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bobyqa_ArraysSpecified.f
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bobyqa_ArraysSpecified.f
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c 6 Aug 2011
c I have gone through and given bounds on dummy arrays when possible to try and determine
c memeory overwrite problem. Have basically replaced assumed size arrays with explicit-shaped arrays.
c 10 Mar 2012
c Recalling from aug 2011, I believe the problem was not this routine, but rather the TSPACK routine used to
C determine spline fits.
SUBROUTINE BOBYQA (N,NPT,X,XL,XU,RHOBEG,RHOEND,IPRINT,
1 MAXFUN,W)
IMPLICIT REAL*8 (A-H,O-Z)
c DIMENSION X(*),XL(*),XU(*),W(*)
DIMENSION X(n),XL(n),XU(n),W(*)
C
C This subroutine seeks the least value of a function of many variables,
C by applying a trust region method that forms quadratic models by
C interpolation. There is usually some freedom in the interpolation
C conditions, which is taken up by minimizing the Frobenius norm of
C the change to the second derivative of the model, beginning with the
C zero matrix. The values of the variables are constrained by upper and
C lower bounds. The 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
C the interval [N+2,(N+1)(N+2)/2]. Choices that exceed 2*N+1 are not
C recommended.
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 For I=1,2,...,N, XL(I) and XU(I) must provide the lower and upper
C bounds, respectively, on X(I). The construction of quadratic models
C requires XL(I) to be strictly less than XU(I) for each I. Further,
C the contribution to a model from changes to the I-th variable is
C damaged severely by rounding errors if XU(I)-XL(I) is too small.
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 no greater than
C RHOBEG. Typically, RHOBEG should be about one tenth of the greatest
C expected change to a variable, while RHOEND should indicate the
C accuracy that is required in the final values of the variables. An
C error return occurs if any of the differences XU(I)-XL(I), I=1,...,N,
C is less than 2*RHOBEG.
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+5)*(NPT+N)+3*N*(N+5)/2.
C
C SUBROUTINE CALFUN (N,X,F) has to be provided by the user. It must set
C F to the value of the objective function for the current values of the
C variables X(1),X(2),...,X(N), which are generated automatically in a
C way that satisfies the bounds given in XL and XU.
C
C Return if the value of NPT is unacceptable.
C
NP=N+1
IF (NPT .LT. N+2 .OR. NPT .GT. ((N+2)*NP)/2) THEN
PRINT 10
10 FORMAT (/4X,'Return from BOBYQA because NPT is not in',
1 ' the required interval')
GO TO 40
END IF
C
C Partition the working space array, so that different parts of it can
C be treated separately during the calculation of BOBYQB. The partition
C requires the first (NPT+2)*(NPT+N)+3*N*(N+5)/2 elements of W plus the
C space that is taken by the last array in the argument list of BOBYQB.
C
NDIM=NPT+N
IXB=1
IXP=IXB+N
IFV=IXP+N*NPT
IXO=IFV+NPT
IGO=IXO+N
IHQ=IGO+N
IPQ=IHQ+(N*NP)/2
IBMAT=IPQ+NPT
IZMAT=IBMAT+NDIM*N
ISL=IZMAT+NPT*(NPT-NP)
ISU=ISL+N
IXN=ISU+N
IXA=IXN+N
ID=IXA+N
IVL=ID+N
IW=IVL+NDIM
C
C Return if there is insufficient space between the bounds. Modify the
C initial X if necessary in order to avoid conflicts between the bounds
C and the construction of the first quadratic model. The lower and upper
C bounds on moves from the updated X are set now, in the ISL and ISU
C partitions of W, in order to provide useful and exact information about
C components of X that become within distance RHOBEG from their bounds.
C
ZERO=0.0D0
DO 30 J=1,N
TEMP=XU(J)-XL(J)
IF (TEMP .LT. RHOBEG+RHOBEG) THEN
PRINT 20
20 FORMAT (/4X,'Return from BOBYQA because one of the',
1 ' differences XU(I)-XL(I)'/6X,' is less than 2*RHOBEG.')
GO TO 40
END IF
JSL=ISL+J-1
JSU=JSL+N
W(JSL)=XL(J)-X(J)
W(JSU)=XU(J)-X(J)
IF (W(JSL) .GE. -RHOBEG) THEN
IF (W(JSL) .GE. ZERO) THEN
X(J)=XL(J)
W(JSL)=ZERO
W(JSU)=TEMP
ELSE
X(J)=XL(J)+RHOBEG
W(JSL)=-RHOBEG
W(JSU)=DMAX1(XU(J)-X(J),RHOBEG)
END IF
ELSE IF (W(JSU) .LE. RHOBEG) THEN
IF (W(JSU) .LE. ZERO) THEN
X(J)=XU(J)
W(JSL)=-TEMP
W(JSU)=ZERO
ELSE
X(J)=XU(J)-RHOBEG
W(JSL)=DMIN1(XL(J)-X(J),-RHOBEG)
W(JSU)=RHOBEG
END IF
END IF
30 CONTINUE
C
C Make the call of BOBYQB.
C
CALL BOBYQB (N,NPT,X,XL,XU,RHOBEG,RHOEND,IPRINT,MAXFUN,W(IXB),
1 W(IXP),W(IFV),W(IXO),W(IGO),W(IHQ),W(IPQ),W(IBMAT),W(IZMAT),
2 NDIM,W(ISL),W(ISU),W(IXN),W(IXA),W(ID),W(IVL),W(IW))
40 RETURN
END
SUBROUTINE BOBYQB (N,NPT,X,XL,XU,RHOBEG,RHOEND,IPRINT,
1 MAXFUN,XBASE,XPT,FVAL,XOPT,GOPT,HQ,PQ,BMAT,ZMAT,NDIM,
2 SL,SU,XNEW,XALT,D,VLAG,W)
IMPLICIT REAL*8 (A-H,O-Z)
c DIMENSION X(*),XL(*),XU(*),XBASE(*),XPT(NPT,*),FVAL(*),
c 1 XOPT(*),GOPT(*),HQ(*),PQ(*),BMAT(NDIM,*),ZMAT(NPT,*),
c 2 SL(*),SU(*),XNEW(*),XALT(*),D(*),VLAG(*),W(*)
DIMENSION X(n),XL(n),XU(n),XBASE(*),XPT(NPT,*),FVAL(*),
1 XOPT(*),GOPT(*),HQ(*),PQ(*),BMAT(NDIM,*),ZMAT(NPT,*),
2 SL(*),SU(*),XNEW(*),XALT(*),D(*),VLAG(*),W(*)
C
C The arguments N, NPT, X, XL, XU, RHOBEG, RHOEND, IPRINT and MAXFUN
C are identical to the corresponding arguments in SUBROUTINE BOBYQA.
C XBASE holds a shift of origin that should reduce the contributions
C from rounding errors to values of the model and Lagrange functions.
C XPT is a two-dimensional array that holds the coordinates of the
C interpolation points relative to XBASE.
C FVAL holds the values of F at the interpolation points.
C XOPT is set to the displacement from XBASE of the trust region centre.
C GOPT holds the gradient of the quadratic model at XBASE+XOPT.
C HQ holds the explicit second derivatives of the quadratic model.
C PQ contains the parameters of the implicit second derivatives of the
C quadratic model.
C BMAT holds the last N columns of H.
C ZMAT holds the factorization of the leading NPT by NPT submatrix of H,
C this factorization being ZMAT times ZMAT^T, which provides both the
C correct rank and positive semi-definiteness.
C NDIM is the first dimension of BMAT and has the value NPT+N.
C SL and SU hold the differences XL-XBASE and XU-XBASE, respectively.
C All the components of every XOPT are going to satisfy the bounds
C SL(I) .LEQ. XOPT(I) .LEQ. SU(I), with appropriate equalities when
C XOPT is on a constraint boundary.
