^2*(1-integrate_legendre_sampled(y->(1-y)/y^3,dEta0.*u+Eta0.*du,0.,1.,weights)/(8*pi^3)) +end + +# derivative of u with respect to v in direction g +@everywhere function easyeq_dudv( + g::Function,# should be of the form g(k,x) where x is a parameter + x::Float64, + u::Array{Float64,1}, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + Eta::Array{Array{Float64,1},1}, + Eta0::Array{Float64,1}, + dEta::Array{Array{Float64,1},1}, + dEta0::Array{Float64,1}, + approx::Easyeq_approx, + E::Float64, + S::Array{Float64,1}, + A::Array{Float64,1}, + T::Array{Float64,1}, + B::Array{Float64,1}, + X::Array{Float64,1} +) + # initialize dV and dEta + (dV,dV0)=easyeq_init_v(weights,k->g(k,x)) + + du=-inv(easyeq_dXi(rho*u,Eta,Eta0,weights,rho,approx,E,S,A,T,B,X))*easyeq_dXidv(rho*u,dV,dV0,dEta,dEta0,weights,rho,approx,E,S,A,T,B,X) + # rescale rho + du=du/rho + + return du +end + +# derivative of Xi with respect to potential +@everywhere function easyeq_dXidv( + u::Array{Float64,1}, + dv::Array{Float64,1}, + dv0::Float64, + dEta::Array{Array{Float64,1},1}, + dEta0::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + rho::Float64, + approx::Easyeq_approx, + E::Float64, + S::Array{Float64,1}, + A::Array{Float64,1}, + T::Array{Float64,1}, + B::Array{Float64,1}, + X::Array{Float64,1} +) + order=length(weights[1]) # init out=zeros(Float64,order) - # compute E before running the loop - E=easyeq_en(u,V0,Eta0,rho,weights) - for i in 1:order # k_i k=(1-weights[1][i])/(1+weights[1][i]) - # S_i - S=V[i]-1/(rho*(2*pi)^3)*integrate_legendre_sampled(y->(1-y)/y^3,Eta[i].*u,0,1,weights) - # A_K,i - A=0. - if approx.bK!=0. - A+=approx.bK*S - end - if approx.bK!=1. - A+=(1-approx.bK)*E + dS=dv[i] + dE=dv0 + for j in 1:order + y=(weights[1][j]+1)/2 + dS+=-1/rho*(1-y)*u[j]*dEta[i][j]/(2*(2*pi)^3*y^3)*weights[2][j] + dE+=-1/rho*(1-y)*u[j]*dEta0[j]/(2*(2*pi)^3*y^3)*weights[2][j] end - # T - if approx.bK==1. - T=1. - else - T=S/A - end + out[i]=easyeq_dXi_of_dSdEdU(k,dS,dE,0.,E,S[i],A[i],T[i],B[i],X[i],rho,approx) + end - # B - if approx.bK==approx.bL - B=1. - else - B=(approx.bL*S+(1-approx.bL*E))/(approx.bK*S+(1-approx.bK*E)) - end + return out +end - # X_i - X=k^2/(2*A*rho) +# maximum of 2 point correlation function +@everywhere function easyeq_C2_max( + u::Array{Float64,1}, + x0::Float64, + dx::Float64, + maxstep::Float64, + maxiter::Int64, + tolerance::Float64, + windowL::Float64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + V::Array{Float64,1}, + V0::Float64, + Eta::Array{Array{Float64,1},1}, + Eta0::Array{Float64,1}, + approx::Easyeq_approx +) + # compute some useful terms + (E,S,A,T,B,X)=easyeq_ESATBX(rho*u,V,V0,Eta,Eta0,weights,rho,approx) - out[i]=T/(2*rho*A*(X+1)^2)*Phi(B*T/(X+1)^2)+B*T^2/(rho*A*(X+1)^4)*dPhi(B*T/(X+1)^2) - end + (x,f)=newton_maximum(y->easyeq_C2(y,u,windowL,rho,weights,Eta,Eta0,approx,E,S,A,T,B,X),x0,dx,maxiter,tolerance,maxstep) - return out + return(x,f) end -# energy -@everywhere function easyeq_en(u,V0,Eta0,rho,weights) - return V0-1/(rho*(2*pi)^3)*integrate_legendre_sampled(y->(1-y)/y^3,Eta0.*u,0,1,weights) +# Fourier transform of correlation function +@everywhere function easyeq_C2_fourier( + q::Float64, + u::Array{Float64,1}, + windowL::Float64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + Eta::Array{Array{Float64,1},1}, + Eta0::Array{Float64,1}, + approx::Easyeq_approx, + E::Float64, + S::Array{Float64,1}, + A::Array{Float64,1}, + T::Array{Float64,1}, + B::Array{Float64,1}, + X::Array{Float64,1} +) + # direction in which to differentiate u + g=(r,x)->(r>0. ? (1.)/(2*x*r*windowL)*(gaussian(x-r,(1.)/windowL)-gaussian(x+r,(1.)/windowL)) : gaussian(x,(1.)/windowL)) + (dEta,dEta0)=easyeq_init_H(weights,k->g(k,q)) + du=easyeq_dudv(g,q,u,rho,weights,Eta,Eta0,dEta,dEta0,approx,E,S,A,T,B,X) + + return rho^2*(-integrate_legendre_sampled(y->(1-y)/y^3,dEta0.*u+Eta0.*du,0.,1.,weights)/(8*pi^3)) end -# condensate fraction -@everywhere function easyeq_eta(u,order,rho,v,maxiter,tolerance,weights,approx) - (V,V0)=easyeq_init_v(weights,v) - (Eta,Eta0)=easyeq_init_H(weights,v) - du=-inv(easyeq_dXi(rho*u,V,V0,Eta,Eta0,weights,rho,approx))*easyeq_dXidmu(rho*u,V,V0,Eta,Eta0,weights,rho,approx) +# maximum of Fourier transform of 2 point correlation function +@everywhere function easyeq_C2_fourier_max( + u::Array{Float64,1}, + k0::Float64, + dk::Float64, + maxstep::Float64, + maxiter::Int64, + tolerance::Float64, + windowL::Float64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + V::Array{Float64,1}, + V0::Float64, + Eta::Array{Array{Float64,1},1}, + Eta0::Array{Float64,1}, + approx::Easyeq_approx +) + # compute some useful terms + (E,S,A,T,B,X)=easyeq_ESATBX(rho*u,V,V0,Eta,Eta0,weights,rho,approx) - eta=-1/(2*(2*pi)^3)*integrate_legendre_sampled(y->(1-y)/y^3,Eta0.*du,0,1,weights) + (k,f)=newton_maximum(y->easyeq_C2_fourier(y,u,windowL,rho,weights,Eta,Eta0,approx,E,S,A,T,B,X),k0,dk,maxiter,tolerance,maxstep) - return eta + return (k,f) end -# inverse Fourier transform -@everywhere function easyeq_u_x(x,u,weights) - order=length(weights[1]) - out=integrate_legendre_sampled(y->(1-y)/y^3*sin(x*(1-y)/y)/x/(2*pi^2),u,0,1,weights) - return out + +@everywhere function barEta( + q::Float64, + y::Float64, + t::Float64 +) + return (t>=abs(y-q) && t<=y+q ? pi/y/q : 0.) end diff --git a/src/integration.jl b/src/integration.jl index 9be4641..223d6cc 100644 --- a/src/integration.jl +++ b/src/integration.jl @@ -1,4 +1,4 @@ -## Copyright 2021 Ian Jauslin +## Copyright 2021-2023 Ian Jauslin ## ## Licensed under the Apache License, Version 2.0 (the "License"); ## you may not use this file except in compliance with the License. @@ -13,7 +13,12 @@ ## limitations under the License. # approximate \int_a^b f using Gauss-Legendre quadratures -@everywhere function integrate_legendre(f,a,b,weights) +@everywhere function integrate_legendre( + f::Function, + a::Float64, + b::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}} +) out=0 for i in 1:length(weights[1]) out+=(b-a)/2*weights[2][i]*f((b-a)/2*weights[1][i]+(b+a)/2) @@ -21,7 +26,13 @@ return out end # \int f*g where g is sampled at the Legendre nodes -@everywhere function integrate_legendre_sampled(f,g,a,b,weights) +@everywhere function integrate_legendre_sampled( + f::Function, + g::Array{Float64,1}, + a::Float64, + b::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}} +) out=0 for i in 1:length(weights[1]) out+=(b-a)/2*weights[2][i]*f((b-a)/2*weights[1][i]+(b+a)/2)*g[i] @@ -31,7 +42,12 @@ end # approximate \int_a^b f/sqrt((b-x)(x-a)) using Gauss-Chebyshev quadratures -@everywhere function integrate_chebyshev(f,a,b,N) +@everywhere function integrate_chebyshev( + f::Function, + a::Float64, + b::Float64, + N::Int64 +) out=0 for i in 1:N out=out+pi/N*f((b-a)/2*cos((2*i-1)/(2*N)*pi)+(b+a)/2) @@ -40,7 +56,11 @@ end end # approximate \int_0^\infty