Compute 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.
-
On input, it must take a scalar of the same type and kind as the input/output argument
xmin.
-
On output, it must return a scalar of the same type and kind as the input/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
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,out] | xmin | : The input/output scalar of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128).
-
On input, it must contain the initial best guess abscissa at the function minimum.
Note that the specified input value must be between the input arguments xlow and xupp, such that the triplet (xlow, getFunc(xlow)), (xmin, getFunc(fmin)), (xupp, getFunc(fupp)) can be interpolated by a convex parabolic curve.
-
On output, it will contain the inferred abscissa at the function minimum, if the algorithm succeeds.
|
| [in] | xlow | : The input scalar of the same type and kind as the input/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.
|
| [in] | xupp | : The input scalar of the same type and kind as the input/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.
|
| [in,out] | fmin | : The input/output scalar of the same type and kind as the input/output argument xmin.
-
On input, it must contain
getFunc(xmin).
-
On output, it will contain the function value at the identified minimum
xmin.
|
| [in] | tol | : The input positive scalar of the same type and kind as the input/output argument xmin, containing the 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.
A reasonable choice is tol = sqrt(epsilon(1._RKG)).
|
| [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,
-
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.
-
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.
A reasonable choice is niter = int(100 * precision(xmin) / 53.).
|
Possible calling interfaces ⛓
call setMinBrent(getFunc, xmin, xlow, xupp, fmin, tol, niter)
Compute and return the minimum value and the corresponding abscissa xmin of the input 1-dimensional f...
This module contains procedures, generic interfaces, and types for numerical optimizations of mathema...
- Warning
- The condition
xlow < xmin .and. xmin < xupp must hold for the corresponding input arguments.
The condition getFunc(xlow) > fmin .or. fmin < getFunc(xupp) must hold for the corresponding input arguments.
The condition fmin == getFunc(xmin) must hold for the corresponding input arguments.
The condition 0 < niter must hold for the corresponding input arguments.
The condition 0 < tol must hold for the corresponding input arguments.
- See also
- getMinBrent
setMinBrent
Example usage ⛓
11 integer(IK) :: niter, retin
12 real(RKG) :: xlow, xmin, xupp, fmin, tol
13 type(display_type) :: disp
17 call disp%show(
"getSq(x) = (x - 1)**2")
18 call disp%show(
"xlow = -3; xupp = -1; tol = epsilon(xmin)**.8; retin = 100; niter = 100")
19 xlow
= -3; xupp
= -1; tol
= epsilon(xmin)
**.
8; retin
= 100; niter
= 100
20 call disp%show(
"call setBracketMin(getSq, retin, xmin, xlow, xupp, fmin) ! find a good bracket, though here the choice is obvious.")
22 call disp%show(
"if (100 < retin) error stop 'Bracketing failed.'")
23 if (
100 < retin)
error stop 'Bracketing failed.'
24 call disp%show(
"call setMinBrent(getSq, xmin, xlow, xupp, fmin, tol, niter)")
25 call setMinBrent(getSq, xmin, xlow, xupp, fmin, tol, niter)
28 call disp%show(
"if (niter > 100) error stop 'minimization failed.'")
29 if (niter
> 100)
error stop 'minimization failed.'
35 call disp%show(
"getSq(x) = (x - 1)**2")
36 call disp%show(
"xlow = 1; xmin = 1; xupp = 3; tol = epsilon(xmin)**.8; niter = 100")
37 xlow
= 1; xmin
= 1; xupp
= 3; tol
= epsilon(xmin)
**.
8; niter
= 100
38 call disp%show(
"call setMinBrent(getSq, xmin, xlow, xupp, fmin, tol, niter)")
39 call setMinBrent(getSq, xmin, xlow, xupp, fmin, tol, niter)
42 call disp%show(
"if (niter > 100) error stop 'minimization failed.'")
43 if (niter
> 100)
error stop 'minimization failed.'