C XNEW is chosen by SUBROUTINE TRSBOX or ALTMOV. Usually XBASE+XNEW is the
C vector of variables for the next call of CALFUN. XNEW also satisfies
C the SL and SU constraints in the way that has just been mentioned.
C XALT is an alternative to XNEW, chosen by ALTMOV, that may replace XNEW
C in order to increase the denominator in the updating of UPDATE.
C D is reserved for a trial step from XOPT, which is usually XNEW-XOPT.
C VLAG contains 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 W is a one-dimensional array that is used for working space. Its length
C must be at least 3*NDIM = 3*(NPT+N).
C
C Set some constants.
C
HALF=0.5D0
ONE=1.0D0
TEN=10.0D0
TENTH=0.1D0
TWO=2.0D0
ZERO=0.0D0
NP=N+1
NPTM=NPT-NP
NH=(N*NP)/2
C
C The call of PRELIM sets the elements of XBASE, XPT, FVAL, GOPT, HQ, PQ,
C BMAT and ZMAT for the first iteration, with the corresponding values of
C of NF and KOPT, which are the number of calls of CALFUN so far and the
C index of the interpolation point at the trust region centre. Then the
C initial XOPT is set too. The branch to label 720 occurs if MAXFUN is
C less than NPT. GOPT will be updated if KOPT is different from KBASE.
C
CALL PRELIM (N,NPT,X,XL,XU,RHOBEG,IPRINT,MAXFUN,XBASE,XPT,
1 FVAL,GOPT,HQ,PQ,BMAT,ZMAT,NDIM,SL,SU,NF,KOPT)
XOPTSQ=ZERO
DO 10 I=1,N
XOPT(I)=XPT(KOPT,I)
10 XOPTSQ=XOPTSQ+XOPT(I)**2
FSAVE=FVAL(1)
IF (NF .LT. NPT) THEN
IF (IPRINT .GT. 0) PRINT 390
GOTO 720
END IF
KBASE=1
C
C Complete the settings that are required for the iterative procedure.
C
RHO=RHOBEG
DELTA=RHO
NRESC=NF
NTRITS=0
DIFFA=ZERO
DIFFB=ZERO
ITEST=0
NFSAV=NF
C
C Update GOPT if necessary before the first iteration and after each
C call of RESCUE that makes a call of CALFUN.
C
20 IF (KOPT .NE. KBASE) THEN
IH=0
DO 30 J=1,N
DO 30 I=1,J
IH=IH+1
IF (I .LT. J) GOPT(J)=GOPT(J)+HQ(IH)*XOPT(I)
30 GOPT(I)=GOPT(I)+HQ(IH)*XOPT(J)
IF (NF .GT. NPT) THEN
DO 50 K=1,NPT
TEMP=ZERO
DO 40 J=1,N
40 TEMP=TEMP+XPT(K,J)*XOPT(J)
TEMP=PQ(K)*TEMP
DO 50 I=1,N
50 GOPT(I)=GOPT(I)+TEMP*XPT(K,I)
END IF
END IF
C
C Generate the next point in the trust region that provides a small value
C of the quadratic model subject to the constraints on the variables.
C The integer NTRITS is set to the number "trust region" iterations that
C have occurred since the last "alternative" iteration. If the length
C of XNEW-XOPT is less than HALF*RHO, however, then there is a branch to
C label 650 or 680 with NTRITS=-1, instead of calculating F at XNEW.
C
60 CALL TRSBOX (N,NPT,XPT,XOPT,GOPT,HQ,PQ,SL,SU,DELTA,XNEW,D,
1 W,W(NP),W(NP+N),W(NP+2*N),W(NP+3*N),DSQ,CRVMIN)
DNORM=DMIN1(DELTA,DSQRT(DSQ))
IF (DNORM .LT. HALF*RHO) THEN
NTRITS=-1
DISTSQ=(TEN*RHO)**2
IF (NF .LE. NFSAV+2) GOTO 650
C
C The following choice between labels 650 and 680 depends on whether or
C not our work with the current RHO seems to be complete. Either RHO is
C decreased or termination occurs if the errors in the quadratic model at
C the last three interpolation points compare favourably with predictions
C of likely improvements to the model within distance HALF*RHO of XOPT.
