ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
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pm_optimization::getMinBrent Interface Reference

Generate and return the minimum value and the corresponding abscissa xmin of the input 1-dimensional function isolated to a fractional precision of about tol using the Brent method.
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Detailed Description

Generate and return the minimum value and the corresponding abscissa xmin of the input 1-dimensional function isolated to a fractional precision of about tol using the Brent method.

Parameters
[in]getFunc: The scalar function be minimized.
  1. On input, it must take a scalar of the same type and kind as the output argument xmin.
  2. On output, it must return a scalar of the same type and kind as the output argument xmin, containing the function value at the specified input scalar point.
The following demonstrates the interface of getFunc,
function getFunc(x) result(func)
real(RKG), intent(in) :: x
real(RKG) :: func
end function
where RKG can refer to any real type kind parameter any supported by the processor (e.g., RK, RK32, RK64, or RK128) supported by the library.
[in]xlow: The input scalar of the same type and kind as the output argument xmin, containing the lower bound of the search interval for the function minimum abscissa.
The condition xlow < xmin .and. getFunc(xmin) < getFunc(xlow) must hold for the corresponding input arguments.
(optional, default = 0.1. This requires 0.1 to be within the support of getFunc())
[in]xupp: The input scalar of the same type and kind as the output argument xmin, containing the upper bound of the search interval for the function minimum abscissa.
The condition xmin < xupp .and. getFunc(xmin) < getFunc(xupp) must hold for the corresponding input arguments.
(optional, default = 0.9. This requires 0.9 to be within the support of getFunc())
[out]fmin: The output scalar of the same type and kind as the output argument xmin.
On output, it contains the function value at the identified minimum xmin.
(optional. If missing, the function value at the minimum will not be returned.)
[in]tol: The input positive scalar of the same type and kind as the output argument xmin, containing the fractional minimum distance that a new function evaluation point xmin can have from any previously evaluated point.
Values smaller than the suggestion below might lead to algorithm failure due to roundoff error accumulation.
(optional. default = sqrt(epsilon(xmin)).)
[in,out]niter: The input/output positive scalar of type integer of default kind IK containing the maximum number of allowed iterations in the algorithm in search of the minimum.
On output,
  1. If the algorithm succeeds, niter will be set to the actual number of iterations taken to find the minimum which is by definition, less than or equal to the input value.
  2. If the algorithm fails to converge, it will be a number larger than the input value (by only 1 unit).
The value of niter is effectively the number of calls to the user-specified input function.
(optional. default = int(100 * precision(xmin) / 53.).)
Returns
xmin : The output scalar of,
  1. type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128),
containing the inferred abscissa at the function minimum, if the algorithm succeeds.


Possible calling interfaces

xmin = getMinBrent(getFunc, xlow = xlow, xupp = xupp, fmin = fmin, tol = tol, niter = niter)
Generate and return the minimum value and the corresponding abscissa xmin of the input 1-dimensional ...
This module contains procedures, generic interfaces, and types for numerical optimizations of mathema...
Warning
The condition 0 < niter must hold for the corresponding input arguments.
The condition 0 < tol must hold for the corresponding input arguments.
Remarks
The procedures under discussion are impure.
The procedures under discussion are recursive.
See also
getMinBrent


Example usage

1program example
2
3 use pm_kind, only: SK, IK, LK
4 use pm_kind, only: RKG => RKH ! all processor kinds are supported.
5 use pm_io, only: display_type
7
8 implicit none
9
10 integer(IK) :: niter
11 real(RKG) :: xmin, fmin
12 type(display_type) :: disp
13 disp = display_type(file = "main.out.F90")
14
15 call disp%skip()
16 call disp%show("getSq(x) = (x - 1)**2")
17 call disp%show("xmin = getMinBrent(getSq)")
18 xmin = getMinBrent(getSq)
19 call disp%show("xmin")
20 call disp%show( xmin )
21 call disp%skip()
22
23 call disp%skip()
24 call disp%show("getSq(x) = (x - 1)**2")
25 call disp%show("niter = 10")
26 niter = 10
27 call disp%show("xmin = getMinBrent(getSq, fmin = fmin, tol = epsilon(xmin)**.8, niter = niter)")
28 xmin = getMinBrent(getSq, fmin = fmin, tol = epsilon(xmin)**.8, niter = niter)
29 call disp%show("niter")
30 call disp%show( niter )
31 call disp%show("if (niter > 10) error stop 'minimization failed.'")
32 if (niter > 10) error stop 'minimization failed.'
33 call disp%show("[xmin, fmin]")
34 call disp%show( [xmin, fmin] )
35 call disp%skip()
36
37contains
38
39 function getSq(x) result(func)
40 real(RKG), intent(in) :: x
41 real(RKG) :: func
42 func = (x - 1)**2
43 end function
44
45end program example
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11726
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11508
This module contains classes and procedures for input/output (IO) or generic display operations on st...
Definition: pm_io.F90:252
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
Definition: pm_io.F90:11393
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
Definition: pm_kind.F90:268
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
Definition: pm_kind.F90:541
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in C-Fortran Interoper...
Definition: pm_kind.F90:540
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
Definition: pm_kind.F90:539
integer, parameter RKH
The scalar integer constant of intrinsic default kind, representing the highest-precision real kind t...
Definition: pm_kind.F90:858
Generate and return an object of type display_type.
Definition: pm_io.F90:10282

Example Unix compile command via Intel ifort compiler
1#!/usr/bin/env sh
2rm main.exe
3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example Windows Batch compile command via Intel ifort compiler
1del main.exe
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
4main.exe

Example Unix / MinGW compile command via GNU gfortran compiler
1#!/usr/bin/env sh
2rm main.exe
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example output
1
2getSq(x) = (x - 1)**2
3xmin = getMinBrent(getSq)
4xmin
5+1.00000000000000000000000000000000000
6
7
8getSq(x) = (x - 1)**2
9niter = 10
10xmin = getMinBrent(getSq, fmin = fmin, tol = epsilon(xmin)**.8, niter = niter)
11niter
12+10
13if (niter > 10) error stop 'minimization failed.'
14[xmin, fmin]
15+1.00000000000000000000000000000000000, +0.00000000000000000000000000000000000
16
17
Test:
test_pm_optimization


Final Remarks


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For details on the naming conventions, see this page.
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Author:
Amir Shahmoradi, Tuesday March 7, 2017, 3:50 AM, Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin

Definition at line 768 of file pm_optimization.F90.


The documentation for this interface was generated from the following file: