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|
/*@@
@file AHFinder_pow.F
@date April 1998
@author Miguel Alcubierre
@desc
Multidimensional minimization routines taken
from Numerical Recipes. The routines have some
modifications to adapt them to the specific problem
we are trying to solve.
@enddesc
@@*/
#include "cctk.h"
#include "cctk_Parameters.h"
#include "cctk_Arguments.h"
module F1COM
! Data to be shared by minimization routines.
implicit none
logical error
integer NCOM
CCTK_REAL XMIN
CCTK_REAL, allocatable, dimension(:) :: PCOM,XICOM
! ***************
! *** END ***
! ***************
end module F1COM
***********************************************************************
***********************************************************************
subroutine POWELL(CCTK_ARGUMENTS,P,XI,N,FTOL,ITER,ITMAX,FRET,found)
! Minimization of a function FUNC on N variables (FUNC is
! not an argument, it is a fixed function name). Input consists
! of an initial starting point P that is a vector of length N;
! an initial matrix XI of dimensions NxN and whose columns contain
! the initial set of directions (usually the N unit vectors); and
! FTOL, the fractional tolerance in the function value such that
! failure to decrease by more than this ammount on one iteration
! signals doneness. On output, P is set to the best point found,
! XI is the then-current direction set, FRET is the returned
! function value at P, and ITER is the number of iterations taken.
!
! Extra output added by myself (Miguel Alcubierre) is a logical
! variable "found" that is set to true is a minimum was indeed
! found, and to false if it was not.
use AHFinder_dat
implicit none
DECLARE_CCTK_ARGUMENTS
logical found
integer i,j,N
integer ITER,ITMAX,IBIG
CCTK_REAL FUNC,DEL,FP,FPTT,FRET,FTOL,T,TOL1,ZEPS
CCTK_REAL, dimension(1:N) :: P,PT,PTT,XIT
CCTK_REAL, dimension(1:N,1:N) :: XI
character(len=200) :: logf
parameter (ZEPS=1.0D-10)
! *************************
! *** START ROUTINE ***
! *************************
FRET = FUNC(CCTK_ARGUMENTS,P,N)
! Save initial point.
do j=1,N
PT(j) = P(j)
end do
ITER = 0
10 ITER = ITER + 1
if ((myproc.eq.0).and.veryver) then
write(*,*)
write(*,"(A20,I2)") 'POWELL ITERATION = ',ITER
end if
if ((myproc.eq.0).and.logfile) then
logf = filestr(1:nfile)//"/ahf_logfile"
open(11,file=logf,form='formatted',status='old',
. position='append')
write(11,*)
write(11,"(A20,I2)") 'POWELL ITERATION = ',ITER
close(11)
end if
FP = FRET
IBIG = 0
! Initialize DEL. It will be the biggest function decrease.
DEL = 0.0D0
! In each iteration, loop over all directions in the set.
do i=1,N
! Copy the direction.
do j=1,N
XIT(j) = XI(j,i)
end do
FPTT = FRET
! Minimize along it.
call LINMIN(CCTK_ARGUMENTS,P,XIT,N,FRET,FTOL)
! And record it if it is the largest decrease so far.
if (abs(FPTT-FRET).gt.DEL) then
DEL = abs(FPTT-FRET)
IBIG = I
end if
20 continue
end do
! Termination criterion.
TOL1 = 0.5D0*FTOL*(abs(FP)+abs(FRET)) + ZEPS
if (abs(FP-FRET).le.TOL1) then
found = .true.
return
else if (abs(FRET).lt.ZEPS) then
found = .true.
return
end if
! Too many iterations?
if (ITER.ge.ITMAX) return
! Construct the extrapolated point and the average direction.
do j=1,N
PTT(j) = 2.0D0*P(j) - PT(j)
XIT(j) = P(j) - PT(j)
PT(j) = P(j)
end do
! Function value at extrapolated point.
FPTT = FUNC(CCTK_ARGUMENTS,PTT,N)
! First reason not to use new direction.
if (FPTT.ge.FP) goto 10
! Second reason not to use new direction.
