Update to v0.4v0.4

  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 files changed, 94 insertions, 0 deletions

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
+