dr f(r)*exp(-a*r) using Gauss-Chebyshev quadratures -@everywhere function integrate_laguerre(f,a,weights_gL) +@everywhere function integrate_laguerre( + f::Function, + a::Float64, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}} +) out=0. for i in 1:length(weights_gL[1]) out+=1/a*f(weights_gL[1][i]/a)*weights_gL[2][i] @@ -49,10 +69,28 @@ end end # Hann window -@everywhere function hann(x,L) +@everywhere function hann( + x::Float64, + L::Float64 +) if abs(x)<L/2 return cos(pi*x/L)^2 else return 0. end end +# Fourier transform (in 3d) +@everywhere function hann_fourier( + k::Float64, + L::Float64 +) + return L^2*4*pi^3/k*(((k*L)^3-4*k*L*pi^2)*cos(k*L/2)-2*(3*(k*L)^2-4*pi^2)*sin(k*L/2))/((k*L)^3-4*k*L*pi^2)^2 +end + +# normalized Gaussian (in 3d) +@everywhere function gaussian( + k::Float64, + L::Float64 +) + return exp(-k^2/(2*L))/sqrt(8*pi^3*L^3) +end diff --git a/src/interpolation.jl b/src/interpolation.jl index fa3bcdb..066bc20 100644 --- a/src/interpolation.jl +++ b/src/interpolation.jl @@ -1,4 +1,4 @@ -## Copyright 2021 Ian Jauslin +## Copyright 2021-2023 Ian Jauslin ## ## Licensed under the Apache License, Version 2.0 (the "License"); ## you may not use this file except in compliance with the License. @@ -14,7 +14,11 @@ # linear interpolation: given vectors x,y, compute a linear interpolation for y(x0) # assume x is ordered -@everywhere function linear_interpolation(x0,x,y) +@everywhere function linear_interpolation( + x0::Float64, + x::Array{Float64,1}, + y::Array{Float64,1} +) # if x0 is beyond all x's, then return the corresponding boundary value. if x0>x[length(x)] return y[length(y)] @@ -28,7 +32,12 @@ # interpolate return y[i]+(y[i+1]-y[i])*(x0-x[i])/(x[i+1]-x[i]) end -@everywhere function bracket(x0,x,a,b) +@everywhere function bracket( + x0::Float64, + x::Array{Float64,1}, + a::Int64, + b::Int64 +) i=floor(Int64,(a+b)/2) if x0<x[i] return bracket(x0,x,a,i) @@ -41,15 +50,18 @@ end # polynomial interpolation of a family of points -@everywhere function poly_interpolation(x,y) +@everywhere function poly_interpolation( + x::Array{Float64,1}, + y::Array{Float64,1} +) # init for recursion - rec=Array{Polynomial{Float64}}(undef,length(x)) + rec=Array{Polynomial{Float64},1}(undef,length(x)) for i in 1:length(x) rec[i]=Polynomial([1.]) end # compute \prod (x-x_i) - poly_interpolation_rec(rec,x,1,length(x)) + poly_interpolation_rec(rec,x,1.,length(x)) # sum terms together out=0. @@ -60,7 +72,12 @@ end return out end # recursive helper function -@everywhere function poly_interpolation_rec(out,x,a,b) +@everywhere function poly_interpolation_rec( + out::Array{Float64,1}, + x::Array{Float64,1}, + a::Float64, + b::Float64 +) if a==b return end @@ -91,7 +108,10 @@ end return end ## the following does the same, but has complexity N^2, the function above has N*log(N) -#@everywhere function poly_interpolation(x,y) +#@everywhere function poly_interpolation( +# x::Array{Float64,1}, +# y::Array{Float64,1} +#) # out=Polynomial([0.]) # for i in 1:length(x) # prod=Polynomial([1.]) diff --git a/src/main.jl b/src/main.jl index 382fe6b..28fc2be 100644 --- a/src/main.jl +++ b/src/main.jl @@ -1,4 +1,4 @@ -## Copyright 2021 Ian Jauslin +## Copyright 2021-2023 Ian Jauslin ## ## Licensed under the Apache License, Version 2.0 (the "License"); ## you may not use this file except in compliance with the License. @@ -22,6 +22,8 @@ include("chebyshev.jl") include("integration.jl") include("interpolation.jl") include("tools.jl") +include("multithread.jl") +include("optimization.jl") include("potentials.jl") include("print.jl") include("easyeq.jl") @@ -36,9 +38,6 @@ function main() rho=1e-6 e=1e-4 - # incrementally initialize Newton algorithm - nlrho_init=1 - # potential v=k->v_exp(k,1.) a0=a0_exp(1.) @@ -49,19 +48,29 @@ function main() v_param_e=1. # plot range when plotting in rho - minlrho=-6 - maxlrho=2 + # linear + minrho=1e-6 + maxrho=1e2 + nrho=0 + # logarithmic + minlrho=-6. + maxlrho=2. nlrho=100 - rhos=Array{Float64}(undef,0) + # list + rhos=Array{Float64,1}(undef,0) # plot range when plotting in e - minle=-6 - maxle=2 + minle=-6. + maxle=2. nle=100 - es=Array{Float64}(undef,0) + es=Array{Float64,1}(undef,0) # plot range when plotting in x - xmin=0 - xmax=100 + xmin=0. + xmax=100. nx=100 + # plot range when plotting in k + kmin=0. + kmax=10. + nk=100 # cutoffs tolerance=1e-11 @@ -77,8 +86,8 @@ function main() J=10 # starting rho from which to incrementally initialize Newton algorithm - # default must be set after reading rho, if not set explicitly - minlrho_init=nothing + minlrho_init=-6. + nlrho_init=0 # Hann window for Fourier transforms windowL=1e3 @@ -98,6 +107,18 @@ function main() anyeq_compleq_approx=Anyeq_approx(1.,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.) anyeq_approx=anyeq_bigeq_approx + # numerical approximations of derivatives + dx=1e-7 + dk=1e-7 + + # initial guess for 2pt_max + x0=1. + k0=1. + # maximum step in 2pt_max + maxstep=Inf + # tolerance for max search + tolerance_max=Inf + # read cli arguments (params,potential,method,savefile,command)=read_args(ARGS) @@ -109,8 +130,8 @@ function main() print(stderr,"error: could not read parameter '",param,"'.\n") exit(-1) end - lhs=terms[1] - rhs=terms[2] + lhs=string(terms[1]) + rhs=string(terms[2]) if lhs=="rho" rho=parse(Float64,rhs) elseif lhs=="minlrho_init" @@ -133,6 +154,12 @@ function main() maxlrho=parse(Float64,rhs) elseif lhs=="nlrho" nlrho=parse(Int64,rhs) + elseif lhs=="minrho" + minrho=parse(Float64,rhs) + elseif lhs=="maxrho" + maxrho=parse(Float64,rhs) + elseif lhs=="nrho" + nrho=parse(Int64,rhs) elseif lhs=="es" es=parse_list(rhs) elseif lhs=="minle" @@ -147,6 +174,12 @@ function main() xmax=parse(Float64,rhs) elseif lhs=="nx" nx=parse(Int64,rhs) + elseif lhs=="kmin" + kmin=parse(Float64,rhs) + elseif lhs=="kmax" + kmax=parse(Float64,rhs) + elseif lhs=="nk" + nk=parse(Int64,rhs) elseif lhs=="P" P=parse(Int64,rhs) elseif lhs=="N" @@ -155,6 +188,18 @@ function main() J=parse(Int64,rhs) elseif lhs=="window_L" windowL=parse(Float64,rhs) + elseif lhs=="dx" + dx=parse(Float64,rhs) + elseif lhs=="x0" + x0=parse(Float64,rhs) + elseif lhs=="dk" + dk=parse(Float64,rhs) + elseif lhs=="k0" + k0=parse(Float64,rhs) + elseif lhs=="maxstep" + maxstep=parse(Float64,rhs) + elseif lhs=="tolerance_max" + tolerance_max=parse(Float64,rhs) elseif lhs=="aK" anyeq_approx.aK=parse(Float64,rhs) elseif lhs=="bK" @@ -242,37 +287,48 @@ function main() ## set parameters # rhos if length(rhos)==0 - rhos=Array{Float64}(undef,nlrho) - for j in 0:nlrho-1 - rhos[j+1]=(nlrho==1 ? 10^minlrho : 10^(minlrho+(maxlrho-minlrho)/(nlrho-1)*j)) + # linear only if nrho is specified + if nrho>0 + rhos=Array{Float64,1}(undef,nrho) + for j in 0:nrho-1 + rhos[j+1]=(nrho==1 ? minrho : minrho+(maxrho-minrho)/(nrho-1)*j) + end + else + rhos=Array{Float64,1}(undef,nlrho) + for j in 0:nlrho-1 + rhos[j+1]=(nlrho==1 ? 10^minlrho : 10^(minlrho+(maxlrho-minlrho)/(nlrho-1)*j)) + end end end + # es if length(es)==0 - es=Array{Float64}(undef,nle) + es=Array{Float64,1}(undef,nle) for j in 0:nle-1 es[j+1]=(nle==1 ? 10^minle : 10^(minle+(maxle-minle)/(nle-1)*j)) end end - # default minlrho_init - if (minlrho_init==nothing) - minlrho_init=log10(rho) - end - # splines - taus=Array{Float64}(undef,J+1) + taus=Array{Float64,1}(undef,J+1) for j in 0:J taus[j+1]=-1+2*j/J end + # tolerance_max + if tolerance_max==Inf + tolerance_max=tolerance + end + ## run command - if method=="easyeq" + if command=="scattering_length" + @printf("% .15e\n",a0) + elseif method=="easyeq" if command=="energy" easyeq_energy(minlrho_init,nlrho_init,order,rho,a0,v,maxiter,tolerance,easyeq_approx) # e(rho) elseif command=="energy_rho" - easyeq_energy_rho(rhos,order,a0,v,maxiter,tolerance,easyeq_approx) + easyeq_energy_rho(rhos,minlrho_init,nlrho_init,order,a0,v,maxiter,tolerance,easyeq_approx) # u(k) elseif command=="uk" easyeq_uk(minlrho_init,nlrho_init,order,rho,a0,v,maxiter,tolerance,easyeq_approx) @@ -285,7 +341,28 @@ function main() elseif command=="condensate_fraction" easyeq_condensate_fraction(minlrho_init,nlrho_init,order,rho,a0,v,maxiter,tolerance,easyeq_approx) elseif command=="condensate_fraction_rho" - easyeq_condensate_fraction_rho(rhos,order,a0,v,maxiter,tolerance,easyeq_approx) + easyeq_condensate_fraction_rho(rhos,minlrho_init,nlrho_init,order,a0,v,maxiter,tolerance,easyeq_approx) + # 2pt correlation + elseif command=="2pt" + easyeq_2pt(xmin,xmax,nx,minlrho_init,nlrho_init,order,windowL,rho,a0,v,maxiter,tolerance,easyeq_approx) + # max of 2pt correlation + elseif command=="2pt_max" + easyeq_2pt_max(dx,x0,minlrho_init,nlrho_init,order,windowL,rho,a0,v,maxstep,maxiter,tolerance,tolerance_max,easyeq_approx) + # max of 2pt correlation as a function of rho + elseif command=="2pt_max_rho" + easyeq_2pt_max_rho(rhos,dx,x0,minlrho_init,nlrho_init,order,windowL,a0,v,maxstep,maxiter,tolerance,tolerance_max,easyeq_approx) + # fourier transform of 2pt correlation + elseif command=="2pt_fourier" + easyeq_2pt_fourier(kmin,kmax,nk,minlrho_init,nlrho_init,order,windowL,rho,a0,v,maxiter,tolerance,easyeq_approx) + # max of fourier transform of 2pt correlation + elseif command=="2pt_fourier_max" + easyeq_2pt_fourier_max(dk,k0,minlrho_init,nlrho_init,order,windowL,rho,a0,v,maxstep,maxiter,tolerance,tolerance_max,easyeq_approx) + # max of 2pt correlation as a function of rho + elseif command=="2pt_fourier_max_rho" + easyeq_2pt_fourier_max_rho(rhos,dk,k0,minlrho_init,nlrho_init,order,windowL,a0,v,maxstep,maxiter,tolerance,tolerance_max,easyeq_approx) + # momentum distribution + elseif command=="momentum_distribution" + easyeq_momentum_distribution(kmin,kmax,minlrho_init,nlrho_init,order,windowL,rho,a0,v,maxiter,tolerance,easyeq_approx) else print(stderr,"unrecognized command '",command,"'.\n") exit(-1) @@ -316,7 +393,7 @@ function main() anyeq_energy(rho,minlrho_init,nlrho_init,taus,P,N,J,a0,v,maxiter,tolerance,anyeq_approx,savefile) # e(rho) elseif command=="energy_rho" - anyeq_energy_rho(rhos,taus,P,N,J,a0,v,maxiter,tolerance,anyeq_approx,savefile) + anyeq_energy_rho(rhos,minlrho_init,nlrho_init,taus,P,N,J,a0,v,maxiter,tolerance,anyeq_approx,savefile) elseif command=="energy_rho_init_prevrho" anyeq_energy_rho_init_prevrho(rhos,taus,P,N,J,a0,v,maxiter,tolerance,anyeq_approx,savefile) elseif command=="energy_rho_init_nextrho" @@ -331,12 +408,29 @@ function main() elseif command=="condensate_fraction" anyeq_condensate_fraction(rho,minlrho_init,nlrho_init,taus,P,N,J,a0,v,maxiter,tolerance,anyeq_approx,savefile) elseif command=="condensate_fraction_rho" - anyeq_condensate_fraction_rho(rhos,taus,P,N,J,a0,v,maxiter,tolerance,anyeq_approx,savefile) + anyeq_condensate_fraction_rho(rhos,minlrho_init,nlrho_init,taus,P,N,J,a0,v,maxiter,tolerance,anyeq_approx,savefile) # momentum distribution elseif command=="momentum_distribution" - anyeq_momentum_distribution(rho,minlrho_init,nlrho_init,taus,P,N,J,a0,v,maxiter,tolerance,anyeq_approx,savefile) + anyeq_momentum_distribution(kmin,kmax,rho,minlrho_init,nlrho_init,taus,P,N,J,windowL,a0,v,maxiter,tolerance,anyeq_approx,savefile) elseif command=="2pt" anyeq_2pt_correlation(minlrho_init,nlrho_init,taus,P,N,J,windowL,rho,a0,v,maxiter,tolerance,xmin,xmax,nx,anyeq_approx,savefile) + elseif command=="2pt_max" + anyeq_2pt_correlation_max(rho,minlrho_init,nlrho_init,dx,x0,maxstep,taus,P,N,J,windowL,a0,v,maxiter,tolerance,tolerance_max,anyeq_approx,savefile) + elseif command=="2pt_max_rho" + anyeq_2pt_correlation_max_rho(rhos,minlrho_init,nlrho_init,dx,x0,maxstep,taus,P,N,J,windowL,a0,v,maxiter,tolerance,tolerance_max,anyeq_approx,savefile) + elseif command=="2pt_fourier" + anyeq_2pt_correlation_fourier(minlrho_init,nlrho_init,taus,P,N,J,windowL,rho,a0,v,maxiter,tolerance,kmin,kmax,nk,anyeq_approx,savefile) + elseif command=="2pt_fourier_test" + anyeq_2pt_correlation_fourier_test(minlrho_init,nlrho_init,taus,P,N,J,windowL,rho,a0,v,maxiter,tolerance,xmax,kmin,kmax,nk,anyeq_approx,savefile) + elseif command=="2pt_fourier_max" + anyeq_2pt_correlation_fourier_max(rho,minlrho_init,nlrho_init,dk,k0,maxstep,taus,P,N,J,windowL,a0,v,maxiter,tolerance,tolerance_max,anyeq_approx,savefile) + elseif command=="2pt_fourier_max_rho" + anyeq_2pt_correlation_fourier_max_rho(rhos,minlrho_init,nlrho_init,dk,k0,maxstep,taus,P,N,J,windowL,a0,v,maxiter,tolerance,tolerance_max,anyeq_approx,savefile) + elseif command=="uncondensed_2pt" + anyeq_uncondensed_2pt_correlation(minlrho_init,nlrho_init,taus,P,N,J,windowL,rho,a0,v,maxiter,tolerance,xmin,xmax,nx,anyeq_approx,savefile) + # compressibility + elseif command=="compressibility_rho" + anyeq_compressibility_rho(rhos,minlrho_init,nlrho_init,taus,P,N,J,a0,v,maxiter,tolerance,anyeq_approx,savefile) else print(stderr,"unrecognized command: '",command,"'\n") exit(-1) @@ -360,9 +454,11 @@ function main() end # parse a comma separated list as an array of Float64 -function parse_list(str) +function parse_list( + str::String +) elems=split(str,",") - out=Array{Float64}(undef,length(elems)) + out=Array{Float64,1}(undef,length(elems)) for i in 1:length(elems) out[i]=parse(Float64,elems[i]) end @@ -370,7 +466,9 @@ function parse_list(str) end # read cli arguments -function read_args(ARGS) +function read_args( + ARGS +) # flag flag="" diff --git a/src/multithread.jl b/src/multithread.jl new file mode 100644 index 0000000..f61cdd7 --- /dev/null +++ b/src/multithread.jl @@ -0,0 +1,16 @@ +# split up 1...n among workers +function spawn_workers(n::Int64) + # number of workers + nw=nworkers() + # split jobs among workers + work=Array{Array{Int64,1},1}(undef,nw) + # init empty arrays + for p in 1:nw + work[p]=Int64[] + end + for i in 1:n + append!