T = 2.0D0*(FP-2.0D0*FRET+FPTT)*(FP-FRET-DEL)**2
. - DEL*(FP-FPTT)**2
if (T.gt.0.0D0) goto 10
! Move to the minimum of the new direction.
call LINMIN(CCTK_ARGUMENTS,P,XIT,N,FRET,FTOL)
! Save new direction.
do j=1,N
XI(j,IBIG) = XIT(j)
end do
! Back for another iteration.
goto 10
! ***************
! *** END ***
! ***************
end subroutine POWELL
***********************************************************************
***********************************************************************
subroutine LINMIN(CCTK_ARGUMENTS,P,XI,N,FRET,TOL)
! Given an N dimensional point P and an N dimensional direction XI,
! moves and resets P to where the function FUNC takes on a
! minimum along the direction XI from P, and replaces XI by the
! actual vector displacement that P was moved. Also returns as
! FRET the value of FUNC at the returned location P. This is
! actually all acomplished by calling the routines MNBRAK and
! PARABOLA.
use F1COM
implicit none
DECLARE_CCTK_ARGUMENTS
integer i,j,N
integer l,m
CCTK_REAL XA,XB,XX,FA,FB,FX
CCTK_REAL FRET,PARABOLA,TOL
CCTK_REAL zero
CCTK_REAL dx,dy,dz
CCTK_REAL, dimension(1:N) :: P,XI
CCTK_REAL F1DIM
EXTERNAL F1DIM
! **********************************************
! *** ALLOCATE STORAGE FOR SHARED ARRAYS ***
! **********************************************
zero = 0.0d0
allocate(PCOM(1:N),XICOM(1:N))
! *************************
! *** START ROUTINE ***
! *************************
dx = cctk_delta_space(1)
dy = cctk_delta_space(2)
dz = cctk_delta_space(3)
NCOM = N
do j=1,N
PCOM(j) = P(j)
XICOM(j) = XI(j)
end do
! Initial guess for brackets.
XA = 0.0D0
XX = 0.1D0*min(dx,dy,dz)
! Bracket the minimum.
call MNBRAK(CCTK_ARGUMENTS,XA,XX,XB,FA,FX,FB,F1DIM)
! Check if function is constant.
if ((FA.ne.FX).or.(FB.ne.FX)) then
! Call 1D minimization.
FRET = PARABOLA(CCTK_ARGUMENTS,XA,XX,XB,FA,FX,FB,F1DIM,TOL)
! Construct the vector results to return.
XI = XMIN*XI
P = P + XI
end if
! ******************************
! *** DEALLOCATE STORAGE ***
! ******************************
deallocate(PCOM,XICOM)
! ***************
! *** END ***
! ***************
end subroutine LINMIN
***********************************************************************
***********************************************************************
CCTK_REAL function F1DIM(CCTK_ARGUMENTS,XX)
! Must accompany LINMIN
use F1COM
implicit none
DECLARE_CCTK_ARGUMENTS
integer j
CCTK_REAL XX,FUNC
CCTK_REAL, dimension(1:NCOM) :: XT
! *************************
! *** START ROUTINE ***
! *************************
do j=1,NCOM
XT(j) = PCOM(j) + XX*XICOM(j)
end do
F1DIM = FUNC(CCTK_ARGUMENTS,XT,NCOM)
! ***************
! *** END ***
! ***************
end function F1DIM
***********************************************************************
***********************************************************************
subroutine MNBRAK(CCTK_ARGUMENTS,XA,XB,XC,FA,FB,FC,FUNC)
! Given a function FUNC, and given distinct initial points
! XA and XB, this routine searches in the downhill direction
! (defined by the function as evaluated at the initial points)
! and returns new points XA, XB, XC which bracket a minimum
! of the function. Also returned are the function values
! at the three points, FA, FB and FC.
implicit none
DECLARE_CCTK_ARGUMENTS
CCTK_REAL XA,XB,XC
CCTK_REAL FA,FB,FC,FU,FUNC
CCTK_REAL DUM,Q,S,U,ULIM
CCTK_REAL GOLD,GLIMIT,TINY
parameter (GOLD=1.618034D0,GLIMIT=4.0D0,TINY=1.0D-20)
! *************************
! *** START ROUTINE ***
! *************************
FA = FUNC(CCTK_ARGUMENTS,XA)
FB = FUNC(CCTK_ARGUMENTS,XB)
! Switch roles of A and B so that we can go downhill
! in the direction from A to B.
if (FB.gt.FA) then
DUM = XA
XA = XB
XB = DUM
DUM = FB
FB = FA
FA = DUM
end if
! First guess for C.
XC = XB + GOLD*(XB-XA)
FC = FUNC(CCTK_ARGUMENTS,XC)
! Check if the function is constant.
if ((FA.eq.FB).and.(FB.eq.FC)) then
return
end if
! Keep returning here until we bracket.
do while(FB.gt.FC)
! Compute U by parabollic extrapolation from A, B, C.
! TINY is used to prevent any possible division by zero.
