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C version modif by F. HECHT
C ---------------------------
C The code in fortran is totally free, and the interface is under LGPL
C http://www.inrialpes.fr/bipop/people/guilbert/newuoa/newuoa.pdf
C original source :
C http://www.inrialpees.fr/bipop/people/guilbert/newuoa/newuoa.tar.gz
C --------------------
SUBROUTINE BIGDEN (N,NPT,XOPT,XPT,BMAT,ZMAT,IDZ,NDIM,KOPT,
1 KNEW,D,W,VLAG,BETA,S,WVEC,PROD)
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION XOPT(*),XPT(NPT,*),BMAT(NDIM,*),ZMAT(NPT,*),D(*),
1 W(*),VLAG(*),S(*),WVEC(NDIM,*),PROD(NDIM,*)
DIMENSION DEN(9),DENEX(9),PAR(9)
C
C N is the number of variables.
C NPT is the number of interpolation equations.
C XOPT is the best interpolation point so far.
C XPT contains the coordinates of the current interpolation points.
C BMAT provides the last N columns of H.
C ZMAT and IDZ give a factorization of the first NPT by NPT submatrix of H.
C NDIM is the first dimension of BMAT and has the value NPT+N.
C KOPT is the index of the optimal interpolation point.
C KNEW is the index of the interpolation point that is going to be moved.
C D will be set to the step from XOPT to the new point, and on entry it
C should be the D that was calculated by the last call of BIGLAG. The
C length of the initial D provides a trust region bound on the final D.
C W will be set to Wcheck for the final choice of D.
C VLAG will be set to Theta*Wcheck+e_b for the final choice of D.
C BETA will be set to the value that will occur in the updating formula
C when the KNEW-th interpolation point is moved to its new position.
C S, WVEC, PROD and the private arrays DEN, DENEX and PAR will be used
C for working space.
C
C D is calculated in a way that should provide a denominator with a large
C modulus in the updating formula when the KNEW-th interpolation point is
C shifted to the new position XOPT+D.
C
C Set some constants.
C
HALF=0.5D0
ONE=1.0D0
QUART=0.25D0
TWO=2.0D0
ZERO=0.0D0
TWOPI=8.0D0*DATAN(ONE)
NPTM=NPT-N-1
C
C Store the first NPT elements of the KNEW-th column of H in W(N+1)
C to W(N+NPT).
C
DO 10 K=1,NPT
10 W(N+K)=ZERO
DO 20 J=1,NPTM
TEMP=ZMAT(KNEW,J)
IF (J .LT. IDZ) TEMP=-TEMP
DO 20 K=1,NPT
20 W(N+K)=W(N+K)+TEMP*ZMAT(K,J)
ALPHA=W(N+KNEW)
C
C The initial search direction D is taken from the last call of BIGLAG,
C and the initial S is set below, usually to the direction from X_OPT
C to X_KNEW, but a different direction to an interpolation point may
C be chosen, in order to prevent S from being nearly parallel to D.
C
DD=ZERO
DS=ZERO
SS=ZERO
XOPTSQ=ZERO
DO 30 I=1,N
DD=DD+D(I)**2
S(I)=XPT(KNEW,I)-XOPT(I)
DS=DS+D(I)*S(I)
SS=SS+S(I)**2
30 XOPTSQ=XOPTSQ+XOPT(I)**2
IF (DS*DS .GT. 0.99D0*DD*SS) THEN
KSAV=KNEW
DTEST=DS*DS/SS
DO 50 K=1,NPT
IF (K .NE. KOPT) THEN
DSTEMP=ZERO
SSTEMP=ZERO
DO 40 I=1,N
DIFF=XPT(K,I)-XOPT(I)
DSTEMP=DSTEMP+D(I)*DIFF
40 SSTEMP=SSTEMP+DIFF*DIFF
IF (DSTEMP*DSTEMP/SSTEMP .LT. DTEST) THEN
KSAV=K
DTEST=DSTEMP*DSTEMP/SSTEMP
DS=DSTEMP
SS=SSTEMP
END IF
END IF
50 CONTINUE
DO 60 I=1,N
60 S(I)=XPT(KSAV,I)-XOPT(I)
END IF
SSDEN=DD*SS-DS*DS
ITERC=0
DENSAV=ZERO
C
C Begin the iteration by overwriting S with a vector that has the
C required length and direction.
C
70 ITERC=ITERC+1
TEMP=ONE/DSQRT(SSDEN)
XOPTD=ZERO
XOPTS=ZERO
DO 80 I=1,N
S(I)=TEMP*(DD*S(I)-DS*D(I))
XOPTD=XOPTD+XOPT(I)*D(I)
80 XOPTS=XOPTS+XOPT(I)*S(I)
C
C Set the coefficients of the first two terms of BETA.
C
TEMPA=HALF*XOPTD*XOPTD
TEMPB=HALF*XOPTS*XOPTS
DEN(1)=DD*(XOPTSQ+HALF*DD)+TEMPA+TEMPB
DEN(2)=TWO*XOPTD*DD
DEN(3)=TWO*XOPTS*DD
DEN(4)=TEMPA-TEMPB
DEN(5)=XOPTD*XOPTS
DO 90 I=6,9
90 DEN(I)=ZERO
C
C Put the coefficients of Wcheck in WVEC.
C
DO 110 K=1,NPT
TEMPA=ZERO
TEMPB=ZERO
TEMPC=ZERO
DO 100 I=1,N
TEMPA=TEMPA+XPT(K,I)*D(I)
TEMPB=TEMPB+XPT(K,I)*S(I)
100 TEMPC=TEMPC+XPT(K,I)*XOPT(I)
WVEC(K,1)=QUART*(TEMPA*TEMPA+TEMPB*TEMPB)
WVEC(K,2)=TEMPA*TEMPC
WVEC(K,3)=TEMPB*TEMPC
WVEC(K,4)=QUART*(TEMPA*TEMPA-TEMPB*TEMPB)
110 WVEC(K,5)=HALF*TEMPA*TEMPB
DO 120 I=1,N
IP=I+NPT
WVEC(IP,1)=ZERO
WVEC(IP,2)=D(I)
WVEC(IP,3)=S(I)
WVEC(IP,4)=ZERO
120 WVEC(IP,5)=ZERO
C
C Put the coefficents of THETA*Wcheck in PROD.
C
DO 190 JC=1,5
NW=NPT
IF (JC .EQ. 2 .OR. JC .EQ. 3) NW=NDIM
DO 130 K=1,NPT
130 PROD(K,JC)=ZERO
DO 150 J=1,NPTM
SUM=ZERO
DO 140 K=1,NPT
140 SUM=SUM+ZMAT(K,J)*WVEC(K,JC)
IF (J .LT. IDZ) SUM=-SUM
DO 150 K=1,NPT
150 PROD(K,JC)=PROD(K,JC)+SUM*ZMAT(K,J)
IF (NW .EQ. NDIM) THEN
DO 170 K=1,NPT
SUM=ZERO
DO 160 J=1,N
160 SUM=SUM+BMAT(K,J)*WVEC(NPT+J,JC)
170 PROD(K,JC)=PROD(K,JC)+SUM
END IF
DO 190 J=1,N
SUM=ZERO
DO 180 I=1,NW
180 SUM=SUM+BMAT(I,J)*WVEC(I,JC)
190 PROD(NPT+J,JC)=SUM
C
C Include in DEN the part of BETA that depends on THETA.