(work[(i-1)%nw+1],[i]) + end + + return work +end diff --git a/src/optimization.jl b/src/optimization.jl new file mode 100644 index 0000000..bacead7 --- /dev/null +++ b/src/optimization.jl @@ -0,0 +1,94 @@ +# gradient descent: find local minimum of function of one variable from initial guess +# numerically estimate the derivative +@everywhere function gradient_descent( + f::Function, + x0::Float64, + delta::Float64, # shift is delta*df + dx::Float64, # finite difference for numerical derivative evaluation + maxiter::Int64 # interrupt and fail after maxiter steps +) + counter=0 + + # init + x=x0 + + while counter<maxiter + # value at x and around + val=f(x) + valm=f(x-dx) + valp=f(x+dx) + # quit if minimum + if(val<valm && val<valp) + return(x,val) + end + + # derivative + df=(valp-val)/dx + # step + x=x-delta*df + counter+=1 + end + + # fail + return(Inf,Inf) +end + +# Newton algorithm to compute extrema +# numerically estimate the derivatives +@everywhere function newton_maximum( + f::Function, + x0::Float64, + dx::Float64, # finite difference for numerical derivative evaluation + maxiter::Int64, + tolerance::Float64, + maxstep::Float64 # maximal size of step +) + counter=0 + + # init + x=x0 + + while counter<maxiter + # value at x and around + val=f(x) + valm=f(x-dx) + valp=f(x+dx) + + # derivative + dfp=(valp-val)/dx + dfm=(val-valm)/dx + # second derivative + ddf=(dfp-dfm)/dx + + #@printf(stderr,"% .15e % .15e % .15e % .15e\n",x,val,dfp,ddf) + + if abs(dfp/ddf)<tolerance + # check it is a local maximum + if ddf<0 + return(x,val) + else + return(Inf,Inf) + end + end + + # step + step=dfp/abs(ddf) + + # step too large + if abs(step)>maxstep + step=maxstep*sign(step) + end + + x=x+step + + # fail if off to infinity + if x==Inf || x==-Inf + return(x,val) + end + counter+=1 + end + + # fail + return(Inf,Inf) +end + diff --git a/src/potentials.jl b/src/potentials.jl index 46cafc0..b480db0 100644 --- a/src/potentials.jl +++ b/src/potentials.jl @@ -1,4 +1,4 @@ -## Copyright 2021 Ian Jauslin +## Copyright 2021-2023 Ian Jauslin ## ## Licensed under the Apache License, Version 2.0 (the "License"); ## you may not use this file except in compliance with the License. @@ -13,10 +13,15 @@ ## limitations under the License. # exponential potential in 3 dimensions -@everywhere function v_exp(k,a) +@everywhere function v_exp( + k::Float64, + a::Float64 +) return 8*pi/(1+k^2)^2*a end -@everywhere function a0_exp(a) +@everywhere function a0_exp( + a::Float64 +) if a>0. return log(a)+2*MathConstants.eulergamma+2*besselk(0,2*sqrt(a))/besseli(0,2*sqrt(a)) elseif a<0. @@ -27,15 +32,24 @@ end end # exp(-x)-a*exp(-b*x) in 3 dimensions -@everywhere function v_expcry(k,a,b) +@everywhere function v_expcry( + k::Float64, + a::Float64, + b::Float64 +) return 8*pi*((1+k^2)^(-2)-a*b*(b^2+k^2)^(-2)) end -@everywhere function a0_expcry(a,b) +@everywhere function a0_expcry( + a::Float64, + b::Float64 +) return 1.21751642717932720441274114683413710125487579284827462 #ish end # x^2*exp(-|x|) in 3 dimensions -@everywhere function v_npt(k) +@everywhere function v_npt( + k::Float64 +) return 96*pi*(1-k^2)/(1+k^2)^4 end @everywhere function a0_npt() @@ -43,7 +57,9 @@ end end # 1/(1+x^4/4) potential in 3 dimensions -@everywhere function v_alg(k) +@everywhere function v_alg( + k::Float64 +) if(k==0) return 4*pi^2 else @@ -53,32 +69,50 @@ end a0_alg=1. #ish # (1+a x^4)/(1+x^2)^4 potential in 3 dimensions -@everywhere function v_algwell(k) +@everywhere function v_algwell( + k::Float64 +) a=4 return pi^2/24*exp(-k)*(a*(k^2-9*k+15)+k^2+3*k+3) end a0_algwell=1. #ish # potential corresponding to the exact solution c/(1+b^2x^2)^2 -@everywhere function v_exact(k,b,c,e) +@everywhere function v_exact( + k::Float64, + b::Float64, + c::Float64, + e::Float64 +) if k!=0 return 48*pi^2*((18+3*sqrt(c)-(4-3*e/b^2)*c-(1-2*e/b^2)*c^1.5)/(4*(3+sqrt(c))^2*sqrt(c))*exp(-sqrt(1-sqrt(c))*k/b)+(-18+3*sqrt(c)+(4-3*e/b^2)*c-(1-2*e/b^2)*c^1.5)/(4*(3-sqrt(c))^2*sqrt(c))*exp(-sqrt(1+sqrt(c))*k/b)+(1-k/b)/2*exp(-k/b)-c*e/b^2*(3*(9-c)*k/b+8*c)/(8*(9-c)^2)*exp(-2*k/b))/k else return 48*pi^2*(-sqrt(1-sqrt(c))/b*(18+3*sqrt(c)-(4-3*e/b^2)*c-(1-2*e/b^2)*c^1.5)/(4*(3+sqrt(c))^2*sqrt(c))-sqrt(1+sqrt(c))/b*(-18+3*sqrt(c)+(4-3*e/b^2)*c-(1-2*e/b^2)*c^1.5)/(4*(3-sqrt(c))^2*sqrt(c))-1/b-c*e/b^2*(27-19*c)/(8*(9-c)^2)) end end -@everywhere function a0_exact(b,c,e) +@everywhere function a0_exact( + b::Float64, + c::Float64, + e::Float64 +) return 1. #ish end # tent potential (convolution of soft sphere with itself): a*pi/12*(2*|x|/b-2)^2*(2*|x|/b+4) for |x|<b -@everywhere function v_tent(k,a,b) +@everywhere function v_tent( + k::Float64, + a::Float64, + b::Float64 +) if k!=0 return (b/2)^3*a*(4*pi*(sin(k*b/2)-k*b/2*cos(k*b/2))/(k*b/2)^3)^2 else return (b/2)^3*a*(4*pi/3)^2 end end -@everywhere function a0_tent(a,b) +@everywhere function a0_tent( + a::Float64, + b::Float64 +) return b #ish end diff --git a/src/print.jl b/src/print.jl index bef1c4d..3587728 100644 --- a/src/print.jl +++ b/src/print.jl @@ -1,4 +1,4 @@ -## Copyright 2021 Ian Jauslin +## Copyright 2021-2023 Ian Jauslin ## ## Licensed under the Apache License, Version 2.0 (the "License"); ## you may not use this file except in compliance with the License. diff --git a/src/simpleq-Kv.jl b/src/simpleq-Kv.jl index 8789656..5a6579c 100644 --- a/src/simpleq-Kv.jl +++ b/src/simpleq-Kv.jl @@ -1,4 +1,4 @@ -## Copyright 2021 Ian Jauslin +## Copyright 2021-2023 Ian Jauslin ## ## Licensed under the Apache License, Version 2.0 (the "License"); ## you may not use this file except in compliance with the License. @@ -13,12 +13,27 @@ ## limitations under the License. # Compute Kv=(-\Delta+v+4e(1-\rho u*))^{-1}v -function anyeq_Kv(minlrho,nlrho,taus,P,N,J,rho,a0,v,maxiter,tolerance,xmin,xmax,nx) +function anyeq_Kv( + minlrho::Float64, + nlrho::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + xmin::Float64, + xmax::Float64, + nx::Int64 +) # init vectors (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) # compute initial guess from medeq - rhos=Array{Float64}(undef,nlrho) + rhos=Array{Float64,1}(undef,nlrho) for j in 0:nlrho-1 rhos[j+1]=(nlrho==1 ? rho : 10^(minlrho+(log10(rho)-minlrho)/(nlrho-1)*j)) end @@ -44,7 +59,17 @@ function anyeq_Kv(minlrho,nlrho,taus,P,N,J,rho,a0,v,maxiter,tolerance,xmin,xmax, end # Compute the condensate fraction for simpleq using Kv -function simpleq_Kv_condensate_fraction(rhos,taus,P,N,J,a0,v,maxiter,tolerance) +function simpleq_Kv_condensate_fraction( + rhos::Array{Float64,1}, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64 +) # init vectors (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) @@ -67,19 +92,35 @@ function simpleq_Kv_condensate_fraction(rhos,taus,P,N,J,a0,v,maxiter,tolerance) end # Compute the two-point correlation function for simpleq