Nelder-Mead algorithm. More...

Data Types
interface	nelminrange
	Minimizes a user-specified function using the Nelder-Mead algorithm. More...

Functions/Subroutines
real(dp) function, dimension(size(pstart)), public	nelmin (func, pstart, varmin, step, konvge, maxeval, funcmin, neval, numrestart, ierror, history)
	Minimizes a user-specified function using the Nelder-Mead algorithm.

real(dp) function, dimension(size(pstart)), public	nelminxy (func, pstart, xx, yy, varmin, step, konvge, maxeval, funcmin, neval, numrestart, ierror, history)
	Minimizes a user-specified function using the Nelder-Mead algorithm.

real(dp) function, dimension(size(pstart))	nelminrange_dp (func, pstart, prange, varmin, step, konvge, maxeval, funcmin, neval, numrestart, ierror, history)

real(sp) function, dimension(size(pstart))	nelminrange_sp (func, pstart, prange, varmin, step, konvge, maxeval, funcmin, neval, numrestart, ierror, history)

Detailed Description

Nelder-Mead algorithm.

This module provides NELMIN, which minimizes a function using the Nelder-Mead algorithm with the Applied Statistics algorithms No. 047. Original FORTRAN77 version by R ONeill. FORTRAN90 version by John Burkardt.

Author: Matthias Cuntz

Date: 2012

Copyright: Copyright 2005-2024, the CHS Developers, Sabine Attinger: All rights reserved. FORCES is released under the LGPLv3+ license

The UFZ (CHS) FORCES code is free software. You can redistribute it and/or modify it under the terms of the GNU General Public License as published by the free Software Foundation either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You received a copy of the GNU Lesser General Public License along with the UFZ CHS FORCES code. It can be found in the files COPYING and COPYING.LESSER provided with this software. The complete GNU license text can also be found at http://www.gnu.org/licenses/.

Function/Subroutine Documentation

◆ nelmin()

real(dp) function, dimension(size(pstart)), public mo_nelmin::nelmin	(	external real(dp) function(real(dp), dimension(:), intent(in) pp)	func,
		real(dp), dimension(:), intent(in)	pstart,
		real(dp), intent(in), optional	varmin,
		real(dp), dimension(:), intent(in), optional	step,
		integer(i4), intent(in), optional	konvge,
		integer(i4), intent(in), optional	maxeval,
		real(dp), intent(out), optional	funcmin,
		integer(i4), intent(out), optional	neval,
		integer(i4), intent(out), optional	numrestart,
		integer(i4), intent(out), optional	ierror,
		real(dp), dimension(:), intent(out), optional, allocatable	history
	)

Minimizes a user-specified function using the Nelder-Mead algorithm.

Simplex function minimisation procedure due to Nelder and Mead (1965), as implemented by O'Neill(1971, Appl.Statist. 20, 338-45), with subsequent comments by Chambers+Ertel(1974, 23, 250-1), Benyon(1976, 25, 97) and Hill(1978, 27, 380-2)

The function to be minimized is the first argument of nelmin and must be defined as

FUNCTION func(p)
  USE mo_kind, ONLY: dp
  IMPLICIT NONE
  REAL(dp), DIMENSION(:), INTENT(IN) :: p
  REAL(dp) :: func
END FUNCTION func

and for nelminxy

FUNCTION func(p,x,y)
  USE mo_kind, ONLY: dp
  IMPLICIT NONE
  REAL(dp), DIMENSION(:), INTENT(IN) :: p
  REAL(dp), DIMENSION(:), INTENT(IN) :: x
  REAL(dp), DIMENSION(:), INTENT(IN) :: y
  REAL(dp) :: func
END FUNCTION func

This routine does not include a termination test using the fitting of a quadratic surface.

Calling sequence

pmin = nelmin(func, pstart, funcmin, varmin, step, konvge, maxeval, &
              neval, numrestart, ierror)
pmin = nelminxy(func, pstart, xx, yy, funcmin, varmin, step, konvge, maxeval, &
                neval, numrestart, ierror)
pmin = nelmin(func, pstart, prange, funcmin, varmin, step, konvge, maxeval, &
              neval, numrestart, ierror)

Original FORTRAN77 version by R ONeill. FORTRAN90 version by John Burkardt.

Example

pstart(1:n) = (/ -1.2_dp, 1.0_dp /)
step(1:n) = (/ 1.0_dp, 1.0_dp /)
pmin = nelmin(rosenbrock, pstart, step=step, ierror=ierror)

-> see also example in test directory

Literature

Simplex function minimisation procedure due to Nelder and Mead (1965), as implemented by O'Neill(1971, Appl.Statist. 20, 338-45), with subsequent comments by Chambers+Ertel(1974, 23, 250-1), Benyon(1976, 25, 97) and Hill(1978, 27, 380-2)
John Nelder, Roger Mead, A simplex method for function minimization, Computer Journal, Volume 7, 1965, pages 308-313.
R ONeill, Algorithm AS 47: Function Minimization Using a Simplex Procedure, Applied Statistics, Volume 20, Number 3, 1971, pages 338-345.