S = (XB-XA)*(FB-FC)
Q = (XB-XC)*(FB-FA)
U = XB - ((XB-XC)*Q - (XB-XA)*S)
. / (2.0D0*SIGN(MAX(ABS(Q-S),TINY),Q-S))
! We will not go further than this.
ULIM = XB + GLIMIT*(XC-XB)
! Now to test various possibilities.
if ((XB-U)*(U-XC).gt.0.0D0) then
! Parabolic U is between B and C, try it.
FU = FUNC(CCTK_ARGUMENTS,U)
if (FU.lt.FC) then
! Got a minimum between B and C.
XA = XB
FA = FB
XB = U
FB = FU
return
else if (FU.gt.FB) then
! Got a minimum between A and U.
XC = U
FC = FU
return
end if
! Parabolic fit was no use. Use default magnification.
U = XC + GOLD*(XC-XB)
FU = FUNC(CCTK_ARGUMENTS,U)
else if ((XC-U)*(U-ULIM).gt.0.0D0) then
! Parabolic fit is between C and its allowed limit.
FU = FUNC(CCTK_ARGUMENTS,U)
if (FU.lt.FC) then
XB = XC
XC = U
U = XC + GOLD*(XC-XB)
FB = FC
FC = FU
FU = FUNC(CCTK_ARGUMENTS,U)
end if
else if ((U-ULIM)*(ULIM-XC).ge.0.0D0) then
! Limit parabolic U to maximum allowed value.
U = ULIM
FU = FUNC(CCTK_ARGUMENTS,U)
else
! Reject parabolic U, use default magnification.
U = XC + GOLD*(XC-XB)
FU = FUNC(CCTK_ARGUMENTS,U)
end if
! Eliminate oldest point and continue.
XA = XB
XB = XC
XC = U
FA = FB
FB = FC
FC = FU
end do
! ***************
! *** END ***
! ***************
end subroutine MNBRAK
***********************************************************************
***********************************************************************
CCTK_REAL function PARABOLA(CCTK_ARGUMENTS,XA,XB,XC,FXA,FXB,FXC,
. FUNC,TOL)
! Given a function F, and given a bracketing triplet of abscissas
! XA, XB, XC (such that XB is between XA and XC, and F(XB) is less
! than both F(XA) and F(XC)), this routine isolates the minimum
! to a fractional precision of about TOL using inverse parabolic
! interpolation. The abscissa of the minimum is returned as XMIN,
! and the minimum function value is returned as PARABOLA, the
! returned function value.
use F1COM
implicit none
DECLARE_CCTK_ARGUMENTS
integer ITER,ITMAX
CCTK_REAL XA,XB,XC,FXA,FXB,FXC,FUNC,TOL
CCTK_REAL A,B,D,P,Q,S,U,XX,XM,TOL1
CCTK_REAL FA,FB,FU,FX,FP,ZEPS
parameter (ITMAX=100,ZEPS=1.0D-10)
! *************************
! *** START ROUTINE ***
! *************************
error = .false.
! A and B must be in ascending order, though the input
! abscissas need not be.
if (XC.lt.XA) then
A = XC
B = XA
FA = FXC
FB = FXA
else
A = XA
B = XC
FA = FXA
FB = FXC
end if
XX = XB
FX = FXB
FP = FA
! Main program loop.
do ITER=1,ITMAX
XM = 0.5D0*(A + B)
! Test for done here.
TOL1 = 0.5D0*TOL*(abs(FP)+abs(FX)) + ZEPS
if (abs(FP-FX).le.TOL1) goto 3
! Construct a trial parabolic fit.
S = (XX-A)*(FX-FB)
Q = (XX-B)*(FX-FA)
P = (XX-B)*Q - (XX-A)*S
Q = 2.0D0*(Q-S)
! Check if Q is zero. This can only happen if the
! three points are colinear (in which case they where
! not bracketing a minimum in the first place), or
! if two of them are in fact the same point.
if (Q.eq.0.0D0) then
write(*,*) 'AHFinder: problem in PARABOLA'
error = .true.
return
end if
! Find new point.
D = - P/Q
U = XX + D
! Check if X was already the minimum.
if (abs(U-XX).lt.ZEPS) goto 3
! This is the one function evaluation per iteration.
FU = FUNC(CCTK_ARGUMENTS,U)
! And now we have to decide what to do with our function
! evaluation.
if (FU.le.FX) then
if (U.ge.XX) then
A = XX
FA = FX
else
B = XX
FB = FX
end if
XX = U
FP = FX
FX = FU
else
if (U.lt.XX) then
A = U
FA = FU
else
B = U
FB = FU
end if
end if
end do
write(*,*) 'PARABOLA exceeded maximum number of iterations.'