using Kv -function simpleq_Kv_2pt(minlrho,nlrho,taus,P,N,J,rho,a0,v,maxiter,tolerance,xmin,xmax,nx) +function simpleq_Kv_2pt( + minlrho::Float64, + nlrho::Int64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + rho::Float64, + a0::Float64, + v::Function, + maxiter::Int64, + tolerance::Float64, + xmin::Float64, + xmax::Float64, + nx::Int64 +) # init vectors (weights,T,k,V,V0,A,Upsilon,Upsilon0)=anyeq_init(taus,P,N,J,v) # compute initial guess from medeq - rhos=Array{Float64}(undef,nlrho) + rhos=Array{Float64,1}(undef,nlrho) for j in 0:nlrho-1 rhos[j+1]=(nlrho==1 ? rho : 10^(minlrho+(log10(rho)-minlrho)/(nlrho-1)*j)) end u0s=anyeq_init_medeq(rhos,N,J,k,a0,v,maxiter,tolerance) u0=u0s[nlrho] - (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,nothing,Upsilon,Upsilon0,v,maxiter,tolerance,Anyeq_approx(0.,0.,1.,0.,0.,0.,0.,0.,0.,0.,0.)) + Abar=Array{Float64,5}(undef,0,0,0,0,0) + (u,E,error)=anyeq_hatu(u0,P,N,J,rho,a0,weights,k,taus,V,V0,A,Abar,Upsilon,Upsilon0,v,maxiter,tolerance,Anyeq_approx(0.,0.,1.,0.,0.,0.,0.,0.,0.,0.,0.)) # Kv in Fourier space Kvk=simpleq_Kv_Kvk(u,V,E,rho,Upsilon,k,taus,weights,N,J) @@ -103,7 +144,18 @@ function simpleq_Kv_2pt(minlrho,nlrho,taus,P,N,J,rho,a0,v,maxiter,tolerance,xmin end # Kv -function simpleq_Kv_Kvk(u,V,E,rho,Upsilon,k,taus,weights,N,J) +function simpleq_Kv_Kvk( + u::Array{Float64,1}, + V::Array{Float64,1}, + E::Float64, + rho::Float64, + Upsilon::Array{Array{Float64,1},1}, + k::Array{Float64,1}, + taus::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + N::Int64, + J::Int64 +) # (-Delta+v+4e(1-\rho u*)) in Fourier space M=Array{Float64,2}(undef,N*J,N*J) for zetapp in 0:J-1 diff --git a/src/simpleq-hardcore.jl b/src/simpleq-hardcore.jl index ca64f78..398ac06 100644 --- a/src/simpleq-hardcore.jl +++ b/src/simpleq-hardcore.jl @@ -1,4 +1,4 @@ -## Copyright 2021 Ian Jauslin +## Copyright 2021-2023 Ian Jauslin ## ## Licensed under the Apache License, Version 2.0 (the "License"); ## you may not use this file except in compliance with the License. @@ -13,28 +13,26 @@ ## limitations under the License. # compute energy as a function of rho -function simpleq_hardcore_energy_rho(rhos,taus,P,N,J,maxiter,tolerance) - ## spawn workers - # number of workers - nw=nworkers() - # split jobs among workers - work=Array{Array{Int64,1},1}(undef,nw) - # init empty arrays - for p in 1:nw - work[p]=zeros(0) - end - for j in 1:length(rhos) - append!(work[j%nw+1],j) - end +function simpleq_hardcore_energy_rho( + rhos::Array{Float64,1}, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + maxiter::Int64, + tolerance::Float64 +) + # spawn workers + work=spawn_workers(length(rhos)) # initialize vectors (weights,weights_gL,r,T)=simpleq_hardcore_init(taus,P,N,J) # initial guess - u0s=Array{Array{Float64}}(undef,length(rhos)) - e0s=Array{Float64}(undef,length(rhos)) + u0s=Array{Array{Float64,1}}(undef,length(rhos)) + e0s=Array{Float64,1}(undef,length(rhos)) for j in 1:length(rhos) - u0s[j]=Array{Float64}(undef,N*J) + u0s[j]=Array{Float64,1}(undef,N*J) for i in 1:N*J u0s[j][i]=1/(1+r[i]^2)^2 end @@ -43,17 +41,17 @@ function simpleq_hardcore_energy_rho(rhos,taus,P,N,J,maxiter,tolerance) # save result from each task - us=Array{Array{Float64}}(undef,length(rhos)) - es=Array{Float64}(undef,length(rhos)) - err=Array{Float64}(undef,length(rhos)) + us=Array{Array{Float64,1}}(undef,length(rhos)) + es=Array{Float64,1}(undef,length(rhos)) + err=Array{Float64,1}(undef,length(rhos)) count=0 # for each worker - @sync for p in 1:nw + @sync for p in 1:length(work) # for each task @async for j in work[p] count=count+1 - if count>=nw + if count>=length(work) progress(count,length(rhos),10000) end # run the task @@ -67,12 +65,20 @@ function simpleq_hardcore_energy_rho(rhos,taus,P,N,J,maxiter,tolerance) end # compute u(x) -function simpleq_hardcore_ux(rho,taus,P,N,J,maxiter,tolerance) +function simpleq_hardcore_ux( + rho::Float64, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + maxiter::Int64, + tolerance::Float64 +) # initialize vectors (weights,weights_gL,r,T)=simpleq_hardcore_init(taus,P,N,J) # initial guess - u0=Array{Float64}(undef,N*J) + u0=Array{Float64,1}(undef,N*J) for i in 1:N*J u0[i]=1/(1+r[i]^2)^2 end @@ -86,28 +92,26 @@ function simpleq_hardcore_ux(rho,taus,P,N,J,maxiter,tolerance) end # compute condensate fraction as a function of rho -function simpleq_hardcore_condensate_fraction_rho(rhos,taus,P,N,J,maxiter,tolerance) - ## spawn workers - # number of workers - nw=nworkers() - # split jobs among workers - work=Array{Array{Int64,1},1}(undef,nw) - # init empty arrays - for p in 1:nw - work[p]=zeros(0) - end - for j in 1:length(rhos) - append!(work[j%nw+1],j) - end +function simpleq_hardcore_condensate_fraction_rho( + rhos::Array{Float64,1}, + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64, + maxiter::Int64, + tolerance::Float64 +) + # spawn workers + work=spawn_workers(length(rhos)) # initialize vectors (weights,weights_gL,r,T)=simpleq_hardcore_init(taus,P,N,J) # initial guess - u0s=Array{Array{Float64}}(undef,length(rhos)) - e0s=Array{Float64}(undef,length(rhos)) + u0s=Array{Array{Float64,1}}(undef,length(rhos)) + e0s=Array{Float64,1}(undef,length(rhos)) for j in 1:length(rhos) - u0s[j]=Array{Float64}(undef,N*J) + u0s[j]=Array{Float64,1}(undef,N*J) for i in 1:N*J u0s[j][i]=1/(1+r[i]^2)^2 end @@ -116,17 +120,17 @@ function simpleq_hardcore_condensate_fraction_rho(rhos,taus,P,N,J,maxiter,tolera # save result from each task - us=Array{Array{Float64}}(undef,length(rhos)) - es=Array{Float64}(undef,length(rhos)) - err=Array{Float64}(undef,length(rhos)) + us=Array{Array{Float64,1}}(undef,length(rhos)) + es=Array{Float64,1}(undef,length(rhos)) + err=Array{Float64,1}(undef,length(rhos)) count=0 # for each worker - @sync for p in 1:nw + @sync for p in 1:length(work) # for each task @async for j in work[p] count=count+1 - if count>=nw + if count>=length(work) progress(count,length(rhos),10000) end # run the task @@ -142,13 +146,18 @@ end # initialize computation -@everywhere function simpleq_hardcore_init(taus,P,N,J) +@everywhere function simpleq_hardcore_init( + taus::Array{Float64,1}, + P::Int64, + N::Int64, + J::Int64 +) # Gauss-Legendre weights weights=gausslegendre(N) weights_gL=gausslaguerre(N) # r - r=Array{Float64}(undef,J*N) + r=Array{Float64,1}(undef,J*N) for zeta in 0:J-1 for j in 1:N xj=weights[1][j] @@ -164,9 +173,23 @@ end end # compute u using chebyshev expansions -@everywhere function simpleq_hardcore_hatu(u0,e0,rho,r,taus,T,weights,weights_gL,P,N,J,maxiter,tolerance) +@everywhere function simpleq_hardcore_hatu( + u0::Array{Float64,1}, + e0::Float64, + rho::Float64, + r::Array{Float64,1}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64, + maxiter::Int64, + tolerance::Float64 +) # init - vec=Array{Float64}(undef,J*N+1) + vec=Array{Float64,1}(undef,J*N+1) for