pm_optimization::setBracketMax Interface Reference

Refine an initial input interval such that the final returned interval is guaranteed to contain the maximum of the user-specified input function.
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Detailed Description

Refine an initial input interval such that the final returned interval is guaranteed to contain the maximum of the user-specified input function.

Given two initial input values [xlow, xupp] representing the best-guess interval to hopefully contain the maximum of the input function, compute and return the refined interval and an interior point xmax within the interval such that the triple [xlow, xmax, xupp] are guaranteed to pass through a convex parabola.
In addition, return the function values at the final computed triplet.
The output refined interval [xlow, xupp] is guaranteed to contain a maximum of the user-specified input function.

Warning

The final output xmax is not the location of a function maximum.
It merely satisfies the condition xlow < xmax .and. xmax < xupp.

Parameters

[in]	getFunc	: The scalar function be maximized. On input, it must take a scalar of the same type and kind as the output argument xmax. On output, it must return a scalar of the same type and kind as the output argument xmax, 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,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 bracket. On output, If the algorithm succeeds, niter will be set to the actual number of iterations taken to find the bracket 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 almost twice the maximum allowed number of calls to the user-specified input function. The reasonable choice can be 1000.
[out]	xmax	: The output scalar of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), containing the abscissa corresponding to the maximum function value among the triplet [xlow, xmax, xupp].
[in,out]	xlow	: The input/output scalar of the same type and kind as the output argument xmax, containing the lower bound of the search interval for the function maximum abscissa. The condition xlow < xmax .and. getFunc(xmax) < getFunc(xlow) must hold for the corresponding input arguments.
[in,out]	xupp	: The input/output scalar of the same type and kind as the output argument xmax, containing the upper bound of the search interval for the function maximum abscissa. The condition xmax < xupp .and. getFunc(xmax) < getFunc(xupp) must hold for the corresponding input arguments.
[out]	fmax	: The output scalar of the same type and kind as the output argument xmax, containing the user-specified function value at xmax.
[out]	flow	: The output scalar of the same type and kind as the output argument xmax, containing the user-specified function value at the final refined xlow. (optional.)
[out]	fupp	: The output scalar of the same type and kind as the output argument xmax, containing the user-specified function value at the final refined xupp. (optional.)

Possible calling interfaces ⛓

use pm_optimization, only: setBracketMax
 
call setBracketMax(getFunc, niter, xmax, xlow, xupp, fmax, flow, fupp)

Warning

The condition 0 < niter must hold for the corresponding input arguments.
The condition xlow < xupp must hold for the corresponding input arguments.
The input function getFunc() must be concave (with negative second derivative) at least in one interval within its domain, otherwise the procedure will enter a semi-infinite loop in search of the maximum.

Remarks

The procedures under discussion are impure.

The procedures under discussion are recursive.

See also

getMinBrent
setMinBrent

Example usage ⛓

program example
 
    use pm_kind, only: SK, IK, LK
    use pm_kind, only: RKG => RKH ! all processor kinds are supported.
    use pm_io, only: display_type
    use pm_optimization, only: setBracketMax, isBracketMax
    use pm_err, only: setAsserted
    use pm_val2str, only: getStr
 
    implicit none
 
    integer(IK) :: niter, minniter = 1000
    real(RKG) :: xlow, xmax, xupp, flow, fmax, fupp
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    call disp%skip()
    call disp%show("getSqNeg(x) = (x - 1)**2")
    call disp%show("xlow = -3; xupp = -1; niter = minniter")
                    xlow = -3; xupp = -1; niter = minniter
    call disp%show("call setBracketMax(getSqNeg, niter, xmax, xlow, xupp, fmax)")
                    call setBracketMax(getSqNeg, niter, xmax, xlow, xupp, fmax)
    call disp%show("if (niter > minniter) error stop 'Bracketing failed.'")
                    if (niter > minniter) error stop 'Bracketing failed.'
    call disp%show("[xlow, xmax, xupp, getSqNeg(xlow), fmax, getSqNeg(xupp)]")
    call disp%show( [xlow, xmax, xupp, getSqNeg(xlow), fmax, getSqNeg(xupp)] )
    call disp%show("niter")
    call disp%show( niter )
    call disp%show("call setAsserted(isBracketMax(xmax, xlow, xupp, fmax, getSqNeg(xlow), getSqNeg(xupp)))")
                    call setAsserted(isBracketMax(xmax, xlow, xupp, fmax, getSqNeg(xlow), getSqNeg(xupp)))
    call disp%skip()
 
    call disp%skip()
    call disp%show("getSqNeg(x) = (x - 1)**2")
    call disp%show("xlow = -3; xupp = -1; niter = minniter")
                    xlow = -3; xupp = -1; niter = minniter
    call disp%show("call setBracketMax(getSqNeg, niter, xmax, xlow, xupp, fmax, flow, fupp)")
                    call setBracketMax(getSqNeg, niter, xmax, xlow, xupp, fmax, flow, fupp)
    call disp%show("if (niter > minniter) error stop 'Bracketing failed.'")
                    if (niter > minniter) error stop 'Bracketing failed.'
    call disp%show("[xlow, xmax, xupp, flow, fmax, fupp]")
    call disp%show( [xlow, xmax, xupp, flow, fmax, fupp] )
    call disp%show("niter")
    call disp%show( niter )
    call disp%show("call setAsserted(isBracketMax(xmax, xlow, xupp, flow, fmax, fupp))")
                    call setAsserted(isBracketMax(xmax, xlow, xupp, fmax, flow, fupp))
    call disp%skip()
 
