{Optimization utilities
author: Nikolai V. Shokhirev http://www.shokhirev.com/nikolai.html
created: 2003.06.06
last modified: 2004.12.23
ŠNikolai V. Shokhirev, 2001-2003 }
unit uOptimUtils;

interface

uses
  uMatTypes;

{ 
PURPOSE:
  Bracketing the minimum of a function within given limits
INPUT:
  f(x) - function
  xmin, xmax - search limits
  x0 - start ( xmin < x0 < xmax )
  tol - accuracy
  step - initial step
OUTPUT
  step - final step
  x1, x2 - brackets

  ErrCode = -5 - small interval: |xmax - xmin| < |tol|
  ErrCode =  5 - xmin > xmax
  ErrCode = -4 - x0 < xmin
  ErrCode =  4 - x0 > xmax
  ErrCode = -3 - flat function: |f(x)-f(x+step)| < dfmin
  ErrCode =  3 -
  ErrCode = -2 - x0 = xmin and f(x0) < f(x0+|step|)
  ErrCode =  2 - x0 = xmax and f(x0) < f(x0-|step|)
  ErrCode = -1 - minimum at x = xmin
  ErrCode =  1 - minimum at x = xmax
  ErrCode =  0 - OK                            }
procedure Bracketing(f: FloatFunc1D; x0, tol: TFloat;
               var step, x1, x2: TFloat; var ErrCode: TInt; xmin, xmax: TFloat);


{
PURPOSE:
  Bracketing the minimum of a function in the positive semi-axis
INPUT:
  f(x) - function
  xmax - search limit ( < MaxFloat )
  tol - accuracy
  step - initial step
OUTPUT
  step - final step
  x1, x2 - brackets

  ErrCode = -5 - small interval: |xmax - xmin| < |tol|
  ErrCode = -3 - flat function: |f(x)-f(x+step)| < dfmin
  ErrCode = -2 - x0 = xmin and f(x0) < f(x0+|step|)
  ErrCode =  1 - minimum at x = xmax
  ErrCode =  0 - OK                                      }

procedure Bracketing0(f: FloatFunc1D; tol: TFloat; var step, x1, x2: TFloat;
                                               var ErrCode: TInt; xmax: TFloat);

{Pascal translation of the minimization FORTRAN routine from
  "Computer Methods for Mathematical Computations", by George E. Forsythe,
      Michael A. Malcolm, and Cleve B. Moler. Prentice-Hall, 1977.

 An approximation x to the point where f attains a minimum on
 the interval (ax,bx) is determined.

INPUT:
  ax  left endpoint of initial interval
  bx  right endpoint of initial interval
  f   function subprogram which evaluates f(x) for any x in the interval (ax,bx)
  tol  desired length of the interval of uncertainty of the final result (.ge. 0.0)

OUTPUT:
  fmin abcissa approximating the point where f attains a minimum

    The method used is a combination of golden section search and successive
  parabolic interpolation. Convergence is never much slower than that for
  a fibonacci search. If f has a continuous second derivative which is positive
  at the minimum (which is not at ax or bx), then convergence is superlinear,
  and usually of the order of about 1.324....
    The function f is never evaluated at two points closer together than
  eps*abs(fmin) + (tol/3), where eps is approximately the square root of
  the relative machine precision.  if  f  is a unimodal function and the computed
  values of  f  are always unimodal when separated by at least
  eps*abs(x) + (tol/3), then fmin approximates the abscissa of the global minimum
  of f on the interval ax,bx with  an error less than 3*eps*abs(fmin) + tol.
    If  f  is not unimodal, then fmin may approximate a local, but perhaps
  non-global, minimum to the same accuracy.
    This function subprogram is a slightly modified version of the algol 60
  procedure localmin given in Richard Brent, algorithms for minimization without
  derivatives, prentice - hall, inc. (1973).                                     }
function fmin(ax, bx: TFloat; f: FloatFunc1D; tol: TFloat): TFloat;

implementation

uses
  math;

end.

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