C
DO 210 K=1,NDIM
SUM=ZERO
DO 200 I=1,5
PAR(I)=HALF*PROD(K,I)*WVEC(K,I)
200 SUM=SUM+PAR(I)
DEN(1)=DEN(1)-PAR(1)-SUM
TEMPA=PROD(K,1)*WVEC(K,2)+PROD(K,2)*WVEC(K,1)
TEMPB=PROD(K,2)*WVEC(K,4)+PROD(K,4)*WVEC(K,2)
TEMPC=PROD(K,3)*WVEC(K,5)+PROD(K,5)*WVEC(K,3)
DEN(2)=DEN(2)-TEMPA-HALF*(TEMPB+TEMPC)
DEN(6)=DEN(6)-HALF*(TEMPB-TEMPC)
TEMPA=PROD(K,1)*WVEC(K,3)+PROD(K,3)*WVEC(K,1)
TEMPB=PROD(K,2)*WVEC(K,5)+PROD(K,5)*WVEC(K,2)
TEMPC=PROD(K,3)*WVEC(K,4)+PROD(K,4)*WVEC(K,3)
DEN(3)=DEN(3)-TEMPA-HALF*(TEMPB-TEMPC)
DEN(7)=DEN(7)-HALF*(TEMPB+TEMPC)
TEMPA=PROD(K,1)*WVEC(K,4)+PROD(K,4)*WVEC(K,1)
DEN(4)=DEN(4)-TEMPA-PAR(2)+PAR(3)
TEMPA=PROD(K,1)*WVEC(K,5)+PROD(K,5)*WVEC(K,1)
TEMPB=PROD(K,2)*WVEC(K,3)+PROD(K,3)*WVEC(K,2)
DEN(5)=DEN(5)-TEMPA-HALF*TEMPB
DEN(8)=DEN(8)-PAR(4)+PAR(5)
TEMPA=PROD(K,4)*WVEC(K,5)+PROD(K,5)*WVEC(K,4)
210 DEN(9)=DEN(9)-HALF*TEMPA
C
C Extend DEN so that it holds all the coefficients of DENOM.
C
SUM=ZERO
DO 220 I=1,5
PAR(I)=HALF*PROD(KNEW,I)**2
220 SUM=SUM+PAR(I)
DENEX(1)=ALPHA*DEN(1)+PAR(1)+SUM
TEMPA=TWO*PROD(KNEW,1)*PROD(KNEW,2)
TEMPB=PROD(KNEW,2)*PROD(KNEW,4)
TEMPC=PROD(KNEW,3)*PROD(KNEW,5)
DENEX(2)=ALPHA*DEN(2)+TEMPA+TEMPB+TEMPC
DENEX(6)=ALPHA*DEN(6)+TEMPB-TEMPC
TEMPA=TWO*PROD(KNEW,1)*PROD(KNEW,3)
TEMPB=PROD(KNEW,2)*PROD(KNEW,5)
TEMPC=PROD(KNEW,3)*PROD(KNEW,4)
DENEX(3)=ALPHA*DEN(3)+TEMPA+TEMPB-TEMPC
DENEX(7)=ALPHA*DEN(7)+TEMPB+TEMPC
TEMPA=TWO*PROD(KNEW,1)*PROD(KNEW,4)
DENEX(4)=ALPHA*DEN(4)+TEMPA+PAR(2)-PAR(3)
TEMPA=TWO*PROD(KNEW,1)*PROD(KNEW,5)
DENEX(5)=ALPHA*DEN(5)+TEMPA+PROD(KNEW,2)*PROD(KNEW,3)
DENEX(8)=ALPHA*DEN(8)+PAR(4)-PAR(5)
DENEX(9)=ALPHA*DEN(9)+PROD(KNEW,4)*PROD(KNEW,5)
C
C Seek the value of the angle that maximizes the modulus of DENOM.
C
SUM=DENEX(1)+DENEX(2)+DENEX(4)+DENEX(6)+DENEX(8)
DENOLD=SUM
DENMAX=SUM
ISAVE=0
IU=49
TEMP=TWOPI/DFLOAT(IU+1)
PAR(1)=ONE
DO 250 I=1,IU
ANGLE=DFLOAT(I)*TEMP
PAR(2)=DCOS(ANGLE)
PAR(3)=DSIN(ANGLE)
DO 230 J=4,8,2
PAR(J)=PAR(2)*PAR(J-2)-PAR(3)*PAR(J-1)
230 PAR(J+1)=PAR(2)*PAR(J-1)+PAR(3)*PAR(J-2)
SUMOLD=SUM
SUM=ZERO
DO 240 J=1,9
240 SUM=SUM+DENEX(J)*PAR(J)
IF (DABS(SUM) .GT. DABS(DENMAX)) THEN
DENMAX=SUM
ISAVE=I
TEMPA=SUMOLD
ELSE IF (I .EQ. ISAVE+1) THEN
TEMPB=SUM
END IF
250 CONTINUE
IF (ISAVE .EQ. 0) TEMPA=SUM
IF (ISAVE .EQ. IU) TEMPB=DENOLD
STEP=ZERO
IF (TEMPA .NE. TEMPB) THEN
TEMPA=TEMPA-DENMAX
TEMPB=TEMPB-DENMAX
STEP=HALF*(TEMPA-TEMPB)/(TEMPA+TEMPB)
END IF
ANGLE=TEMP*(DFLOAT(ISAVE)+STEP)
C
C Calculate the new parameters of the denominator, the new VLAG vector
C and the new D. Then test for convergence.
C
PAR(2)=DCOS(ANGLE)
PAR(3)=DSIN(ANGLE)
DO 260 J=4,8,2
PAR(J)=PAR(2)*PAR(J-2)-PAR(3)*PAR(J-1)
260 PAR(J+1)=PAR(2)*PAR(J-1)+PAR(3)*PAR(J-2)
BETA=ZERO
DENMAX=ZERO
DO 270 J=1,9
BETA=BETA+DEN(J)*PAR(J)
270 DENMAX=DENMAX+DENEX(J)*PAR(J)
DO 280 K=1,NDIM
VLAG(K)=ZERO
DO 280 J=1,5
280 VLAG(K)=VLAG(K)+PROD(K,J)*PAR(J)
TAU=VLAG(KNEW)
DD=ZERO
TEMPA=ZERO
TEMPB=ZERO
DO 290 I=1,N
D(I)=PAR(2)*D(I)+PAR(3)*S(I)
W(I)=XOPT(I)+D(I)
DD=DD+D(I)**2
TEMPA=TEMPA+D(I)*W(I)
290 TEMPB=TEMPB+W(I)*W(I)
IF (ITERC .GE. N) GOTO 340
IF (ITERC .GT. 1) DENSAV=DMAX1(DENSAV,DENOLD)
IF (DABS(DENMAX) .LE. 1.1D0*DABS(DENSAV)) GOTO 340
DENSAV=DENMAX
C
C Set S to half the gradient of the denominator with respect to D.
C Then branch for the next iteration.
C
DO 300 I=1,N
TEMP=TEMPA*XOPT(I)+TEMPB*D(I)-VLAG(NPT+I)
300 S(I)=TAU*BMAT(KNEW,I)+ALPHA*TEMP
DO 320 K=1,NPT
SUM=ZERO
DO 310 J=1,N
310 SUM=SUM+XPT(K,J)*W(J)
TEMP=(TAU*W(N+K)-ALPHA*VLAG(K))*SUM
DO 320 I=1,N
320 S(I)=S(I)+TEMP*XPT(K,I)
SS=ZERO
DS=ZERO
DO 330 I=1,N
SS=SS+S(I)**2
330 DS=DS+D(I)*S(I)
SSDEN=DD*SS-DS*DS
IF (SSDEN .GE. 1.0D-8*DD*SS) GOTO 70
C
C Set the vector W before the RETURN from the subroutine.