    call disp%skip()
    call disp%show("getSqNeg(x) = (x - 1)**2")
    call disp%show("xlow = 3; xupp = 9; niter = minniter")
                    xlow = 3; xupp = 9; niter = minniter
    call disp%show("call setBracketMax(getSqNeg, niter, xmax, xlow, xupp, fmax, flow, fupp)")
                    call setBracketMax(getSqNeg, niter, xmax, xlow, xupp, fmax, flow, fupp)
    call disp%show("if (niter > minniter) error stop 'Bracketing failed.'")
                    if (niter > minniter) error stop 'Bracketing failed.'
    call disp%show("[xlow, xmax, xupp, flow, fmax, fupp]")
    call disp%show( [xlow, xmax, xupp, flow, fmax, fupp] )
    call disp%show("niter")
    call disp%show( niter )
    call disp%show("call setAsserted(isBracketMax(xmax, xlow, xupp, fmax, flow, fupp))")
                    call setAsserted(isBracketMax(xmax, xlow, xupp, fmax, flow, fupp))
    call disp%skip()
 
    call disp%skip()
    call disp%show("getSqNeg(x) = (x - 1)**2")
    call disp%show("xlow = -5; xupp = 4; niter = minniter")
                    xlow = -5; xupp = 4; niter = minniter
    call disp%show("call setBracketMax(getSqNeg, niter, xmax, xlow, xupp, fmax, flow, fupp)")
                    call setBracketMax(getSqNeg, niter, xmax, xlow, xupp, fmax, flow, fupp)
    call disp%show("if (niter > minniter) error stop 'Bracketing failed.'")
                    if (niter > minniter) error stop 'Bracketing failed.'
    call disp%show("[xlow, xmax, xupp, flow, fmax, fupp]")
    call disp%show( [xlow, xmax, xupp, flow, fmax, fupp] )
    call disp%show("niter")
    call disp%show( niter )
    call disp%show("call setAsserted(isBracketMax(xmax, xlow, xupp, fmax, flow, fupp))")
                    call setAsserted(isBracketMax(xmax, xlow, xupp, fmax, flow, fupp))
    call disp%skip()
 
contains
 
    function getSqNeg(x) result(func)
        real(RKG), intent(in) :: x
        real(RKG) :: func
        func = -(x - 1)**2
    end function
 
end program example

Example Unix compile command via Intel ifort compiler ⛓

#!/usr/bin/env sh
rm main.exe
ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓

del main.exe
set PATH=..\..\..\lib;%PATH%
ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓

#!/usr/bin/env sh
rm main.exe
gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example output ⛓

 
getSqNeg(x) = (x - 1)**2
xlow = -3; xupp = -1; niter = minniter
call setBracketMax(getSqNeg, niter, xmax, xlow, xupp, fmax)
if (niter > minniter) error stop 'Bracketing failed.'
[xlow, xmax, xupp, getSqNeg(xlow), fmax, getSqNeg(xupp)]
-1.00000000000000000000000000000000000, +1.00000000000000000000000000000000000, +2.23606797749978969640917366873127632, -4.00000000000000000000000000000000000, -0.00000000000000000000000000000000000, -1.52786404500042060718165266253744775
niter
+1
call setAsserted(isBracketMax(xmax, xlow, xupp, fmax, getSqNeg(xlow), getSqNeg(xupp)))
 
 
getSqNeg(x) = (x - 1)**2
xlow = -3; xupp = -1; niter = minniter
call setBracketMax(getSqNeg, niter, xmax, xlow, xupp, fmax, flow, fupp)
if (niter > minniter) error stop 'Bracketing failed.'
[xlow, xmax, xupp, flow, fmax, fupp]
-1.00000000000000000000000000000000000, +1.00000000000000000000000000000000000, +2.23606797749978969640917366873127632, -4.00000000000000000000000000000000000, -0.00000000000000000000000000000000000, -1.52786404500042060718165266253744775
niter
+1
call setAsserted(isBracketMax(xmax, xlow, xupp, flow, fmax, fupp))
 
 
getSqNeg(x) = (x - 1)**2
xlow = 3; xupp = 9; niter = minniter
call setBracketMax(getSqNeg, niter, xmax, xlow, xupp, fmax, flow, fupp)
if (niter > minniter) error stop 'Bracketing failed.'
[xlow, xmax, xupp, flow, fmax, fupp]
-6.70820393249936908922752100619382819, +3.00000000000000000000000000000000000, +9.00000000000000000000000000000000000, -64.0000000000000000000000000000000000, -4.00000000000000000000000000000000000, -59.4164078649987381784550420123876502
niter
+1
call setAsserted(isBracketMax(xmax, xlow, xupp, fmax, flow, fupp))
 
 
getSqNeg(x) = (x - 1)**2
xlow = -5; xupp = 4; niter = minniter
call setBracketMax(getSqNeg, niter, xmax, xlow, xupp, fmax, flow, fupp)
if (niter > minniter) error stop 'Bracketing failed.'
[xlow, xmax, xupp, flow, fmax, fupp]
-5.00000000000000000000000000000000000, +4.00000000000000000000000000000000000, +18.5623058987490536338412815092907438, -36.0000000000000000000000000000000000, -9.00000000000000000000000000000000000, -308.434588481235804507619222639361176
niter
+1
call setAsserted(isBracketMax(xmax, xlow, xupp, fmax, flow, fupp))
 
 

Test:

test_pm_optimization

Final Remarks ⛓

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For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

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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 Austin

Definition at line 433 of file pm_optimization.F90.

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

• src/fortran/main/pm_optimization.F90