C
ERRBIG=DMAX1(DIFFA,DIFFB,DIFFC)
FRHOSQ=0.125D0*RHO*RHO
IF (CRVMIN .GT. ZERO .AND. ERRBIG .GT. FRHOSQ*CRVMIN)
1 GOTO 650
BDTOL=ERRBIG/RHO
DO 80 J=1,N
BDTEST=BDTOL
IF (XNEW(J) .EQ. SL(J)) BDTEST=W(J)
IF (XNEW(J) .EQ. SU(J)) BDTEST=-W(J)
IF (BDTEST .LT. BDTOL) THEN
CURV=HQ((J+J*J)/2)
DO 70 K=1,NPT
70 CURV=CURV+PQ(K)*XPT(K,J)**2
BDTEST=BDTEST+HALF*CURV*RHO
IF (BDTEST .LT. BDTOL) GOTO 650
END IF
80 CONTINUE
GOTO 680
END IF
NTRITS=NTRITS+1
C
C Severe cancellation is likely to occur if XOPT is too far from XBASE.
C If the following test holds, then XBASE is shifted so that XOPT becomes
C zero. The appropriate changes are made to BMAT and to the second
C derivatives of the current model, beginning with the changes to BMAT
C that do not depend on ZMAT. VLAG is used temporarily for working space.
C
90 IF (DSQ .LE. 1.0D-3*XOPTSQ) THEN
FRACSQ=0.25D0*XOPTSQ
SUMPQ=ZERO
DO 110 K=1,NPT
SUMPQ=SUMPQ+PQ(K)
SUM=-HALF*XOPTSQ
DO 100 I=1,N
100 SUM=SUM+XPT(K,I)*XOPT(I)
W(NPT+K)=SUM
TEMP=FRACSQ-HALF*SUM
DO 110 I=1,N
W(I)=BMAT(K,I)
VLAG(I)=SUM*XPT(K,I)+TEMP*XOPT(I)
IP=NPT+I
DO 110 J=1,I
110 BMAT(IP,J)=BMAT(IP,J)+W(I)*VLAG(J)+VLAG(I)*W(J)
C
C Then the revisions of BMAT that depend on ZMAT are calculated.
C
DO 150 JJ=1,NPTM
SUMZ=ZERO
SUMW=ZERO
DO 120 K=1,NPT
SUMZ=SUMZ+ZMAT(K,JJ)
VLAG(K)=W(NPT+K)*ZMAT(K,JJ)
120 SUMW=SUMW+VLAG(K)
DO 140 J=1,N
SUM=(FRACSQ*SUMZ-HALF*SUMW)*XOPT(J)
DO 130 K=1,NPT
130 SUM=SUM+VLAG(K)*XPT(K,J)
W(J)=SUM
DO 140 K=1,NPT
140 BMAT(K,J)=BMAT(K,J)+SUM*ZMAT(K,JJ)
DO 150 I=1,N
IP=I+NPT
TEMP=W(I)
DO 150 J=1,I
150 BMAT(IP,J)=BMAT(IP,J)+TEMP*W(J)
C
C The following instructions complete the shift, including the changes
C to the second derivative parameters of the quadratic model.
C
IH=0
DO 170 J=1,N
W(J)=-HALF*SUMPQ*XOPT(J)
DO 160 K=1,NPT
W(J)=W(J)+PQ(K)*XPT(K,J)
160 XPT(K,J)=XPT(K,J)-XOPT(J)
DO 170 I=1,J
IH=IH+1
HQ(IH)=HQ(IH)+W(I)*XOPT(J)+XOPT(I)*W(J)
170 BMAT(NPT+I,J)=BMAT(NPT+J,I)
DO 180 I=1,N
XBASE(I)=XBASE(I)+XOPT(I)
XNEW(I)=XNEW(I)-XOPT(I)
SL(I)=SL(I)-XOPT(I)
SU(I)=SU(I)-XOPT(I)
180 XOPT(I)=ZERO
XOPTSQ=ZERO
END IF
IF (NTRITS .EQ. 0) GOTO 210
GOTO 230
C
C XBASE is also moved to XOPT by a call of RESCUE. This calculation is
C more expensive than the previous shift, because new matrices BMAT and
C ZMAT are generated from scratch, which may include the replacement of
C interpolation points whose positions seem to be causing near linear
C dependence in the interpolation conditions. Therefore RESCUE is called
C only if rounding errors have reduced by at least a factor of two the
C denominator of the formula for updating the H matrix. It provides a
C useful safeguard, but is not invoked in most applications of BOBYQA.