C
340 DO 350 K=1,NDIM
W(K)=ZERO
DO 350 J=1,5
350 W(K)=W(K)+WVEC(K,J)*PAR(J)
VLAG(KOPT)=VLAG(KOPT)+ONE
RETURN
END
SUBROUTINE BIGLAG (N,NPT,XOPT,XPT,BMAT,ZMAT,IDZ,NDIM,KNEW,
1 DELTA,D,ALPHA,HCOL,GC,GD,S,W)
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION XOPT(*),XPT(NPT,*),BMAT(NDIM,*),ZMAT(NPT,*),D(*),
1 HCOL(*),GC(*),GD(*),S(*),W(*)
C
C N is the number of variables.
C NPT is the number of interpolation equations.
C XOPT is the best interpolation point so far.
C XPT contains the coordinates of the current interpolation points.
C BMAT provides the last N columns of H.
C ZMAT and IDZ give a factorization of the first NPT by NPT submatrix of H.
C NDIM is the first dimension of BMAT and has the value NPT+N.
C KNEW is the index of the interpolation point that is going to be moved.
C DELTA is the current trust region bound.
C D will be set to the step from XOPT to the new point.
C ALPHA will be set to the KNEW-th diagonal element of the H matrix.
C HCOL, GC, GD, S and W will be used for working space.
C
C The step D is calculated in a way that attempts to maximize the modulus
C of LFUNC(XOPT+D), subject to the bound ||D|| .LE. DELTA, where LFUNC is
C the KNEW-th Lagrange function.
C
C Set some constants.
C
HALF=0.5D0
ONE=1.0D0
ZERO=0.0D0
TWOPI=8.0D0*DATAN(ONE)
DELSQ=DELTA*DELTA
NPTM=NPT-N-1
C
C Set the first NPT components of HCOL to the leading elements of the
C KNEW-th column of H.
C
ITERC=0
DO 10 K=1,NPT
10 HCOL(K)=ZERO
DO 20 J=1,NPTM
TEMP=ZMAT(KNEW,J)
IF (J .LT. IDZ) TEMP=-TEMP
DO 20 K=1,NPT
20 HCOL(K)=HCOL(K)+TEMP*ZMAT(K,J)
ALPHA=HCOL(KNEW)
C
C Set the unscaled initial direction D. Form the gradient of LFUNC at
C XOPT, and multiply D by the second derivative matrix of LFUNC.
C
DD=ZERO
DO 30 I=1,N
D(I)=XPT(KNEW,I)-XOPT(I)
GC(I)=BMAT(KNEW,I)
GD(I)=ZERO
30 DD=DD+D(I)**2
DO 50 K=1,NPT
TEMP=ZERO
SUM=ZERO
DO 40 J=1,N
TEMP=TEMP+XPT(K,J)*XOPT(J)
40 SUM=SUM+XPT(K,J)*D(J)
TEMP=HCOL(K)*TEMP
SUM=HCOL(K)*SUM
DO 50 I=1,N
GC(I)=GC(I)+TEMP*XPT(K,I)
50 GD(I)=GD(I)+SUM*XPT(K,I)
C
C Scale D and GD, with a sign change if required. Set S to another
C vector in the initial two dimensional subspace.
C
GG=ZERO
SP=ZERO
DHD=ZERO
DO 60 I=1,N
GG=GG+GC(I)**2
SP=SP+D(I)*GC(I)
60 DHD=DHD+D(I)*GD(I)
SCALE=DELTA/DSQRT(DD)
IF (SP*DHD .LT. ZERO) SCALE=-SCALE
TEMP=ZERO
IF (SP*SP .GT. 0.99D0*DD*GG) TEMP=ONE
TAU=SCALE*(DABS(SP)+HALF*SCALE*DABS(DHD))
IF (GG*DELSQ .LT. 0.01D0*TAU*TAU) TEMP=ONE
DO 70 I=1,N
D(I)=SCALE*D(I)
GD(I)=SCALE*GD(I)
70 S(I)=GC(I)+TEMP*GD(I)
C
C Begin the iteration by overwriting S with a vector that has the
C required length and direction, except that termination occurs if
C the given D and S are nearly parallel.
C
80 ITERC=ITERC+1
DD=ZERO
SP=ZERO
SS=ZERO
DO 90 I=1,N
DD=DD+D(I)**2
SP=SP+D(I)*S(I)
90 SS=SS+S(I)**2
TEMP=DD*SS-SP*SP
IF (TEMP .LE. 1.0D-8*DD*SS) GOTO 160
DENOM=DSQRT(TEMP)
DO 100 I=1,N
S(I)=(DD*S(I)-SP*D(I))/DENOM
100 W(I)=ZERO
C
C Calculate the coefficients of the objective function on the circle,
C beginning with the multiplication of S by the second derivative matrix.
C
DO 120 K=1,NPT
SUM=ZERO
DO 110 J=1,N
110 SUM=SUM+XPT(K,J)*S(J)
SUM=HCOL(K)*SUM
DO 120 I=1,N
120 W(I)=W(I)+SUM*XPT(K,I)
CF1=ZERO
CF2=ZERO
CF3=ZERO
CF4=ZERO
CF5=ZERO
DO 130 I=1,N
CF1=CF1+S(I)*W(I)
CF2=CF2+D(I)*GC(I)
CF3=CF3+S(I)*GC(I)
CF4=CF4+D(I)*GD(I)
130 CF5=CF5+S(I)*GD(I)
CF1=HALF*CF1
CF4=HALF*CF4-CF1
C
C Seek the value of the angle that maximizes the modulus of TAU.
C
TAUBEG=CF1+CF2+CF4
TAUMAX=TAUBEG
TAUOLD=TAUBEG
ISAVE=0
IU=49
TEMP=TWOPI/DFLOAT(IU+1)
DO 140 I=1,IU
ANGLE=DFLOAT(I)*TEMP
CTH=DCOS(ANGLE)
STH=DSIN(ANGLE)
TAU=CF1+(CF2+CF4*CTH)*CTH+(CF3+CF5*CTH)*STH
IF (DABS(TAU) .GT. DABS(TAUMAX)) THEN
TAUMAX=TAU
ISAVE=I
TEMPA=TAUOLD
ELSE IF (I .EQ. ISAVE+1) THEN
TEMPB=TAU
END IF
140 TAUOLD=TAU
IF (ISAVE .EQ. 0) TEMPA=TAU
IF (ISAVE .EQ. IU) TEMPB=TAUBEG
STEP=ZERO
IF (TEMPA .NE. TEMPB) THEN
TEMPA=TEMPA-TAUMAX
TEMPB=TEMPB-TAUMAX
STEP=HALF*(TEMPA-TEMPB)/(TEMPA+TEMPB)
END IF
ANGLE=TEMP*(DFLOAT(ISAVE)+STEP)
C
C Calculate the new D and GD. Then test for convergence.
C
CTH=DCOS(ANGLE)
STH=DSIN(ANGLE)
TAU=CF1+(CF2+CF4*CTH)*CTH+(CF3+CF5*CTH)*STH
DO 150 I=1,N
D(I)=CTH*D(I)+STH*S(I)
GD(I)=CTH*GD(I)+STH*W(I)
150 S(I)=GC(I)+GD(I)
IF (DABS(TAU) .LE. 1.1D0*DABS(TAUBEG)) GOTO 160
IF (ITERC .LT. N) GOTO 80
160 RETURN
END
FUNCTION NEWUOA (N,NPT,X,RHOBEG,RHOEND,IPRINT,MAXFUN,W,IWF,
1 CALFUN)
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION X(*),W(*),IWF(*)
EXTERNAL CALFUN
real*8 NEWUOB, NEWUOA
C
C This subroutine seeks the least value of a function of many variables,
C by a trust region method that forms quadratic models by interpolation.
C There can be some freedom in the interpolation conditions, which is
C taken up by minimizing the Frobenius norm of the change to the second
C derivative of the quadratic model, beginning with a zero matrix. The
C arguments of the subroutine are as follows.
C
C N must be set to the number of variables and must be at least two.