i in 1:J*N vec[i]=u0[i] end @@ -194,8 +217,20 @@ end end # Xi -@everywhere function simpleq_hardcore_Xi(u,e,rho,r,taus,T,weights,weights_gL,P,N,J) - out=Array{Float64}(undef,J*N+1) +@everywhere function simpleq_hardcore_Xi( + u::Array{Float64,1}, + e::Float64, + rho::Float64, + r::Array{Float64,1}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64 +) + out=Array{Float64,1}(undef,J*N+1) FU=chebyshev(u,taus,weights,P,N,J,4) #D's @@ -215,7 +250,19 @@ end return out end # DXi -@everywhere function simpleq_hardcore_DXi(u,e,rho,r,taus,T,weights,weights_gL,P,N,J) +@everywhere function simpleq_hardcore_DXi( + u::Array{Float64,1}, + e::Float64, + rho::Float64, + r::Array{Float64,1}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64 +) out=Array{Float64,2}(undef,J*N+1,J*N+1) FU=chebyshev(u,taus,weights,P,N,J,4) @@ -232,15 +279,15 @@ end for zetapp in 0:J-1 for n in 1:N - one=zeros(Int64,J*N) - one[zetapp*N+n]=1 + one=zeros(Float64,J*N) + one[zetapp*N+n]=1. Fone=chebyshev(one,taus,weights,P,N,J,4) for i in 1:J*N # du/du out[i,zetapp*N+n]=dot(Fone,d1[i])+2*dot(FU,d2[i]*Fone)-(zetapp*N+n==i ? 1 : 0) # du/de - out[i,J*N+1]=(dsed0[i]+dot(FU,dsed1[i])+dot(FU,dsed2[i]*FU))/(2*sqrt(abs(e)))*(e>=0 ? 1 : -1) + out[i,J*N+1]=(dsed0[i]+dot(FU,dsed1[i])+dot(FU,dsed2[i]*FU))/(2*sqrt(abs(e)))*(e>=0. ? 1. : -1.) end # de/du out[J*N+1,zetapp*N+n]=2*pi*rho* @@ -253,14 +300,26 @@ end #de/de out[J*N+1,J*N+1]=-1+2*pi*rho* (2-dsedgamma0(e,rho,weights)-dot(FU,dsedgamma1(e,rho,taus,T,weights,weights_gL,P,J,4))-dot(FU,dsedgamma2(e,rho,taus,T,weights,weights_gL,P,J,4)*FU))/denom/ - (2*sqrt(abs(e)))*(e>=0 ? 1 : -1) + (2*sqrt(abs(e)))*(e>=0. ? 1. : -1.) return out end # dXi/dmu -@everywhere function simpleq_hardcore_dXidmu(u,e,rho,r,taus,T,weights,weights_gL,P,N,J) - out=Array{Float64}(undef,J*N+1) +@everywhere function simpleq_hardcore_dXidmu( + u::Array{Float64,1}, + e::Float64, + rho::Float64, + r::Array{Float64,1}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64 +) + out=Array{Float64,1}(undef,J*N+1) FU=chebyshev(u,taus,weights,P,N,J,4) #D's @@ -281,17 +340,37 @@ end end # B's -@everywhere function B0(r) +@everywhere function B0( + r::Float64 +) return pi/12*(r-1)^2*(r+5) end -@everywhere function B1(r,zeta,n,taus,T,weights,nu) +@everywhere function B1( + r::Float64, + zeta::Int64, + n::Int64, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + nu::Int64 +) return (taus[zeta+1]>=(2-r)/r || taus[zeta+2]<=-r/(r+2) ? 0 : 8*pi/(r+1)*integrate_legendre(tau-> (1-(r-(1-tau)/(1+tau))^2)/(1+tau)^(3-nu)*T[n+1]((2*tau-(taus[zeta+1]+taus[zeta+2]))/(taus[zeta+2]-taus[zeta+1])),max(taus[zeta+1],-r/(r+2)) ,min(taus[zeta+2],(2-r)/r),weights) ) end -@everywhere function B2(r,zeta,n,zetap,m,taus,T,weights,nu) +@everywhere function B2( + r::Float64, + zeta::Int64, + n::Int64, + zetap::Int64, + m::Int64, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + nu::Int64 +) return 32*pi/(r+1)*integrate_legendre(tau-> 1/(1+tau)^(3-nu)*T[n+1]((2*tau-(taus[zeta+1]+taus[zeta+2]))/(taus[zeta+2]-taus[zeta+1]))* (taus[zetap+1]>=alphap(abs(r-(1-tau)/(1+tau))-2*tau/(1+tau),tau) || taus[zetap+2]<=alpham(1+r,tau) ? 0 : @@ -303,30 +382,49 @@ end end # D's -@everywhere function D0(e,rho,r,weights,N,J) - out=Array{Float64}(undef,J*N) +@everywhere function D0( + e::Float64, + rho::Float64, + r::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + N::Int64, + J::Int64 +) + out=Array{Float64,1}(undef,J*N) for i in 1:J*N out[i]=exp(-2*sqrt(abs(e))*r[i])/(r[i]+1)+ rho*sqrt(abs(e))/(r[i]+1)*integrate_legendre(s-> (s+1)*B0(s)*(exp(-2*sqrt(abs(e))*(r[i]-s))-exp(-2*sqrt(abs(e))*(r[i]+s)))/2 - ,0,min(1,r[i]),weights)+ + ,0.,min(1.,r[i]),weights)+ (r[i]>=1 ? 0 : rho*sqrt(abs(e))/(2*(r[i]+1))*(1-exp(-4*sqrt(abs(e))*r[i]))*integrate_legendre(s-> (s+r[i]+1)*B0(s+r[i])*exp(-2*sqrt(abs(e))*s) - ,0,1-r[i],weights) + ,0.,1. -r[i],weights) ) end return out end -@everywhere function D1(e,rho,r,taus,T,weights,weights_gL,P,N,J,nu) - out=Array{Array{Float64}}(undef,J*N) +@everywhere function D1( + e::Float64, + rho::Float64, + r::Array{Float64,1}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64, + nu::Int64 +) + out=Array{Array{Float64,1},1}(undef,J*N) for i in 1:J*N - out[i]=Array{Float64}(undef,(P+1)*J) + out[i]=Array{Float64,1}(undef,(P+1)*J) for zeta in 0:J-1 for n in 0:P out[i][zeta*(P+1)+n+1]=rho*sqrt(abs(e))/(r[i]+1)*integrate_legendre(s-> (s+1)*B1(s,zeta,n,taus,T,weights,nu)*(exp(-2*sqrt(abs(e))*(r[i]-s))-exp(-2*sqrt(abs(e))*(r[i]+s)))/2 - ,0,r[i],weights)+ + ,0.,r[i],weights)+ rho*sqrt(abs(e))/(2*(r[i]+1))*(1-exp(-4*sqrt(abs(e))*r[i]))*integrate_laguerre(s-> (s+r[i]+1)*B1(s+r[i],zeta,n,taus,T,weights,nu) ,2*sqrt(abs(e)),weights_gL) @@ -336,7 +434,19 @@ end return out end -@everywhere function D2(e,rho,r,taus,T,weights,weights_gL,P,N,J,nu) +@everywhere function D2( + e::Float64, + rho::Float64, + r::Array{Float64,1}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64, + nu::Int64 +) out=Array{Array{Float64,2}}(undef,J*N) for i in 1:J*N out[i]=Array{Float64,2}(undef,(P+1)*J,(P+1)*J) @@ -346,7 +456,7 @@ end for m in 0:P out[i][zeta*(P+1)+n+1,zetap*(P+1)+m+1]=rho*sqrt(abs(e))/(r[i]+1)*integrate_legendre(s-> (s+1)*B2(s,zeta,n,zetap,m,taus,T,weights,nu)*(exp(-2*sqrt(abs(e))*(r[i]-s))-exp(-2*sqrt(abs(e))*(r[i]+s)))/2 - ,0,r[i],weights)+ + ,0.,r[i],weights)+ rho*sqrt(abs(e))/(2*(r[i]+1))*(1-exp(-4*sqrt(abs(e))*r[i]))*integrate_laguerre(s-> (s+r[i]+1)*B2(s+r[i],zeta,n,zetap,m,taus,T,weights,nu) ,2*sqrt(abs(e)),weights_gL) @@ -359,28 +469,47 @@ end end # dD/d sqrt(abs(e))'s -@everywhere function dseD0(e,rho,r,weights,N,J) - out=Array{Float64}(undef,J*N) +@everywhere function dseD0( + e::Float64, + rho::Float64, + r::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + N::Int64, + J::Int64 +) + out=Array{Float64,1}(undef,J*N) for i in 1:J*N out[i]=-2*r[i]*exp(-2*sqrt(abs(e))*r[i])/(r[i]+1)+ rho/(r[i]+1)*integrate_legendre(s-> (s+1)*B0(s)*((1-2*sqrt(abs(e))*r[i])*(exp(-2*sqrt(abs(e))*(r[i]-s))-exp(-2*sqrt(abs(e))*(r[i]+s)))/2+2*sqrt(abs(e))*s*(exp(-2*sqrt(abs(e))*(r[i]-s))+exp(-2*sqrt(abs(e))*(r[i]+s)))/2) - ,0,min(1,r[i]),weights)+ + ,0.,min(1.,r[i]),weights)+ (r[i]>=1 ? 