C
190 NFSAV=NF
KBASE=KOPT
CALL RESCUE (N,NPT,XL,XU,IPRINT,MAXFUN,XBASE,XPT,FVAL,
1 XOPT,GOPT,HQ,PQ,BMAT,ZMAT,NDIM,SL,SU,NF,DELTA,KOPT,
2 VLAG,W,W(N+NP),W(NDIM+NP))
C
C XOPT is updated now in case the branch below to label 720 is taken.
C Any updating of GOPT occurs after the branch below to label 20, which
C leads to a trust region iteration as does the branch to label 60.
C
XOPTSQ=ZERO
IF (KOPT .NE. KBASE) THEN
DO 200 I=1,N
XOPT(I)=XPT(KOPT,I)
200 XOPTSQ=XOPTSQ+XOPT(I)**2
END IF
IF (NF .LT. 0) THEN
NF=MAXFUN
IF (IPRINT .GT. 0) PRINT 390
GOTO 720
END IF
NRESC=NF
IF (NFSAV .LT. NF) THEN
NFSAV=NF
GOTO 20
END IF
IF (NTRITS .GT. 0) GOTO 60
C
C Pick two alternative vectors of variables, relative to XBASE, that
C are suitable as new positions of the KNEW-th interpolation point.
C Firstly, XNEW is set to the point on a line through XOPT and another
C interpolation point that minimizes the predicted value of the next
C denominator, subject to ||XNEW - XOPT|| .LEQ. ADELT and to the SL
C and SU bounds. Secondly, XALT is set to the best feasible point on
C a constrained version of the Cauchy step of the KNEW-th Lagrange
C function, the corresponding value of the square of this function
C being returned in CAUCHY. The choice between these alternatives is
C going to be made when the denominator is calculated.
C
210 CALL ALTMOV (N,NPT,XPT,XOPT,BMAT,ZMAT,NDIM,SL,SU,KOPT,
1 KNEW,ADELT,XNEW,XALT,ALPHA,CAUCHY,W,W(NP),W(NDIM+1))
DO 220 I=1,N
220 D(I)=XNEW(I)-XOPT(I)
C
C Calculate VLAG and BETA for the current choice of D. The scalar
C product of D with XPT(K,.) is going to be held in W(NPT+K) for
C use when VQUAD is calculated.
C
230 DO 250 K=1,NPT
SUMA=ZERO
SUMB=ZERO
SUM=ZERO
DO 240 J=1,N
SUMA=SUMA+XPT(K,J)*D(J)
SUMB=SUMB+XPT(K,J)*XOPT(J)
240 SUM=SUM+BMAT(K,J)*D(J)
W(K)=SUMA*(HALF*SUMA+SUMB)
VLAG(K)=SUM
250 W(NPT+K)=SUMA
BETA=ZERO
DO 270 JJ=1,NPTM
SUM=ZERO
DO 260 K=1,NPT
260 SUM=SUM+ZMAT(K,JJ)*W(K)
BETA=BETA-SUM*SUM
DO 270 K=1,NPT
270 VLAG(K)=VLAG(K)+SUM*ZMAT(K,JJ)
DSQ=ZERO
BSUM=ZERO
DX=ZERO
DO 300 J=1,N
DSQ=DSQ+D(J)**2
SUM=ZERO
DO 280 K=1,NPT
280 SUM=SUM+W(K)*BMAT(K,J)
BSUM=BSUM+SUM*D(J)
JP=NPT+J
DO 290 I=1,N
290 SUM=SUM+BMAT(JP,I)*D(I)
VLAG(JP)=SUM
BSUM=BSUM+SUM*D(J)
300 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 NTRITS is zero, the denominator may be increased by replacing
C the step D of ALTMOV by a Cauchy step. Then RESCUE may be called if
C rounding errors have damaged the chosen denominator.