C NPT is the number of interpolation conditions. Its value must be in the
C interval [N+2,(N+1)(N+2)/2].
C Initial values of the variables must be set in X(1),X(2),...,X(N). They
C will be changed to the values that give the least calculated F.
C RHOBEG and RHOEND must be set to the initial and final values of a trust
C region radius, so both must be positive with RHOEND<=RHOBEG. Typically
C RHOBEG should be about one tenth of the greatest expected change to a
C variable, and RHOEND should indicate the accuracy that is required in
C the final values of the variables.
C The value of IPRINT should be set to 0, 1, 2 or 3, which controls the
C amount of printing. Specifically, there is no output if IPRINT=0 and
C there is output only at the return if IPRINT=1. Otherwise, each new
C value of RHO is printed, with the best vector of variables so far and
C the corresponding value of the objective function. Further, each new
C value of F with its variables are output if IPRINT=3.
C MAXFUN must be set to an upper bound on the number of calls of CALFUN.
C The array W will be used for working space. Its length must be at least
C (NPT+13)*(NPT+N)+3*N*(N+3)/2.
C
C SUBROUTINE CALFUN (N,X,F) must be provided by the user. It must set F to
C the value of the objective function for the variables X(1),X(2),...,X(N).
C
C Partition the working space array, so that different parts of it can be
C treated separately by the subroutine that performs the main calculation.
C
NP=N+1
NPTM=NPT-NP
IF (NPT .LT. N+2 .OR. NPT .GT. ((N+2)*NP)/2) THEN
PRINT 10
10 FORMAT (/4X,'Return from NEWUOA because NPT is not in',
1 ' the required interval')
GO TO 20
END IF
NDIM=NPT+N
IXB=1
IXO=IXB+N
IXN=IXO+N
IXP=IXN+N
IFV=IXP+N*NPT
IGQ=IFV+NPT
IHQ=IGQ+N
IPQ=IHQ+(N*NP)/2
IBMAT=IPQ+NPT
IZMAT=IBMAT+NDIM*N
ID=IZMAT+NPT*NPTM
IVL=ID+N
IW=IVL+NDIM
C
C The above settings provide a partition of W for subroutine NEWUOB.
C The partition requires the first NPT*(NPT+N)+5*N*(N+3)/2 elements of
C W plus the space that is needed by the last array of NEWUOB.
C
NEWUOA= NEWUOB (N,NPT,X,RHOBEG,RHOEND,IPRINT,MAXFUN,W(IXB),
1 W(IXO),W(IXN),W(IXP),W(IFV),W(IGQ),W(IHQ),W(IPQ),W(IBMAT),
2 W(IZMAT),NDIM,W(ID),W(IVL),W(IW),
3 IWF, CALFUN)
20 RETURN
END
FUNCTION NEWUOB (N,NPT,X,RHOBEG,RHOEND,IPRINT,MAXFUN,XBASE,
1 XOPT,XNEW,XPT,FVAL,GQ,HQ,PQ,BMAT,ZMAT,NDIM,D,VLAG,W,IWF,CALFUN)
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION X(*),XBASE(*),XOPT(*),XNEW(*),XPT(NPT,*),FVAL(*),
1 GQ(*),HQ(*),PQ(*),BMAT(NDIM,*),ZMAT(NPT,*),D(*),VLAG(*),W(*),
2 IWF(*)
EXTERNAL CALFUN
REAL*8 NEWUOB
C
C The arguments N, NPT, X, RHOBEG, RHOEND, IPRINT and MAXFUN are identical
C to the corresponding arguments in SUBROUTINE NEWUOA.
C XBASE will hold a shift of origin that should reduce the contributions
C from rounding errors to values of the model and Lagrange functions.
C XOPT will be set to the displacement from XBASE of the vector of
C variables that provides the least calculated F so far.
C XNEW will be set to the displacement from XBASE of the vector of
C variables for the current calculation of F.
C XPT will contain the interpolation point coordinates relative to XBASE.
C FVAL will hold the values of F at the interpolation points.
C GQ will hold the gradient of the quadratic model at XBASE.
C HQ will hold the explicit second derivatives of the quadratic model.
C PQ will contain the parameters of the implicit second derivatives of
C the quadratic model.
C BMAT will hold the last N columns of H.
C ZMAT will hold the factorization of the leading NPT by NPT submatrix of
C H, this factorization being ZMAT times Diag(DZ) times ZMAT^T, where
C the elements of DZ are plus or minus one, as specified by IDZ.
C NDIM is the first dimension of BMAT and has the value NPT+N.
C D is reserved for trial steps from XOPT.
C VLAG will contain the values of the Lagrange functions at a new point X.
C They are part of a product that requires VLAG to be of length NDIM.
C The array W will be used for working space. Its length must be at least
C 10*NDIM = 10*(NPT+N).
C
C Set some constants.
C
HALF=0.5D0
ONE=1.0D0
TENTH=0.1D0
ZERO=0.0D0
NP=N+1
NH=(N*NP)/2
NPTM=NPT-NP
NFTEST=MAX0(MAXFUN,1)
C
C Set the initial elements of XPT, BMAT, HQ, PQ and ZMAT to zero.
C
DO 20 J=1,N
XBASE(J)=X(J)
DO 10 K=1,NPT
10 XPT(K,J)=ZERO
DO 20 I=1,NDIM
20 BMAT(I,J)=ZERO
DO 30 IH=1,NH
30 HQ(IH)=ZERO
DO 40 K=1,NPT
PQ(K)=ZERO
DO 40 J=1,NPTM
40 ZMAT(K,J)=ZERO
C
C Begin the initialization procedure. NF becomes one more than the number
C of function values so far. The coordinates of the displacement of the
C next initial interpolation point from XBASE are set in XPT(NF,.).
C
RHOSQ=RHOBEG*RHOBEG
RECIP=ONE/RHOSQ
RECIQ=DSQRT(HALF)/RHOSQ
NF=0
50 NFM=NF
NFMM=NF-N
NF=NF+1
IF (NFM .LE. 2*N) THEN
IF (NFM .GE. 1 .AND. NFM .LE. N) THEN
XPT(NF,NFM)=RHOBEG
ELSE IF (NFM .GT. N) THEN
XPT(NF,NFMM)=-RHOBEG
END IF
ELSE
ITEMP=(NFMM-1)/N
JPT=NFM-ITEMP*N-N
IPT=JPT+ITEMP
IF (IPT .GT. N) THEN
ITEMP=JPT
JPT=IPT-N
IPT=ITEMP
END IF
XIPT=RHOBEG
IF (FVAL(IPT+NP) .LT. FVAL(IPT+1)) XIPT=-XIPT
XJPT=RHOBEG
IF (FVAL(JPT+NP) .LT. FVAL(JPT+1)) XJPT=-XJPT
XPT(NF,IPT)=XIPT
XPT(NF,JPT)=XJPT
END IF
C
C Calculate the next value of F, label 70 being reached immediately
C after this calculation. The least function value so far and its index
C are required.
C
DO 60 J=1,N
60 X(J)=XPT(NF,J)+XBASE(J)
GOTO 310
70 FVAL(NF)=F
IF (NF .EQ. 1) THEN
FBEG=F
FOPT=F
KOPT=1
ELSE IF (F .LT. FOPT) THEN
FOPT=F
KOPT=NF
END IF
C
C Set the nonzero initial elements of BMAT and the quadratic model in
C the cases when NF is at most 2*N+1.