0 : - rho/(2*(r[i]+1))*integrate_legendre(s->(s+r[i]+1)*B0(s+r[i])*((1-2*sqrt(abs(e))*s)*(1-exp(-4*sqrt(abs(e))*r[i]))*exp(-2*sqrt(abs(e))*s)+4*sqrt(abs(e))*r[i]*exp(-2*sqrt(abs(e))*(2*r[i]+s))),0,1-r[i],weights) + rho/(2*(r[i]+1))*integrate_legendre(s->(s+r[i]+1)*B0(s+r[i])*((1-2*sqrt(abs(e))*s)*(1-exp(-4*sqrt(abs(e))*r[i]))*exp(-2*sqrt(abs(e))*s)+4*sqrt(abs(e))*r[i]*exp(-2*sqrt(abs(e))*(2*r[i]+s))),0.,1. -r[i],weights) ) end return out end -@everywhere function dseD1(e,rho,r,taus,T,weights,weights_gL,P,N,J,nu) - out=Array{Array{Float64}}(undef,J*N) +@everywhere function dseD1( + e::Float64, + rho::Float64, + r::Array{Float64,1}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64, + nu::Int64 +) + out=Array{Array{Float64,1},1}(undef,J*N) for i in 1:J*N - out[i]=Array{Float64}(undef,(P+1)*J) + out[i]=Array{Float64,1}(undef,(P+1)*J) for zeta in 0:J-1 for n in 0:P out[i][zeta*(P+1)+n+1]=rho/(r[i]+1)*integrate_legendre(s-> (s+1)*B1(s,zeta,n,taus,T,weights,nu)*((1-2*sqrt(abs(e))*r[i])*(exp(-2*sqrt(abs(e))*(r[i]-s))-exp(-2*sqrt(abs(e))*(r[i]+s)))/2+2*sqrt(abs(e))*s*(exp(-2*sqrt(abs(e))*(r[i]-s))+exp(-2*sqrt(abs(e))*(r[i]+s)))/2) - ,0,r[i],weights)+ + ,0.,r[i],weights)+ rho/(2*(r[i]+1))*integrate_laguerre(s-> (s+r[i]+1)*B1(s+r[i],zeta,n,taus,T,weights,nu)*((1-2*sqrt(abs(e))*s)*(1-exp(-4*sqrt(abs(e))*r[i]))+4*sqrt(abs(e))*r[i]*exp(-4*sqrt(abs(e))*r[i])) ,2*sqrt(abs(e)),weights_gL) @@ -389,7 +518,19 @@ end end return out end -@everywhere function dseD2(e,rho,r,taus,T,weights,weights_gL,P,N,J,nu) +@everywhere function dseD2( + e::Float64, + rho::Float64, + r::Array{Float64,1}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64, + nu::Int64 +) out=Array{Array{Float64,2}}(undef,J*N) for i in 1:J*N out[i]=Array{Float64,2}(undef,(P+1)*J,(P+1)*J) @@ -399,7 +540,7 @@ end for m in 0:P out[i][zeta*(P+1)+n+1,zetap*(P+1)+m+1]=rho/(r[i]+1)*integrate_legendre(s-> (s+1)*B2(s,zeta,n,zetap,m,taus,T,weights,nu)*((1-2*sqrt(abs(e))*r[i])*(exp(-2*sqrt(abs(e))*(r[i]-s))-exp(-2*sqrt(abs(e))*(r[i]+s)))/2+2*sqrt(abs(e))*s*(exp(-2*sqrt(abs(e))*(r[i]-s))+exp(-2*sqrt(abs(e))*(r[i]+s)))/2) - ,0,r[i],weights)+ + ,0.,r[i],weights)+ rho/(2*(r[i]+1))*integrate_laguerre(s-> (s+r[i]+1)*B2(s+r[i],zeta,n,zetap,m,taus,T,weights,nu)*((1-2*sqrt(abs(e))*s)*(1-exp(-4*sqrt(abs(e))*r[i]))+4*sqrt(abs(e))*r[i]*exp(-4*sqrt(abs(e))*r[i])) ,2*sqrt(abs(e)),weights_gL) @@ -412,28 +553,47 @@ end end # dD/d sqrt(abs(e+mu/2))'s -@everywhere function dsmuD0(e,rho,r,weights,N,J) - out=Array{Float64}(undef,J*N) +@everywhere function dsmuD0( + e::Float64, + rho::Float64, + r::Array{Float64,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + N::Int64, + J::Int64 +) + out=Array{Float64,1}(undef,J*N) for i in 1:J*N out[i]=-2*r[i]*exp(-2*sqrt(abs(e))*r[i])/(r[i]+1)+ rho/(r[i]+1)*integrate_legendre(s-> (s+1)*B0(s)*((-1-2*sqrt(abs(e))*r[i])*(exp(-2*sqrt(abs(e))*(r[i]-s))-exp(-2*sqrt(abs(e))*(r[i]+s)))/2+2*sqrt(abs(e))*s*(exp(-2*sqrt(abs(e))*(r[i]-s))+exp(-2*sqrt(abs(e))*(r[i]+s)))/2) - ,0,min(1,r[i]),weights)+ + ,0.,min(1.,r[i]),weights)+ (r[i]>=1 ? 0 : - rho/(2*(r[i]+1))*integrate_legendre(s->(s+r[i]+1)*B0(s+r[i])*((-1-2*sqrt(abs(e))*s)*(1-exp(-4*sqrt(abs(e))*r[i]))*exp(-2*sqrt(abs(e))*s)+4*sqrt(abs(e))*r[i]*exp(-2*sqrt(abs(e))*(2*r[i]+s))),0,1-r[i],weights) + rho/(2*(r[i]+1))*integrate_legendre(s->(s+r[i]+1)*B0(s+r[i])*((-1-2*sqrt(abs(e))*s)*(1-exp(-4*sqrt(abs(e))*r[i]))*exp(-2*sqrt(abs(e))*s)+4*sqrt(abs(e))*r[i]*exp(-2*sqrt(abs(e))*(2*r[i]+s))),0.,1. -r[i],weights) ) end return out end -@everywhere function dsmuD1(e,rho,r,taus,T,weights,weights_gL,P,N,J,nu) - out=Array{Array{Float64}}(undef,J*N) +@everywhere function dsmuD1( + e::Float64, + rho::Float64, + r::Array{Float64,1}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64, + nu::Int64 +) + out=Array{Array{Float64,1},1}(undef,J*N) for i in 1:J*N - out[i]=Array{Float64}(undef,(P+1)*J) + out[i]=Array{Float64,1}(undef,(P+1)*J) for zeta in 0:J-1 for n in 0:P out[i][zeta*(P+1)+n+1]=rho/(r[i]+1)*integrate_legendre(s-> (s+1)*B1(s,zeta,n,taus,T,weights,nu)*((-1-2*sqrt(abs(e))*r[i])*(exp(-2*sqrt(abs(e))*(r[i]-s))-exp(-2*sqrt(abs(e))*(r[i]+s)))/2+2*sqrt(abs(e))*s*(exp(-2*sqrt(abs(e))*(r[i]-s))+exp(-2*sqrt(abs(e))*(r[i]+s)))/2) - ,0,r[i],weights)+ + ,0.,r[i],weights)+ rho/(2*(r[i]+1))*integrate_laguerre(s-> (s+r[i]+1)*B1(s+r[i],zeta,n,taus,T,weights,nu)*((-1-2*sqrt(abs(e))*s)*(1-exp(-4*sqrt(abs(e))*r[i]))+4*sqrt(abs(e))*r[i]*exp(-4*sqrt(abs(e))*r[i])) ,2*sqrt(abs(e)),weights_gL) @@ -442,7 +602,19 @@ end end return out end -@everywhere function dsmuD2(e,rho,r,taus,T,weights,weights_gL,P,N,J,nu) +@everywhere function dsmuD2( + e::Float64, + rho::Float64, + r::Array{Float64,1}, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + N::Int64, + J::Int64, + nu::Int64 +) out=Array{Array{Float64,2}}(undef,J*N) for i in 1:J*N out[i]=Array{Float64,2}(undef,(P+1)*J,(P+1)*J) @@ -452,7 +624,7 @@ end for m in 0:P out[i][zeta*(P+1)+n+1,zetap*(P+1)+m+1]=rho/(r[i]+1)*integrate_legendre(s-> (s+1)*B2(s,zeta,n,zetap,m,taus,T,weights,nu)*((-1-2*sqrt(abs(e))*r[i])*(exp(-2*sqrt(abs(e))*(r[i]-s))-exp(-2*sqrt(abs(e))*(r[i]+s)))/2+2*sqrt(abs(e))*s*(exp(-2*sqrt(abs(e))*(r[i]-s))+exp(-2*sqrt(abs(e))*(r[i]+s)))/2) - ,0,r[i],weights)+ + ,0.,r[i],weights)+ rho/(2*(r[i]+1))*integrate_laguerre(s-> (s+r[i]+1)*B2(s+r[i],zeta,n,zetap,m,taus,T,weights,nu)*((-1-2*sqrt(abs(e))*s)*(1-exp(-4*sqrt(abs(e))*r[i]))+4*sqrt(abs(e))*r[i]*exp(-4*sqrt(abs(e))*r[i])) ,2*sqrt(abs(e)),weights_gL) @@ -465,11 +637,25 @@ end end # gamma's -@everywhere function gamma0(e,rho,weights) - return 2*rho*e*integrate_legendre(s->(s+1)*B0(s)*exp(-2*sqrt(abs(e))*s),0,1,weights) +@everywhere function gamma0( + e::Float64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}} +) + return 2*rho*e*integrate_legendre(s->(s+1)*B0(s)*exp(-2*sqrt(abs(e))*s),0.,1.,weights) end -@everywhere function gamma1(e,rho,taus,T,weights,weights_gL,P,J,nu) - out=Array{Float64}(undef,J*(P+1)) +@everywhere function gamma1( + e::Float64, + rho::Float64, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + J::Int64, + nu::Int64 +) + out=Array{Float64,1}(undef,J*(P+1)) for zeta in 0:J-1 for n in 0:P out[zeta*(P+1)+n+1]=2*rho*e*integrate_laguerre(s->(s+1)*B1(s,zeta,n,taus,T,weights,nu),2*sqrt(abs(e)),weights_gL) @@ -477,7 +663,17 @@ end end return out end -@everywhere function gamma2(e,rho,taus,T,weights,weights_gL,P,J,nu) +@everywhere function gamma2( + e::Float64, + rho::Float64, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + J::Int64, + nu::Int64 +) out=Array{Float64,2}(undef,J*(P+1),J*(P+1)) for zeta in 0:J-1 for n in 0:P @@ -492,11 +688,25 @@ end end # dgamma/d sqrt(abs(e))'s -@everywhere function dsedgamma0(e,rho,weights) - return 4*rho*sqrt(abs(e))*integrate_legendre(s->(s+1)*B0(s)*(1-sqrt(abs(e))*s)*exp(-2*sqrt(abs(e))*s),0,1,weights) +@everywhere function dsedgamma0( + e::Float64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}} +) + return 4*rho*sqrt(abs(e))*integrate_legendre(s->(s+1)*B0(s)*(1-sqrt(abs(e))*s)*exp(-2*sqrt(abs(e))*s),0.,1.,weights) end -@everywhere function dsedgamma1(e,rho,taus,T,weights,weights_gL,P,J,nu) - out=Array{Float64}(undef,J*(P+1)) +@everywhere function dsedgamma1( + e::Float64, + rho::Float64, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + J::Int64, + nu::Int64 +) + out=Array{Float64,1}(undef,J*(P+1)) for zeta in 0:J-1 for n in 0:P out[zeta*(P+1)+n+1]=4*rho*e*integrate_laguerre(s->(s+1)*B1(s,zeta,n,taus,T,weights,nu)*(1-sqrt(abs(e))*s),2*sqrt(abs(e)),weights_gL) @@ -504,7 +714,17 @@ end end return out end -@everywhere function dsedgamma2(e,rho,taus,T,weights,weights_gL,P,J,nu) +@everywhere function dsedgamma2( + e::Float64, + rho::Float64, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + J::Int64, + nu::Int64 +) out=Array{Float64,2}(undef,J*(P+1),J*(P+1)) for zeta in 0:J-1 for n in 0:P @@ -519,11 +739,25 @@ end end # dgamma/d sqrt(e+mu/2)'s -@everywhere function dsmudgamma0(e,rho,weights) - return -4*rho*e*integrate_legendre(s->(s+1)*s*B0(s)*exp(-2*sqrt(abs(e))*s),0,1,weights) +@everywhere function dsmudgamma0( + e::Float64, + rho::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}} +) + return -4*rho*e*integrate_legendre(s->(s+1)*s*B0(s)*exp(-2*sqrt(abs(e))*s),0.,1.,weights) end -@everywhere function dsmudgamma1(e,rho,taus,T,weights,weights_gL,P,J,nu) - out=Array{Float64}(undef,J*(P+1)) +@everywhere function dsmudgamma1( + e::Float64, + rho::Float64, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + J::Int64, + nu::Int64 +) + out=Array{Float64,1}(undef,J*(P+1)) for zeta in 0:J-1 for n in 0:P out[zeta*(P+1)+n+1]=-4*rho*e*integrate_laguerre(s->s*(s+1)*B1(s,zeta,n,taus,T,weights,nu),2*sqrt(abs(e)),weights_gL) @@ -531,7 +765,17 @@ end end return out end -@everywhere function dsmudgamma2(e,rho,taus,T,weights,weights_gL,P,J,nu) +@everywhere function dsmudgamma2( + e::Float64, + rho::Float64, + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + J::Int64, + nu::Int64 +) out=Array{Float64,2}(undef,J*(P+1),J*(P+1)) for zeta in 0:J-1 for n in 0:P @@ -546,25 +790,41 @@ end end # \bar gamma's -@everywhere function gammabar0(weights) - return 4*pi*integrate_legendre(s->s^2*B0(s-1),0,1,weights) +@everywhere function gammabar0( + weights::Tuple{Array{Float64,1},Array{Float64,1}} +) + return 4*pi*integrate_legendre(s->s^2*B0(s-1),0.,1.,weights) end -@everywhere function gammabar1(taus,T,weights,P,J,nu) - out=Array{Float64}(undef,J*(P+1)) +@everywhere function gammabar1( + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + J::Int64, + nu::Int64 +) + out=Array{Float64,1}(undef,J*(P+1)) for zeta in 0:J-1 for n in 0:P - out[zeta*(P+1)+n+1]=4*pi*integrate_legendre(s->s^2*B1(s-1,zeta,n,taus,T,weights,nu),0,1,weights) + out[zeta*(P+1)+n+1]=4*pi*integrate_legendre(s->s^2*B1(s-1,zeta,n,taus,T,weights,nu),0.,1.,weights) end end return out end -@everywhere function gammabar2(taus,T,weights,P,J,nu) +@everywhere function gammabar2( + taus::Array{Float64,1}, + T::Array{Polynomial,1}, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + P::Int64, + J::Int64, + nu::Int64 +) out=Array{Float64,2}(undef,J*(P+1),J*(P+1)) for zeta in 0:J-1 for n in 0:P for zetap in 0:J-1 for m in 0:P - out[zeta*(P+1)+n+1,zetap*(P+1)+m+1]=4*pi*integrate_legendre(s->s^2*B2(s-1,zeta,n,zetap,m,taus,T,weights,nu),0,1,weights) + out[zeta*(P+1)+n+1,zetap*(P+1)+m+1]=4*pi*integrate_legendre(s->s^2*B2(s-1,zeta,n,zetap,m,taus,T,weights,nu),0.,1.,weights) end end end diff --git a/src/simpleq-iteration.jl b/src/simpleq-iteration.jl index 98977b8..4c4de07 100644 --- a/src/simpleq-iteration.jl +++ b/src/simpleq-iteration.jl @@ -1,4 +1,4 @@ -## Copyright 2021 Ian Jauslin +## Copyright 2021-2023 Ian Jauslin ## ## Licensed under the Apache License, Version 2.0 (the "License"); ## you may not use this file except in compliance with the License. @@ -13,7 +13,12 @@ ## limitations under the License. # compute rho(e) using the iteration -function simpleq_iteration_rho_e(es,order,v,maxiter) +function simpleq_iteration_rho_e( + es::Array{Float64,1}, + order::Int64, + v::Function, + maxiter::Int64 +) for j in 1:length(es) (u,rho)=simpleq_iteration_hatun(es[j],order,v,maxiter) @printf("% .15e % .15e\n",es[j],real(rho[maxiter+1])) @@ -21,7 +26,15 @@ function simpleq_iteration_rho_e(es,order,v,maxiter) end # compute u(x) using the iteration and print at every step -function simpleq_iteration_ux(order,e,v,maxiter,xmin,xmax,nx) +function simpleq_iteration_ux( + order::Int64, + e::Float64, + v::Function, + maxiter::Int64, + xmin::Float64, + xmax::Float64, + nx::Int64 +) (u,rho)=simpleq_iteration_hatun(e,order,v,maxiter) weights=gausslegendre(order) @@ -37,7 +50,12 @@ end # \hat u_n -function simpleq_iteration_hatun(e,order,v,maxiter) +function simpleq_iteration_hatun( + e::Float64, + order::Int64, + v::Function, + maxiter::Int64 +) # gauss legendre weights weights=gausslegendre(order) @@ -46,7 +64,7 @@ function simpleq_iteration_hatun(e,order,v,maxiter) (Eta,Eta0)=easyeq_init_H(weights,v) # init u and rho - u=Array{Array{Float64}}(undef,maxiter+1) + u=Array{Array{Float64,1},1}(undef,maxiter+1) u[1]=zeros(Float64,order) rho=zeros(Float64,maxiter+1) @@ -60,7 +78,11 @@ function simpleq_iteration_hatun(e,order,v,maxiter) end # A -function simpleq_iteration_A(e,weights,Eta) +function simpleq_iteration_A( + e::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + Eta::Array{Float64,1} +) N=length(weights[1]) out=zeros(Float64,N,N) for i in 1:N @@ -75,7 +97,12 @@ function simpleq_iteration_A(e,weights,Eta) end # b -function simpleq_iteration_b(u,e,rho,V) +function simpleq_iteration_b( + u::Array{Float64,1}, + e::Float64, + rho::Float64, + V::Array{Float64,1} +) out=zeros(Float64,length(V)) for i in 1:length(V) out[i]=V[i]+2*e*rho*u[i]^2 @@ -84,7 +111,13 @@ function simpleq_iteration_b(u,e,rho,V) end # rho_n -function simpleq_iteration_rhon(u,e,weights,V0,Eta0) +function simpleq_iteration_rhon( + u::Array{Float64,1}, + e::Float64, + weights::Tuple{Array{Float64,1},Array{Float64,1}}, + V0::Float64, + Eta0::Float64 +) S=V0 for i in 1:length(weights[1]) y=(weights[1][i]+1)/2 diff --git a/src/tools.jl b/src/tools.jl index 0d3dc7f..5d1678d 100644 --- a/src/tools.jl +++ b/src/tools.jl @@ -1,4 +1,4 @@ -## Copyright 2021 Ian Jauslin +## Copyright 2021-2023 Ian Jauslin ## ## Licensed under the Apache License, Version 2.0 (the "License"); ## you may not use this file except in compliance with the License. @@ -13,7 +13,9 @@ ## limitations under the License. # \Phi(x)=2*(1-sqrt(1-x))/x -@everywhere function Phi(x) +@everywhere function Phi( + x::Float64 +) if abs(x)>1e-5 return 2*(1-sqrt(abs(1-x)))/x else @@ -21,7 +23,9 @@ end end # \partial\Phi -@everywhere function dPhi(x) +@everywhere function dPhi( + x::Float64 +) #if abs(x-1)<1e-5 # @printf(stderr,"warning: dPhi is singular at 1, and evaluating it at (% .8e+i% .8e)\n",real(x),imag(x)) #end @@ -33,17 +37,25 @@ end end # apply Phi to every element of a vector -@everywhere function dotPhi(v) +@everywhere function dotPhi( + v::Array{Float64,1} +) out=zeros(Float64,length(v)) for i in 1:length(v) out[i]=Phi(v[i]) end return out end -@everywhere function dotdPhi(v) +@everywhere function dotdPhi( + v::Array{Float64,1} +) out=zeros(Float64,length(v)) for i in 1:length(v) out[i]=dPhi(v[i]) end return out end + +@everywhere function sinc(x::Float64) + return (x == 0 ? 1 : sin(x)/x) +end |