C
IF (NTRITS .EQ. 0) THEN
DENOM=VLAG(KNEW)**2+ALPHA*BETA
IF (DENOM .LT. CAUCHY .AND. CAUCHY .GT. ZERO) THEN
DO 310 I=1,N
XNEW(I)=XALT(I)
310 D(I)=XNEW(I)-XOPT(I)
CAUCHY=ZERO
GO TO 230
END IF
IF (DENOM .LE. HALF*VLAG(KNEW)**2) THEN
IF (NF .GT. NRESC) GOTO 190
IF (IPRINT .GT. 0) PRINT 320
320 FORMAT (/5X,'Return from BOBYQA because of much',
1 ' cancellation in a denominator.')
GOTO 720
END IF
C
C Alternatively, if NTRITS is positive, then set KNEW to the index of
C the next interpolation point to be deleted to make room for a trust
C region step. Again RESCUE may be called if rounding errors have damaged
C the chosen denominator, which is the reason for attempting to select
C KNEW before calculating the next value of the objective function.
C
ELSE
DELSQ=DELTA*DELTA
SCADEN=ZERO
BIGLSQ=ZERO
KNEW=0
DO 350 K=1,NPT
IF (K .EQ. KOPT) GOTO 350
HDIAG=ZERO
DO 330 JJ=1,NPTM
330 HDIAG=HDIAG+ZMAT(K,JJ)**2
DEN=BETA*HDIAG+VLAG(K)**2
DISTSQ=ZERO
DO 340 J=1,N
340 DISTSQ=DISTSQ+(XPT(K,J)-XOPT(J))**2
TEMP=DMAX1(ONE,(DISTSQ/DELSQ)**2)
IF (TEMP*DEN .GT. SCADEN) THEN
SCADEN=TEMP*DEN
KNEW=K
DENOM=DEN
END IF
BIGLSQ=DMAX1(BIGLSQ,TEMP*VLAG(K)**2)
350 CONTINUE
IF (SCADEN .LE. HALF*BIGLSQ) THEN
IF (NF .GT. NRESC) GOTO 190
IF (IPRINT .GT. 0) PRINT 320
GOTO 720
END IF
END IF
C
C Put the variables for the next calculation of the objective function
C in XNEW, with any adjustments for the bounds.
C
C
C Calculate the value of the objective function at XBASE+XNEW, unless
C the limit on the number of calculations of F has been reached.
C
360 DO 380 I=1,N
X(I)=DMIN1(DMAX1(XL(I),XBASE(I)+XNEW(I)),XU(I))
IF (XNEW(I) .EQ. SL(I)) X(I)=XL(I)
IF (XNEW(I) .EQ. SU(I)) X(I)=XU(I)
380 CONTINUE
IF (NF .GE. MAXFUN) THEN
IF (IPRINT .GT. 0) PRINT 390
390 FORMAT (/4X,'Return from BOBYQA because CALFUN has been',
1 ' called MAXFUN times.')
GOTO 720
END IF
NF=NF+1
CALL CALFUN (N,X,F)
IF (IPRINT .EQ. 3) THEN
PRINT 400, NF,F,(X(I),I=1,N)
400 FORMAT (/4X,'Function number',I6,' F =',1PD18.10,
1 ' The corresponding X is:'/(2X,5D15.6))
END IF
IF (NTRITS .EQ. -1) THEN
FSAVE=F
GOTO 720
END IF
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
FOPT=FVAL(KOPT)
VQUAD=ZERO
IH=0
DO 410 J=1,N
VQUAD=VQUAD+D(J)*GOPT(J)
DO 410 I=1,J
IH=IH+1
TEMP=D(I)*D(J)
IF (I .EQ. J) TEMP=HALF*TEMP
410 VQUAD=VQUAD+HQ(IH)*TEMP
DO 420 K=1,NPT
420 VQUAD=VQUAD+HALF*PQ(K)*W(NPT+K)**2
DIFF=F-FOPT-VQUAD
DIFFC=DIFFB
DIFFB=DIFFA
DIFFA=DABS(DIFF)
IF (DNORM .GT. RHO) NFSAV=NF
C
C Pick the next value of DELTA after a trust region step.