C
IF (NFM .LE. 2*N) THEN
IF (NFM .GE. 1 .AND. NFM .LE. N) THEN
GQ(NFM)=(F-FBEG)/RHOBEG
IF (NPT .LT. NF+N) THEN
BMAT(1,NFM)=-ONE/RHOBEG
BMAT(NF,NFM)=ONE/RHOBEG
BMAT(NPT+NFM,NFM)=-HALF*RHOSQ
END IF
ELSE IF (NFM .GT. N) THEN
BMAT(NF-N,NFMM)=HALF/RHOBEG
BMAT(NF,NFMM)=-HALF/RHOBEG
ZMAT(1,NFMM)=-RECIQ-RECIQ
ZMAT(NF-N,NFMM)=RECIQ
ZMAT(NF,NFMM)=RECIQ
IH=(NFMM*(NFMM+1))/2
TEMP=(FBEG-F)/RHOBEG
HQ(IH)=(GQ(NFMM)-TEMP)/RHOBEG
GQ(NFMM)=HALF*(GQ(NFMM)+TEMP)
END IF
C
C Set the off-diagonal second derivatives of the Lagrange functions and
C the initial quadratic model.
C
ELSE
IH=(IPT*(IPT-1))/2+JPT
IF (XIPT .LT. ZERO) IPT=IPT+N
IF (XJPT .LT. ZERO) JPT=JPT+N
ZMAT(1,NFMM)=RECIP
ZMAT(NF,NFMM)=RECIP
ZMAT(IPT+1,NFMM)=-RECIP
ZMAT(JPT+1,NFMM)=-RECIP
HQ(IH)=(FBEG-FVAL(IPT+1)-FVAL(JPT+1)+F)/(XIPT*XJPT)
END IF
IF (NF .LT. NPT) GOTO 50
C
C Begin the iterative procedure, because the initial model is complete.
C
RHO=RHOBEG
DELTA=RHO
IDZ=1
DIFFA=ZERO
DIFFB=ZERO
ITEST=0
XOPTSQ=ZERO
DO 80 I=1,N
XOPT(I)=XPT(KOPT,I)
80 XOPTSQ=XOPTSQ+XOPT(I)**2
90 NFSAV=NF
C
C Generate the next trust region step and test its length. Set KNEW
C to -1 if the purpose of the next F will be to improve the model.
C
100 KNEW=0
CALL TRSAPP (N,NPT,XOPT,XPT,GQ,HQ,PQ,DELTA,D,W,W(NP),
1 W(NP+N),W(NP+2*N),CRVMIN)
DSQ=ZERO
DO 110 I=1,N
110 DSQ=DSQ+D(I)**2
DNORM=DMIN1(DELTA,DSQRT(DSQ))
IF (DNORM .LT. HALF*RHO) THEN
KNEW=-1
DELTA=TENTH*DELTA
RATIO=-1.0D0
IF (DELTA .LE. 1.5D0*RHO) DELTA=RHO
IF (NF .LE. NFSAV+2) GOTO 460
TEMP=0.125D0*CRVMIN*RHO*RHO
IF (TEMP .LE. DMAX1(DIFFA,DIFFB,DIFFC)) GOTO 460
GOTO 490
END IF
C
C Shift XBASE if XOPT may be too far from XBASE. First make the changes
C to BMAT that do not depend on ZMAT.
C
120 IF (DSQ .LE. 1.0D-3*XOPTSQ) THEN
TEMPQ=0.25D0*XOPTSQ
DO 140 K=1,NPT
SUM=ZERO
DO 130 I=1,N
130 SUM=SUM+XPT(K,I)*XOPT(I)
TEMP=PQ(K)*SUM
SUM=SUM-HALF*XOPTSQ
W(NPT+K)=SUM
DO 140 I=1,N
GQ(I)=GQ(I)+TEMP*XPT(K,I)
XPT(K,I)=XPT(K,I)-HALF*XOPT(I)
VLAG(I)=BMAT(K,I)
W(I)=SUM*XPT(K,I)+TEMPQ*XOPT(I)
IP=NPT+I
DO 140 J=1,I
140 BMAT(IP,J)=BMAT(IP,J)+VLAG(I)*W(J)+W(I)*VLAG(J)
C
C Then the revisions of BMAT that depend on ZMAT are calculated.
C
DO 180 K=1,NPTM
SUMZ=ZERO
DO 150 I=1,NPT
SUMZ=SUMZ+ZMAT(I,K)
150 W(I)=W(NPT+I)*ZMAT(I,K)
DO 170 J=1,N
SUM=TEMPQ*SUMZ*XOPT(J)
DO 160 I=1,NPT
160 SUM=SUM+W(I)*XPT(I,J)
VLAG(J)=SUM
IF (K .LT. IDZ) SUM=-SUM
DO 170 I=1,NPT
170 BMAT(I,J)=BMAT(I,J)+SUM*ZMAT(I,K)
DO 180 I=1,N
IP=I+NPT
TEMP=VLAG(I)
IF (K .LT. IDZ) TEMP=-TEMP
DO 180 J=1,I
180 BMAT(IP,J)=BMAT(IP,J)+TEMP*VLAG(J)
C
C The following instructions complete the shift of XBASE, including
C the changes to the parameters of the quadratic model.
C
IH=0
DO 200 J=1,N
W(J)=ZERO
DO 190 K=1,NPT
W(J)=W(J)+PQ(K)*XPT(K,J)
190 XPT(K,J)=XPT(K,J)-HALF*XOPT(J)
DO 200 I=1,J
IH=IH+1
IF (I .LT. J) GQ(J)=GQ(J)+HQ(IH)*XOPT(I)
GQ(I)=GQ(I)+HQ(IH)*XOPT(J)
HQ(IH)=HQ(IH)+W(I)*XOPT(J)+XOPT(I)*W(J)
200 BMAT(NPT+I,J)=BMAT(NPT+J,I)
DO 210 J=1,N
XBASE(J)=XBASE(J)+XOPT(J)
210 XOPT(J)=ZERO
XOPTSQ=ZERO
END IF
C
C Pick the model step if KNEW is positive. A different choice of D
C may be made later, if the choice of D by BIGLAG causes substantial
C cancellation in DENOM.
C
IF (KNEW .GT. 0) THEN
CALL BIGLAG (N,NPT,XOPT,XPT,BMAT,ZMAT,IDZ,NDIM,KNEW,DSTEP,
1 D,ALPHA,VLAG,VLAG(NPT+1),W,W(NP),W(NP+N))
END IF
C
C Calculate VLAG and BETA for the current choice of D. The first NPT
C components of W_check will be held in W.
C
DO 230 K=1,NPT
SUMA=ZERO
SUMB=ZERO
SUM=ZERO
DO 220 J=1,N
SUMA=SUMA+XPT(K,J)*D(J)
SUMB=SUMB+XPT(K,J)*XOPT(J)
220 SUM=SUM+BMAT(K,J)*D(J)
W(K)=SUMA*(HALF*SUMA+SUMB)
230 VLAG(K)=SUM
BETA=ZERO
DO 250 K=1,NPTM
SUM=ZERO
DO 240 I=1,NPT
240 SUM=SUM+ZMAT(I,K)*W(I)
IF (K .LT. IDZ) THEN
BETA=BETA+SUM*SUM
SUM=-SUM
ELSE
BETA=BETA-SUM*SUM
END IF
DO 250 I=1,NPT
250 VLAG(I)=VLAG(I)+SUM*ZMAT(I,K)
BSUM=ZERO
DX=ZERO
DO 280 J=1,N
SUM=ZERO
DO 260 I=1,NPT
260 SUM=SUM+W(I)*BMAT(I,J)
BSUM=BSUM+SUM*D(J)
JP=NPT+J
DO 270 K=1,N
270 SUM=SUM+BMAT(JP,K)*D(K)
VLAG(JP)=SUM
BSUM=BSUM+SUM*D(J)
280 DX=DX+D(J)*XOPT(J)
BETA=DX*DX+DSQ*(XOPTSQ+DX+DX+HALF*DSQ)+BETA-BSUM
VLAG(KOPT)=VLAG(KOPT)+ONE
C
C If KNEW is positive and if the cancellation in DENOM is unacceptable,
C then BIGDEN calculates an alternative model step, XNEW being used for
C working space.