Parameters

[in]	real(dp) :: func(p,xx,yy)	Function on which to search the minimum
[in]	real(dp) :: pstart(:)	Starting point for the iteration.
[in]	real(dp) :: xx	(ONLY nelminxy) First values to pass as function arguments
[in]	real(dp) :: yy	(ONLY nelminxy) Second values to pass as function arguments
[in]	real(dp) :: prange(:,2)	(ONLY nelminrange) Range of parameters (upper and lower bound).
[in]	real(dp), optional :: varmin	the terminating limit for the variance of the function values. varmin>0 is required.
[in]	real(dp), optional :: step(:)	determines the size and shape of the initial simplex. The relative magnitudes of its elements should reflect the units of the variables. size(step)=size(start)
[in]	integer(i4), optional :: konvge	the convergence check is carried out every konvge iterations. konvge>0 is required.
[in]	integer(i4), optional :: maxeval	the maximum number of function evaluations. default: 1000
[out]	real(dp), optional :: funcmin	the minimum value of the function.
[out]	integer(i4), optional :: neval	the number of function evaluations used.
[out]	integer(i4), optional :: numrestart	the number of restarts.
[out]	integer(i4), optional :: ierror	error indicator. 0: no errors detected. 1: varmin or konvge have an illegal value. 2: terminated because maxeval was exceeded without convergence.
[out]	real(dp), optional, allocatable :: history(:)	the history of best function values, history(neval)=funcmin

Returns: real(sp/dp) :: nelmin(size(start)) — coordinates of the point which is estimated to minimize the function.

Author: Matthias Cuntz

Date

Jul 2012

i4, dp, intent
function, optional
nelminxy

Author: Juliane Mai

Date

Aug 2012

nelminrange

Dec 2012

history output

Definition at line 136 of file mo_nelmin.f90.

References mo_kind::dp, and nelmin().

Referenced by nelmin().

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◆ nelminrange_dp()

real(dp) function, dimension(size(pstart)) mo_nelmin::nelminrange_dp	(	external real(dp) function(real(dp), dimension(:), intent(in) pp)	func,
		real(dp), dimension(:), intent(in)	pstart,
		real(dp), dimension(:,:), intent(in)	prange,
		real(dp), intent(in), optional	varmin,
		real(dp), dimension(:), intent(in), optional	step,
		integer(i4), intent(in), optional	konvge,
		integer(i4), intent(in), optional	maxeval,
		real(dp), intent(out), optional	funcmin,
		integer(i4), intent(out), optional	neval,
		integer(i4), intent(out), optional	numrestart,
		integer(i4), intent(out), optional	ierror,
		real(dp), dimension(:), intent(out), optional, allocatable	history
	)

private

Definition at line 817 of file mo_nelmin.f90.

◆ nelminrange_sp()

real(sp) function, dimension(size(pstart)) mo_nelmin::nelminrange_sp	(	external real(sp) function(real(sp), dimension(:), intent(in) pp)	func,
		real(sp), dimension(:), intent(in)	pstart,
		real(sp), dimension(:,:), intent(in)	prange,
		real(sp), intent(in), optional	varmin,
		real(sp), dimension(:), intent(in), optional	step,
		integer(i4), intent(in), optional	konvge,
		integer(i4), intent(in), optional	maxeval,
		real(sp), intent(out), optional	funcmin,
		integer(i4), intent(out), optional	neval,
		integer(i4), intent(out), optional	numrestart,
		integer(i4), intent(out), optional	ierror,
		real(sp), dimension(:), intent(out), optional, allocatable	history
	)

private

Definition at line 1174 of file mo_nelmin.f90.

◆ nelminxy()

real(dp) function, dimension(size(pstart)), public mo_nelmin::nelminxy	(	external real(dp) function(real(dp), dimension(:), intent(in) pp, real(dp), dimension(:), intent(in) xxx, real(dp), dimension(:), intent(in) yyy)	func,
		real(dp), dimension(:), intent(in)	pstart,
		real(dp), dimension(:), intent(in)	xx,
		real(dp), dimension(:), intent(in)	yy,
		real(dp), intent(in), optional	varmin,
		real(dp), dimension(:), intent(in), optional	step,
		integer(i4), intent(in), optional	konvge,
		integer(i4), intent(in), optional	maxeval,
		real(dp), intent(out), optional	funcmin,
		integer(i4), intent(out), optional	neval,
		integer(i4), intent(out), optional	numrestart,
		integer(i4), intent(out), optional	ierror,
		real(dp), dimension(:), intent(out), optional, allocatable	history
	)

Minimizes a user-specified function using the Nelder-Mead algorithm.