C
IF (NTRITS .GT. 0) THEN
IF (VQUAD .GE. ZERO) THEN
IF (IPRINT .GT. 0) PRINT 430
430 FORMAT (/4X,'Return from BOBYQA because a trust',
1 ' region step has failed to reduce Q.')
GOTO 720
END IF
RATIO=(F-FOPT)/VQUAD
IF (RATIO .LE. TENTH) THEN
DELTA=DMIN1(HALF*DELTA,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 Recalculate KNEW and DENOM if the new F is less than FOPT.
C
IF (F .LT. FOPT) THEN
KSAV=KNEW
DENSAV=DENOM
DELSQ=DELTA*DELTA
SCADEN=ZERO
BIGLSQ=ZERO
KNEW=0
DO 460 K=1,NPT
HDIAG=ZERO
DO 440 JJ=1,NPTM
440 HDIAG=HDIAG+ZMAT(K,JJ)**2
DEN=BETA*HDIAG+VLAG(K)**2
DISTSQ=ZERO
DO 450 J=1,N
450 DISTSQ=DISTSQ+(XPT(K,J)-XNEW(J))**2
TEMP=DMAX1(ONE,(DISTSQ/DELSQ)**2)
IF (TEMP*DEN .GT. SCADEN) THEN
SCADEN=TEMP*DEN
KNEW=K
DENOM=DEN
END IF
460 BIGLSQ=DMAX1(BIGLSQ,TEMP*VLAG(K)**2)
IF (SCADEN .LE. HALF*BIGLSQ) THEN
KNEW=KSAV
DENOM=DENSAV
END IF
END IF
END IF
C
C Update BMAT and ZMAT, so that the KNEW-th interpolation point can be
C moved. Also update the second derivative terms of the model.
C
CALL UPDATE (N,NPT,BMAT,ZMAT,NDIM,VLAG,BETA,DENOM,KNEW,W)
IH=0
PQOLD=PQ(KNEW)
PQ(KNEW)=ZERO
DO 470 I=1,N
TEMP=PQOLD*XPT(KNEW,I)
DO 470 J=1,I
IH=IH+1
470 HQ(IH)=HQ(IH)+TEMP*XPT(KNEW,J)
DO 480 JJ=1,NPTM
TEMP=DIFF*ZMAT(KNEW,JJ)
DO 480 K=1,NPT
480 PQ(K)=PQ(K)+TEMP*ZMAT(K,JJ)
C
C Include the new interpolation point, and make the changes to GOPT at
C the old XOPT that are caused by the updating of the quadratic model.
C
FVAL(KNEW)=F
DO 490 I=1,N
XPT(KNEW,I)=XNEW(I)
490 W(I)=BMAT(KNEW,I)
DO 520 K=1,NPT
SUMA=ZERO
DO 500 JJ=1,NPTM
500 SUMA=SUMA+ZMAT(KNEW,JJ)*ZMAT(K,JJ)
SUMB=ZERO
DO 510 J=1,N
510 SUMB=SUMB+XPT(K,J)*XOPT(J)
TEMP=SUMA*SUMB
DO 520 I=1,N
520 W(I)=W(I)+TEMP*XPT(K,I)
DO 530 I=1,N
530 GOPT(I)=GOPT(I)+DIFF*W(I)
C
C Update XOPT, GOPT and KOPT if the new calculated F is less than FOPT.