C
IF (KNEW .GT. 0) THEN
TEMP=ONE+ALPHA*BETA/VLAG(KNEW)**2
IF (DABS(TEMP) .LE. 0.8D0) THEN
CALL BIGDEN (N,NPT,XOPT,XPT,BMAT,ZMAT,IDZ,NDIM,KOPT,
1 KNEW,D,W,VLAG,BETA,XNEW,W(NDIM+1),W(6*NDIM+1))
END IF
END IF
C
C Calculate the next value of the objective function.
C
290 DO 300 I=1,N
XNEW(I)=XOPT(I)+D(I)
300 X(I)=XBASE(I)+XNEW(I)
NF=NF+1
310 IF (NF .GT. NFTEST) THEN
NF=NF-1
IF (IPRINT .GT. 0) PRINT 320
320 FORMAT (/4X,'Return from NEWUOA because CALFUN has been',
1 ' called MAXFUN times.')
GOTO 530
END IF
CALL CALFUN (N,X,F,IWF)
IF (IPRINT .EQ. 3) THEN
PRINT 330, NF,F,(X(I),I=1,N)
330 FORMAT (/4X,'Function number',I6,' F =',1PD18.10,
1 ' The corresponding X is:'/(2X,5D15.6))
END IF
IF (NF .LE. NPT) GOTO 70
IF (KNEW .EQ. -1) GOTO 530
C
C Use the quadratic model to predict the change in F due to the step D,
C and set DIFF to the error of this prediction.
C
VQUAD=ZERO
IH=0
DO 340 J=1,N
VQUAD=VQUAD+D(J)*GQ(J)
DO 340 I=1,J
IH=IH+1
TEMP=D(I)*XNEW(J)+D(J)*XOPT(I)
IF (I .EQ. J) TEMP=HALF*TEMP
340 VQUAD=VQUAD+TEMP*HQ(IH)
DO 350 K=1,NPT
350 VQUAD=VQUAD+PQ(K)*W(K)
DIFF=F-FOPT-VQUAD
DIFFC=DIFFB
DIFFB=DIFFA
DIFFA=DABS(DIFF)
IF (DNORM .GT. RHO) NFSAV=NF
C
C Update FOPT and XOPT if the new F is the least value of the objective
C function so far. The branch when KNEW is positive occurs if D is not
C a trust region step.
C
FSAVE=FOPT
IF (F .LT. FOPT) THEN
FOPT=F
XOPTSQ=ZERO
DO 360 I=1,N
XOPT(I)=XNEW(I)
360 XOPTSQ=XOPTSQ+XOPT(I)**2
END IF
KSAVE=KNEW
IF (KNEW .GT. 0) GOTO 410
C
C Pick the next value of DELTA after a trust region step.
C
IF (VQUAD .GE. ZERO) THEN
IF (IPRINT .GT. 0) PRINT 370
370 FORMAT (/4X,'Return from NEWUOA because a trust',
1 ' region step has failed to reduce Q.')
GOTO 530
END IF
RATIO=(F-FSAVE)/VQUAD
IF (RATIO .LE. TENTH) THEN
DELTA=HALF*DNORM
ELSE IF (RATIO. LE. 0.7D0) THEN
DELTA=DMAX1(HALF*DELTA,DNORM)
ELSE
DELTA=DMAX1(HALF*DELTA,DNORM+DNORM)
END IF
IF (DELTA .LE. 1.5D0*RHO) DELTA=RHO
C
C Set KNEW to the index of the next interpolation point to be deleted.
C
RHOSQ=DMAX1(TENTH*DELTA,RHO)**2
KTEMP=0
DETRAT=ZERO
IF (F .GE. FSAVE) THEN
KTEMP=KOPT
DETRAT=ONE
END IF
DO 400 K=1,NPT
HDIAG=ZERO
DO 380 J=1,NPTM
TEMP=ONE
IF (J .LT. IDZ) TEMP=-ONE
380 HDIAG=HDIAG+TEMP*ZMAT(K,J)**2
TEMP=DABS(BETA*HDIAG+VLAG(K)**2)
DISTSQ=ZERO
DO 390 J=1,N
390 DISTSQ=DISTSQ+(XPT(K,J)-XOPT(J))**2
IF (DISTSQ .GT. RHOSQ) TEMP=TEMP*(DISTSQ/RHOSQ)**3
IF (TEMP .GT. DETRAT .AND. K .NE. KTEMP) THEN
DETRAT=TEMP
KNEW=K
END IF
400 CONTINUE
IF (KNEW .EQ. 0) GOTO 460
C
C Update BMAT, ZMAT and IDZ, so that the KNEW-th interpolation point
C can be moved. Begin the updating of the quadratic model, starting
C with the explicit second derivative term.
C
410 CALL UPDATE (N,NPT,BMAT,ZMAT,IDZ,NDIM,VLAG,BETA,KNEW,W)
FVAL(KNEW)=F
IH=0
DO 420 I=1,N
TEMP=PQ(KNEW)*XPT(KNEW,I)
DO 420 J=1,I
IH=IH+1
420 HQ(IH)=HQ(IH)+TEMP*XPT(KNEW,J)
PQ(KNEW)=ZERO
C
C Update the other second derivative parameters, and then the gradient
C vector of the model. Also include the new interpolation point.
C
DO 440 J=1,NPTM
TEMP=DIFF*ZMAT(KNEW,J)
IF (J .LT. IDZ) TEMP=-TEMP
DO 440 K=1,NPT
440 PQ(K)=PQ(K)+TEMP*ZMAT(K,J)
GQSQ=ZERO
DO 450 I=1,N
GQ(I)=GQ(I)+DIFF*BMAT(KNEW,I)
GQSQ=GQSQ+GQ(I)**2
450 XPT(KNEW,I)=XNEW(I)
C
C If a trust region step makes a small change to the objective function,
C then calculate the gradient of the least Frobenius norm interpolant at
C XBASE, and store it in W, using VLAG for a vector of right hand sides.
C
IF (KSAVE .EQ. 0 .AND. DELTA .EQ. RHO) THEN
IF (DABS(RATIO) .GT. 1.0D-2) THEN
ITEST=0
ELSE
DO 700 K=1,NPT
700 VLAG(K)=FVAL(K)-FVAL(KOPT)
GISQ=ZERO
DO 720 I=1,N
SUM=ZERO
DO 710 K=1,NPT
710 SUM=SUM+BMAT(K,I)*VLAG(K)
GISQ=GISQ+SUM*SUM
720 W(I)=SUM
C
C Test whether to replace the new quadratic model by the least Frobenius
C norm interpolant, making the replacement if the test is satisfied.
C
ITEST=ITEST+1
IF (GQSQ .LT. 1.0D2*GISQ) ITEST=0
IF (ITEST .GE. 3) THEN
DO 730 I=1,N
730 GQ(I)=W(I)
DO 740 IH=1,NH
740 HQ(IH)=ZERO
DO 760 J=1,NPTM
W(J)=ZERO
DO 750 K=1,NPT
750 W(J)=W(J)+VLAG(K)*ZMAT(K,J)
760 IF (J .LT. IDZ) W(J)=-W(J)
DO 770 K=1,NPT
PQ(K)=ZERO
DO 770 J=1,NPTM
770 PQ(K)=PQ(K)+ZMAT(K,J)*W(J)
ITEST=0
END IF
END IF
END IF
IF (F .LT. FSAVE) KOPT=KNEW
C
C If a trust region step has provided a sufficient decrease in F, then
C branch for another trust region calculation. The case KSAVE>0 occurs
C when the new function value was calculated by a model step.