Simplex function minimisation procedure due to Nelder and Mead (1965), as implemented by O'Neill(1971, Appl.Statist. 20, 338-45), with subsequent comments by Chambers+Ertel(1974, 23, 250-1), Benyon(1976, 25, 97) and Hill(1978, 27, 380-2)

The function to be minimized is the first argument of nelmin and must be defined as

FUNCTION func(p)
  USE mo_kind, ONLY: dp
  IMPLICIT NONE
  REAL(dp), DIMENSION(:), INTENT(IN) :: p
  REAL(dp) :: func
END FUNCTION func

and for nelminxy

FUNCTION func(p,x,y)
  USE mo_kind, ONLY: dp
  IMPLICIT NONE
  REAL(dp), DIMENSION(:), INTENT(IN) :: p
  REAL(dp), DIMENSION(:), INTENT(IN) :: x
  REAL(dp), DIMENSION(:), INTENT(IN) :: y
  REAL(dp) :: func
END FUNCTION func

This routine does not include a termination test using the fitting of a quadratic surface.

Calling sequence

pmin = nelmin(func, pstart, funcmin, varmin, step, konvge, maxeval, &
              neval, numrestart, ierror)
pmin = nelminxy(func, pstart, xx, yy, funcmin, varmin, step, konvge, maxeval, &
                neval, numrestart, ierror)
pmin = nelmin(func, pstart, prange, funcmin, varmin, step, konvge, maxeval, &
              neval, numrestart, ierror)

Original FORTRAN77 version by R ONeill. FORTRAN90 version by John Burkardt.

Example

pstart(1:n) = (/ -1.2_dp, 1.0_dp /)
step(1:n) = (/ 1.0_dp, 1.0_dp /)
pmin = nelmin(rosenbrock, pstart, step=step, ierror=ierror)

-> see also example in test directory

Literature

Simplex function minimisation procedure due to Nelder and Mead (1965), as implemented by O'Neill(1971, Appl.Statist. 20, 338-45), with subsequent comments by Chambers+Ertel(1974, 23, 250-1), Benyon(1976, 25, 97) and Hill(1978, 27, 380-2)
John Nelder, Roger Mead, A simplex method for function minimization, Computer Journal, Volume 7, 1965, pages 308-313.
R ONeill, Algorithm AS 47: Function Minimization Using a Simplex Procedure, Applied Statistics, Volume 20, Number 3, 1971, pages 338-345.

Parameters

[in]	real(dp) :: func(p,xx,yy)	Function on which to search the minimum
[in]	real(dp) :: pstart(:)	Starting point for the iteration.
[in]	real(dp) :: xx	(ONLY nelminxy) First values to pass as function arguments
[in]	real(dp) :: yy	(ONLY nelminxy) Second values to pass as function arguments
[in]	real(dp) :: prange(:,2)	(ONLY nelminrange) Range of parameters (upper and lower bound).
[in]	real(dp), optional :: varmin	the terminating limit for the variance of the function values. varmin>0 is required.
[in]	real(dp), optional :: step(:)	determines the size and shape of the initial simplex. The relative magnitudes of its elements should reflect the units of the variables. size(step)=size(start)
[in]	integer(i4), optional :: konvge	the convergence check is carried out every konvge iterations. konvge>0 is required.
[in]	integer(i4), optional :: maxeval	the maximum number of function evaluations. default: 1000
[out]	real(dp), optional :: funcmin	the minimum value of the function.
[out]	integer(i4), optional :: neval	the number of function evaluations used.
[out]	integer(i4), optional :: numrestart	the number of restarts.
[out]	integer(i4), optional :: ierror	error indicator. 0: no errors detected. 1: varmin or konvge have an illegal value. 2: terminated because maxeval was exceeded without convergence.
[out]	real(dp), optional, allocatable :: history(:)	the history of best function values, history(neval)=funcmin

Returns: real(sp/dp) :: nelmin(size(start)) — coordinates of the point which is estimated to minimize the function.

Author: Matthias Cuntz

Date

Jul 2012

i4, dp, intent
function, optional
nelminxy

Author: Juliane Mai

Date

Aug 2012

nelminrange

Dec 2012

history output

Definition at line 475 of file mo_nelmin.f90.

References mo_kind::dp, nelminxy(), and mo_kind::sp.

Referenced by nelminxy().

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Data Types

Functions/Subroutines

Detailed Description

Function/Subroutine Documentation

◆ nelmin()

◆ nelminrange_dp()

◆ nelminrange_sp()

◆ nelminxy()