49 call disp%show(
"getSq(x) = (x - 1)**2")
50 call disp%show(
"xlow = 1; xmin = 1; xupp = 1; tol = epsilon(xmin)**.8; niter = 100")
51 xlow
= 1; xmin
= 1; xupp
= 1; tol
= epsilon(xmin)
**.
8; niter
= 100
52 call disp%show(
"call setMinBrent(getSq, xmin, xlow, xupp, fmin, tol, niter)")
53 call setMinBrent(getSq, xmin, xlow, xupp, fmin, tol, niter)
56 call disp%show(
"if (niter > 100) error stop 'minimization failed.'")
57 if (niter
> 100)
error stop 'minimization failed.'
63 call disp%show(
"getSq(x) = (x - 1)**2")
64 call disp%show(
"xmin = 0; xlow = -1; xupp = 3; fmin = getSq(xmin); tol = epsilon(xmin)**.8; niter = 100")
65 xmin
= 0; xlow
= -1; xupp
= 3; fmin
= getSq(xmin); tol
= epsilon(xmin)
**.
8; niter
= 100
66 call disp%show(
"call setMinBrent(getSq, xmin, xlow, xupp, fmin, tol, niter)")
67 call setMinBrent(getSq, xmin, xlow, xupp, fmin, tol, niter)
70 call disp%show(
"if (niter > 100) error stop 'minimization failed.'")
71 if (niter
> 100)
error stop 'minimization failed.'
78 function getSq(x)
result(func)
79 real(RKG),
intent(in) :: x
This is a generic method of the derived type display_type with pass attribute.
This is a generic method of the derived type display_type with pass attribute.
Refine an initial input interval such that the final returned interval is guaranteed to contain the m...
This module contains classes and procedures for input/output (IO) or generic display operations on st...
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in C-Fortran Interoper...
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
integer, parameter RKH
The scalar integer constant of intrinsic default kind, representing the highest-precision real kind t...
Generate and return an object of type display_type.
Example Unix compile command via Intel ifort compiler ⛓
3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
Example Windows Batch compile command via Intel ifort compiler ⛓
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
Example Unix / MinGW compile command via GNU gfortran compiler ⛓
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
Example output ⛓
3xlow
= -3; xupp
= -1; tol
= epsilon(xmin)
**.
8; retin
= 100; niter
= 100
5if (
100 < retin)
error stop 'Bracketing failed.'
6call setMinBrent(getSq, xmin, xlow, xupp, fmin, tol, niter)
9if (niter
> 100)
error stop 'minimization failed.'
11+1.00000000000000000000000000000000000,
+0.00000000000000000000000000000000000
15xlow
= 1; xmin
= 1; xupp
= 3; tol
= epsilon(xmin)
**.
8; niter
= 100
16call setMinBrent(getSq, xmin, xlow, xupp, fmin, tol, niter)
19if (niter
> 100)
error stop 'minimization failed.'
21+1.00000000000000000000000000000000000,
+0.00000000000000000000000000000000000
25xlow
= 1; xmin
= 1; xupp
= 1; tol
= epsilon(xmin)
**.
8; niter
= 100
26call setMinBrent(getSq, xmin, xlow, xupp, fmin, tol, niter)
29if (niter
> 100)
error stop 'minimization failed.'
31+1.00000000000000000000000000000000000,
+0.00000000000000000000000000000000000
35xmin
= 0; xlow
= -1; xupp
= 3; fmin
= getSq(xmin); tol
= epsilon(xmin)
**.
8; niter
= 100
36call setMinBrent(getSq, xmin, xlow, xupp, fmin, tol, niter)
39if (niter
> 100)
error stop 'minimization failed.'
41+1.00000000000000000000000000000000000,
+0.00000000000000000000000000000000000
- Test:
- test_pm_optimization
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
-
If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
-
If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
- Copyright
- Computational Data Science Lab
- 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 935 of file pm_optimization.F90.
1.9.3