C
IF (F .LE. FSAVE+TENTH*VQUAD) GOTO 100
IF (KSAVE .GT. 0) GOTO 100
C
C Alternatively, find out if the interpolation points are close enough
C to the best point so far.
C
KNEW=0
460 DISTSQ=4.0D0*DELTA*DELTA
DO 480 K=1,NPT
SUM=ZERO
DO 470 J=1,N
470 SUM=SUM+(XPT(K,J)-XOPT(J))**2
IF (SUM .GT. DISTSQ) THEN
KNEW=K
DISTSQ=SUM
END IF
480 CONTINUE
C
C If KNEW is positive, then set DSTEP, and branch back for the next
C iteration, which will generate a "model step".
C
IF (KNEW .GT. 0) THEN
DSTEP=DMAX1(DMIN1(TENTH*DSQRT(DISTSQ),HALF*DELTA),RHO)
DSQ=DSTEP*DSTEP
GOTO 120
END IF
IF (RATIO .GT. ZERO) GOTO 100
IF (DMAX1(DELTA,DNORM) .GT. RHO) GOTO 100
C
C The calculations with the current value of RHO are complete. Pick the
C next values of RHO and DELTA.
C
490 IF (RHO .GT. RHOEND) THEN
DELTA=HALF*RHO
RATIO=RHO/RHOEND
IF (RATIO .LE. 16.0D0) THEN
RHO=RHOEND
ELSE IF (RATIO .LE. 250.0D0) THEN
RHO=DSQRT(RATIO)*RHOEND
ELSE
RHO=TENTH*RHO
END IF
DELTA=DMAX1(DELTA,RHO)
IF (IPRINT .GE. 2) THEN
IF (IPRINT .GE. 3) PRINT 500
500 FORMAT (5X)
PRINT 510, RHO,NF
510 FORMAT (/4X,'New RHO =',1PD11.4,5X,'Number of',
1 ' function values =',I6)
PRINT 520, FOPT,(XBASE(I)+XOPT(I),I=1,N)
520 FORMAT (4X,'Least value of F =',1PD23.15,9X,
1 'The corresponding X is:'/(2X,5D15.6))
END IF
GOTO 90
END IF
C
C Return from the calculation, after another Newton-Raphson step, if
C it is too short to have been tried before.
C
IF (KNEW .EQ. -1) GOTO 290
530 IF (FOPT .LE. F) THEN
DO 540 I=1,N
540 X(I)=XBASE(I)+XOPT(I)
F=FOPT
END IF
IF (IPRINT .GE. 1) THEN
PRINT 550, NF
550 FORMAT (/4X,'At the return from NEWUOA',5X,
1 'Number of function values =',I6)
PRINT 520, F,(X(I),I=1,N)
END IF
NEWUOB =F
RETURN
END
SUBROUTINE TRSAPP (N,NPT,XOPT,XPT,GQ,HQ,PQ,DELTA,STEP,
1 D,G,HD,HS,CRVMIN)
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION XOPT(*),XPT(NPT,*),GQ(*),HQ(*),PQ(*),STEP(*),
1 D(*),G(*),HD(*),HS(*)
C
C N is the number of variables of a quadratic objective function, Q say.
C The arguments NPT, XOPT, XPT, GQ, HQ and PQ have their usual meanings,
C in order to define the current quadratic model Q.
C DELTA is the trust region radius, and has to be positive.
C STEP will be set to the calculated trial step.
C The arrays D, G, HD and HS will be used for working space.
C CRVMIN will be set to the least curvature of H along the conjugate
C directions that occur, except that it is set to zero if STEP goes
C all the way to the trust region boundary.
C
C The calculation of STEP begins with the truncated conjugate gradient
C method. If the boundary of the trust region is reached, then further
C changes to STEP may be made, each one being in the 2D space spanned
C by the current STEP and the corresponding gradient of Q. Thus STEP
C should provide a substantial reduction to Q within the trust region.
C
C Initialization, which includes setting HD to H times XOPT.
C
HALF=0.5D0
ZERO=0.0D0
TWOPI=8.0D0*DATAN(1.0D0)
DELSQ=DELTA*DELTA
ITERC=0
ITERMAX=N
ITERSW=ITERMAX
DO 10 I=1,N
10 D(I)=XOPT(I)
GOTO 170
C
C Prepare for the first line search.
C
20 QRED=ZERO
DD=ZERO
DO 30 I=1,N
STEP(I)=ZERO
HS(I)=ZERO
G(I)=GQ(I)+HD(I)
D(I)=-G(I)
30 DD=DD+D(I)**2
CRVMIN=ZERO
IF (DD .EQ. ZERO) GOTO 160
DS=ZERO
SS=ZERO
GG=DD
GGBEG=GG
C
C Calculate the step to the trust region boundary and the product HD.
C
40 ITERC=ITERC+1
TEMP=DELSQ-SS
BSTEP=TEMP/(DS+DSQRT(DS*DS+DD*TEMP))
GOTO 170
50 DHD=ZERO
DO 60 J=1,N
60 DHD=DHD+D(J)*HD(J)
C
C Update CRVMIN and set the step-length ALPHA.
C
ALPHA=BSTEP
IF (DHD .GT. ZERO) THEN
TEMP=DHD/DD
IF (ITERC .EQ. 1) CRVMIN=TEMP
CRVMIN=DMIN1(CRVMIN,TEMP)
ALPHA=DMIN1(ALPHA,GG/DHD)
END IF
QADD=ALPHA*(GG-HALF*ALPHA*DHD)
QRED=QRED+QADD
C
C Update STEP and HS.
C
GGSAV=GG
GG=ZERO
DO 70 I=1,N
STEP(I)=STEP(I)+ALPHA*D(I)
HS(I)=HS(I)+ALPHA*HD(I)
70 GG=GG+(G(I)+HS(I))**2
C
C Begin another conjugate direction iteration if required.
C
IF (ALPHA .LT. BSTEP) THEN
IF (QADD .LE. 0.01D0*QRED) GOTO 160
IF (GG .LE. 1.0D-4*GGBEG) GOTO 160
IF (ITERC .EQ. ITERMAX) GOTO 160
TEMP=GG/GGSAV
DD=ZERO
DS=ZERO
SS=ZERO
DO 80 I=1,N
D(I)=TEMP*D(I)-G(I)-HS(I)
DD=DD+D(I)**2
DS=DS+D(I)*STEP(I)
80 SS=SS+STEP(I)**2
IF (DS .LE. ZERO) GOTO 160
IF (SS .LT. DELSQ) GOTO 40
END IF
CRVMIN=ZERO
ITERSW=ITERC
C
C Test whether an alternative iteration is required.
C
90 IF (GG .LE. 1.0D-4*GGBEG) GOTO 160
SG=ZERO
SHS=ZERO
DO 100 I=1,N
SG=SG+STEP(I)*G(I)
100 SHS=SHS+STEP(I)*HS(I)
SGK=SG+SHS
ANGTEST=SGK/DSQRT(GG*DELSQ)
IF (ANGTEST .LE. -0.99D0) GOTO 160
C
C Begin the alternative iteration by calculating D and HD and some
C scalar products.
C
ITERC=ITERC+1
TEMP=DSQRT(DELSQ*GG-SGK*SGK)
TEMPA=DELSQ/TEMP
TEMPB=SGK/TEMP
DO 110 I=1,N
110 D(I)=TEMPA*(G(I)+HS(I))-TEMPB*STEP(I)
GOTO 170
120 DG=ZERO
DHD=ZERO
DHS=ZERO
DO 130 I=1,N
DG=DG+D(I)*G(I)
DHD=DHD+HD(I)*D(I)
130 DHS=DHS+HD(I)*STEP(I)
C
C Seek the value of the angle that minimizes Q.