3 XMIN = XX
PARABOLA = FX
! ***************
! *** END ***
! ***************
end function PARABOLA
***********************************************************************
***********************************************************************
CCTK_REAL function FUNC(CCTK_ARGUMENTS,P,N)
! This is a shell to find the surface integral of the expansion.
use AHFinder_dat
implicit none
DECLARE_CCTK_ARGUMENTS
DECLARE_CCTK_PARAMETERS
logical flag
integer i,l,m,N
integer iter
CCTK_REAL zero,half,one
CCTK_REAL aux
CCTK_REAL, dimension(1:N) :: P
character(len=200) :: logf
save iter,logf
! **************************
! *** DEFINE NUMBERS ***
! **************************
zero = 0.0D0
half = 0.5D0
one = 1.0D0
! ************************
! *** OPEN LOGFILE ***
! ************************
logf = filestr(1:nfile)//"/ahf_logfile"
if (logfile.and.(myproc.eq.0)) then
open(12,file=logf,form='formatted',status='old',
. position='append')
end if
! ****************************
! *** FIND {c0,cc,cs} ***
! ****************************
i = 0
! Find {xc,yc,zc}.
if (wander) then
if (.not.refx) then
i = i+1
xc = P(1)
end if
if (.not.refy) then
i = i+1
yc = P(2)
end if
if (.not.refz) then
i = i+1
zc = P(3)
end if
end if
! Find c0(l).
do l=0,lmax,1+stepz
i = i+1
c0(l) = P(i)
end do
! Find {cc,cs}.
if (nonaxi) then
! Find cc(l,m).
do l=1,lmax
do m=1+stepx,l,1+stepx
if (stepz*mod(l-m,2).eq.0) then
i = i+1
cc(l,m) = P(i)
end if
end do
end do
! Find cs(l,m).
if (.not.refy) then
do l=1,lmax
do m=1,l,1+stepx
if (stepz*mod(l-m,2).eq.0) then
i = i+1
cs(l,m) = P(i)
end if
end do
end do
end if
end if
! *********************************************
! *** CALCULATE EXPANSION IF WE NEED TO ***
! *********************************************
! Firstcall?
if (firstfun) then
iter = 1
! Find horizon function and expansion.
call AHFinder_fun(CCTK_ARGUMENTS)
call AHFinder_exp(CCTK_ARGUMENTS)
! Save old values of coefficients.
xc_old = xc
yc_old = yc
zc_old = zc
c0_old = c0
cc_old = cc
cs_old = cs
firstfun = .false.
! Not first call.
else
iter = iter + 1
! Check if coefficients changed from last call.
! I check this because the minimization algorithm
! sometimes reevaluates the function at the same
! point twice. Also, if only c0(0) changed, the
! expansion will not be affected.
flag = .false.
if (wander) then
if (xc.ne.xc_old) flag = .true.
if (yc.ne.yc_old) flag = .true.
if (zc.ne.zc_old) flag = .true.
end if
do l=1+stepz,lmax,1+stepz
if (c0(l).ne.c0_old(l)) flag = .true.
end do
if (nonaxi) then
do l=1,lmax
do m=1+stepx,l,1+stepx
if (stepz*mod(l-m,2).eq.0) then
if (cc(l,m).ne.cc_old(l,m)) flag = .true.
end if
end do
if (.not.refy) then
do m=1,l,1+stepx
if (stepz*mod(l-m,2).eq.0) then
if (cs(l,m).ne.cs_old(l,m)) flag = .true.
end if
end do
end if
end do
end if
! If necessary, find horizon function and expansion.
if (flag) then
call AHFinder_fun(CCTK_ARGUMENTS)
call AHFinder_exp(CCTK_ARGUMENTS)
end if
! Save old values of coefficients.
c0_old = c0
cc_old = cc
cs_old = cs
end if
if (myproc.eq.0) then
if (veryver) then
write(*,*)
write(*,"(A13,I5)") 'FUNC CALL =',iter
end if
if (logfile) then
write(12,*)
write(12,"(A13,I5)") 'FUNC CALL =',iter
end if
end if
! **********************************
! *** FIND SURFACE INTEGRALS ***
! **********************************
call AHFinder_int(CCTK_ARGUMENTS)
! Messages to screen and logfile.
if (myproc.eq.0) then
! Find inverse of area. Be carefull not to divide by zero!