C
CF=HALF*(SHS-DHD)
QBEG=SG+CF
QSAV=QBEG
QMIN=QBEG
ISAVE=0
IU=49
TEMP=TWOPI/DFLOAT(IU+1)
DO 140 I=1,IU
ANGLE=DFLOAT(I)*TEMP
CTH=DCOS(ANGLE)
STH=DSIN(ANGLE)
QNEW=(SG+CF*CTH)*CTH+(DG+DHS*CTH)*STH
IF (QNEW .LT. QMIN) THEN
QMIN=QNEW
ISAVE=I
TEMPA=QSAV
ELSE IF (I .EQ. ISAVE+1) THEN
TEMPB=QNEW
END IF
140 QSAV=QNEW
IF (ISAVE .EQ. ZERO) TEMPA=QNEW
IF (ISAVE .EQ. IU) TEMPB=QBEG
ANGLE=ZERO
IF (TEMPA .NE. TEMPB) THEN
TEMPA=TEMPA-QMIN
TEMPB=TEMPB-QMIN
ANGLE=HALF*(TEMPA-TEMPB)/(TEMPA+TEMPB)
END IF
ANGLE=TEMP*(DFLOAT(ISAVE)+ANGLE)
C
C Calculate the new STEP and HS. Then test for convergence.
C
CTH=DCOS(ANGLE)
STH=DSIN(ANGLE)
REDUC=QBEG-(SG+CF*CTH)*CTH-(DG+DHS*CTH)*STH
GG=ZERO
DO 150 I=1,N
STEP(I)=CTH*STEP(I)+STH*D(I)
HS(I)=CTH*HS(I)+STH*HD(I)
150 GG=GG+(G(I)+HS(I))**2
QRED=QRED+REDUC
RATIO=REDUC/QRED
IF (ITERC .LT. ITERMAX .AND. RATIO .GT. 0.01D0) GOTO 90
160 RETURN
C
C The following instructions act as a subroutine for setting the vector
C HD to the vector D multiplied by the second derivative matrix of Q.
C They are called from three different places, which are distinguished
C by the value of ITERC.
C
170 DO 180 I=1,N
180 HD(I)=ZERO
DO 200 K=1,NPT
TEMP=ZERO
DO 190 J=1,N
190 TEMP=TEMP+XPT(K,J)*D(J)
TEMP=TEMP*PQ(K)
DO 200 I=1,N
200 HD(I)=HD(I)+TEMP*XPT(K,I)
IH=0
DO 210 J=1,N
DO 210 I=1,J
IH=IH+1
IF (I .LT. J) HD(J)=HD(J)+HQ(IH)*D(I)
210 HD(I)=HD(I)+HQ(IH)*D(J)
IF (ITERC .EQ. 0) GOTO 20
IF (ITERC .LE. ITERSW) GOTO 50
GOTO 120
END
SUBROUTINE UPDATE (N,NPT,BMAT,ZMAT,IDZ,NDIM,VLAG,BETA,KNEW,W)
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION BMAT(NDIM,*),ZMAT(NPT,*),VLAG(*),W(*)
C
C The arrays BMAT and ZMAT with IDZ are updated, in order to shift the
C interpolation point that has index KNEW. On entry, VLAG contains the
C components of the vector Theta*Wcheck+e_b of the updating formula
C (6.11), and BETA holds the value of the parameter that has this name.
C The vector W is used for working space.
C
C Set some constants.
C
ONE=1.0D0
ZERO=0.0D0
NPTM=NPT-N-1
C
C Apply the rotations that put zeros in the KNEW-th row of ZMAT.
C
JL=1
DO 20 J=2,NPTM
IF (J .EQ. IDZ) THEN
JL=IDZ
ELSE IF (ZMAT(KNEW,J) .NE. ZERO) THEN
TEMP=DSQRT(ZMAT(KNEW,JL)**2+ZMAT(KNEW,J)**2)
TEMPA=ZMAT(KNEW,JL)/TEMP
TEMPB=ZMAT(KNEW,J)/TEMP
DO 10 I=1,NPT
TEMP=TEMPA*ZMAT(I,JL)+TEMPB*ZMAT(I,J)
ZMAT(I,J)=TEMPA*ZMAT(I,J)-TEMPB*ZMAT(I,JL)
10 ZMAT(I,JL)=TEMP
ZMAT(KNEW,J)=ZERO
END IF
20 CONTINUE
C
C Put the first NPT components of the KNEW-th column of HLAG into W,
C and calculate the parameters of the updating formula.
C
TEMPA=ZMAT(KNEW,1)
IF (IDZ .GE. 2) TEMPA=-TEMPA
IF (JL .GT. 1) TEMPB=ZMAT(KNEW,JL)
DO 30 I=1,NPT
W(I)=TEMPA*ZMAT(I,1)
IF (JL .GT. 1) W(I)=W(I)+TEMPB*ZMAT(I,JL)
30 CONTINUE
ALPHA=W(KNEW)
TAU=VLAG(KNEW)
TAUSQ=TAU*TAU
DENOM=ALPHA*BETA+TAUSQ
VLAG(KNEW)=VLAG(KNEW)-ONE
C
C Complete the updating of ZMAT when there is only one nonzero element
C in the KNEW-th row of the new matrix ZMAT, but, if IFLAG is set to one,
C then the first column of ZMAT will be exchanged with another one later.
C
IFLAG=0
IF (JL .EQ. 1) THEN
TEMP=DSQRT(DABS(DENOM))
TEMPB=TEMPA/TEMP
TEMPA=TAU/TEMP
DO 40 I=1,NPT
40 ZMAT(I,1)=TEMPA*ZMAT(I,1)-TEMPB*VLAG(I)
IF (IDZ .EQ. 1 .AND. TEMP .LT. ZERO) IDZ=2
IF (IDZ .GE. 2 .AND. TEMP .GE. ZERO) IFLAG=1
ELSE
C
C Complete the updating of ZMAT in the alternative case.
C
JA=1
IF (BETA .GE. ZERO) JA=JL
JB=JL+1-JA
TEMP=ZMAT(KNEW,JB)/DENOM
TEMPA=TEMP*BETA
TEMPB=TEMP*TAU
TEMP=ZMAT(KNEW,JA)
SCALA=ONE/DSQRT(DABS(BETA)*TEMP*TEMP+TAUSQ)
SCALB=SCALA*DSQRT(DABS(DENOM))
DO 50 I=1,NPT
ZMAT(I,JA)=SCALA*(TAU*ZMAT(I,JA)-TEMP*VLAG(I))
50 ZMAT(I,JB)=SCALB*(ZMAT(I,JB)-TEMPA*W(I)-TEMPB*VLAG(I))
IF (DENOM .LE. ZERO) THEN
IF (BETA .LT. ZERO) IDZ=IDZ+1
IF (BETA .GE. ZERO) IFLAG=1
END IF
END IF
C
C IDZ is reduced in the following case, and usually the first column
C of ZMAT is exchanged with a later one.
C
IF (IFLAG .EQ. 1) THEN
IDZ=IDZ-1
DO 60 I=1,NPT
TEMP=ZMAT(I,1)
ZMAT(I,1)=ZMAT(I,IDZ)
60 ZMAT(I,IDZ)=TEMP
END IF
C
C Finally, update the matrix BMAT.
C
DO 70 J=1,N
JP=NPT+J
W(JP)=BMAT(KNEW,J)
TEMPA=(ALPHA*VLAG(JP)-TAU*W(JP))/DENOM
TEMPB=(-BETA*W(JP)-TAU*VLAG(JP))/DENOM
DO 70 I=1,JP
BMAT(I,J)=BMAT(I,J)+TEMPA*VLAG(I)+TEMPB*W(I)
IF (I .GT. NPT) BMAT(JP,I-NPT)=BMAT(I,J)
70 CONTINUE
RETURN
END
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