! Also, a value of 1.0D10 indicates an error in the integration
! routine, and all integrals end up with the same value.
! So dont divide by it or I will be confused later.
if ((intarea.ne.zero).and.(intarea.ne.1.0D10)) then
aux = one/intarea
else
aux = one
end if
! Screen.
if (veryver) then
write(*,*)
write(*,"(A21,ES14.6)") ' Surface area =',intarea
write(*,"(A21,ES14.6)") ' Mean value of H =',aux*intexp
write(*,"(A21,ES14.6)") ' Mean value of H^2 =',aux*intexp2
write(*,*) 'Number of interpolated points: ',
. inside_min_count
write(*,*) 'Number of those that are negative: ',
. inside_min_neg_count
end if
! Logfile.
if (logfile) then
write(12,*)
write(12,"(A21,ES14.6)") ' Surface area =',intarea
write(12,"(A21,ES14.6)") ' Mean value of H =',aux*intexp
write(12,"(A21,ES14.6)") ' Mean value of H^2 =',aux*intexp2
write(12,*) 'Number of interpolated points: ',
. inside_min_count
write(12,*) 'Number of those that are negative: ',
. inside_min_neg_count
end if
end if
! ******************************
! *** WRITE COEFFICIENTS ***
! ******************************
if (myproc.eq.0) then
! Headers to screen.
if (veryver) then
write(*,*)
write(*,"(A20)") ' Shape coefficients:'
if (offset) then
write(*,*)
write(*,"(A8,ES14.6)") ' xc =',xc
write(*,"(A8,ES14.6)") ' yc =',yc
write(*,"(A8,ES14.6)") ' zc =',zc
end if
write(*,*)
write(*,*) ' l c0_l'
write(*,*)
write(*,"(I4,A6,ES14.6)") 0,' ',c0(0)
end if
! Headers to logfile.
if (logfile) then
write(12,*)
write(12,"(A20)") ' Shape coefficients:'
if (offset) then
write(12,*)
write(12,"(A8,ES14.6)") ' xc =',xc
write(12,"(A8,ES14.6)") ' yc =',yc
write(12,"(A8,ES14.6)") ' zc =',zc
end if
write(12,*)
write(12,*) ' l c0_l'
write(12,*)
write(12,"(I4,A6,ES14.6)") 0,' ',c0(0)
end if
! c0 coefficients.
do l=1+stepz,lmax,1+stepz
if (veryver) then
write(*,"(I4,A6,ES14.6)") l,' ',c0(l)
end if
if (logfile) then
write(12,"(I4,A6,ES14.6)") l,' ',c0(l)
end if
end do
! cc and cs coefficients (only for nonaxisymmetric case).
if (nonaxi) then
! Reflection symmetry on y: some coefficients are not there.
if (.not.refy) then
if (veryver) then
write(*,*)
write(*,*) ' l m cc_lm cs_lm'
write(*,*)
end if
if (logfile) then
write(12,*)
write(12,*) ' l m cc_lm cs_lm'
write(12,*)
end if
do l=1,lmax
do m=1,l
if (stepz*mod(l-m,2).eq.0) then
if (veryver) then
write(*,10) l,m,' ',cc(l,m),' ',cs(l,m)
end if
if (logfile) then
write(12,10) l,m,' ',cc(l,m),' ',cs(l,m)
end if
10 format(I4,I4,A2,ES14.6,A1,ES14.6)
end if
end do
end do
! No reflection symmetry on y.
else
if (veryver) then
write(*,*)
write(*,*) ' l m cc_lm'
write(*,*)
end if
if (logfile) then
write(12,*)
write(12,*) ' l m cc_lm'
write(12,*)
end if
do l=1,lmax
do m=1+stepx,l,1+stepx
if (stepz*mod(l-m,2).eq.0) then
if (veryver) then
write(*,20) l,m,' ',cc(l,m)
end if
if (logfile) then
write(12,20) l,m,' ',cc(l,m)
end if
20 format(I4,I4,A2,ES14.6)
end if
end do
end do
end if
end if
end if
! *********************
! *** FIND FUNC ***
! *********************
! The value of FUNC will be equal to the surface integral
! of the square of the expansion or to the surface area,
! depending on what we want to minimize.
if (minarea) then
FUNC = intarea
else
if(find_trapped_surface) then
FUNC = intexpdel2
else
FUNC = intexp2
endif
end if
! *************************
! *** CLOSE LOGFILE ***
! *************************
if (logfile.and.(myproc.eq.0)) then
close(12)
end if
! ***************
! *** END ***
! ***************
end function FUNC
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