C
IF (F .LT. FOPT) THEN
KOPT=KNEW
XOPTSQ=ZERO
IH=0
DO 540 J=1,N
XOPT(J)=XNEW(J)
XOPTSQ=XOPTSQ+XOPT(J)**2
DO 540 I=1,J
IH=IH+1
IF (I .LT. J) GOPT(J)=GOPT(J)+HQ(IH)*D(I)
540 GOPT(I)=GOPT(I)+HQ(IH)*D(J)
DO 560 K=1,NPT
TEMP=ZERO
DO 550 J=1,N
550 TEMP=TEMP+XPT(K,J)*D(J)
TEMP=PQ(K)*TEMP
DO 560 I=1,N
560 GOPT(I)=GOPT(I)+TEMP*XPT(K,I)
END IF
C
C Calculate the parameters of the least Frobenius norm interpolant to
C the current data, the gradient of this interpolant at XOPT being put
C into VLAG(NPT+I), I=1,2,...,N.
C
IF (NTRITS .GT. 0) THEN
DO 570 K=1,NPT
VLAG(K)=FVAL(K)-FVAL(KOPT)
570 W(K)=ZERO
DO 590 J=1,NPTM
SUM=ZERO
DO 580 K=1,NPT
580 SUM=SUM+ZMAT(K,J)*VLAG(K)
DO 590 K=1,NPT
590 W(K)=W(K)+SUM*ZMAT(K,J)
DO 610 K=1,NPT
SUM=ZERO
DO 600 J=1,N
600 SUM=SUM+XPT(K,J)*XOPT(J)
W(K+NPT)=W(K)
610 W(K)=SUM*W(K)
GQSQ=ZERO
GISQ=ZERO
DO 630 I=1,N
SUM=ZERO
DO 620 K=1,NPT
620 SUM=SUM+BMAT(K,I)*VLAG(K)+XPT(K,I)*W(K)
IF (XOPT(I) .EQ. SL(I)) THEN
GQSQ=GQSQ+DMIN1(ZERO,GOPT(I))**2
GISQ=GISQ+DMIN1(ZERO,SUM)**2
ELSE IF (XOPT(I) .EQ. SU(I)) THEN
GQSQ=GQSQ+DMAX1(ZERO,GOPT(I))**2
GISQ=GISQ+DMAX1(ZERO,SUM)**2
ELSE
GQSQ=GQSQ+GOPT(I)**2
GISQ=GISQ+SUM*SUM
END IF
630 VLAG(NPT+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. TEN*GISQ) ITEST=0
IF (ITEST .GE. 3) THEN
DO 640 I=1,MAX0(NPT,NH)
IF (I .LE. N) GOPT(I)=VLAG(NPT+I)
IF (I .LE. NPT) PQ(I)=W(NPT+I)
IF (I .LE. NH) HQ(I)=ZERO
ITEST=0
640 CONTINUE
END IF
END IF
C
C If a trust region step has provided a sufficient decrease in F, then
C branch for another trust region calculation. The case NTRITS=0 occurs
C when the new interpolation point was reached by an alternative step.
C
IF (NTRITS .EQ. 0) GOTO 60
IF (F .LE. FOPT+TENTH*VQUAD) GOTO 60
C
C Alternatively, find out if the interpolation points are close enough
C to the best point so far.
C
DISTSQ=DMAX1((TWO*DELTA)**2,(TEN*RHO)**2)
650 KNEW=0
DO 670 K=1,NPT
SUM=ZERO
DO 660 J=1,N
660 SUM=SUM+(XPT(K,J)-XOPT(J))**2
IF (SUM .GT. DISTSQ) THEN
KNEW=K
DISTSQ=SUM
END IF
670 CONTINUE
C
C If KNEW is positive, then ALTMOV finds alternative new positions for
C the KNEW-th interpolation point within distance ADELT of XOPT. It is
C reached via label 90. Otherwise, there is a branch to label 60 for
C another trust region iteration, unless the calculations with the
C current RHO are complete.
C
IF (KNEW .GT. 0) THEN
DIST=DSQRT(DISTSQ)
IF (NTRITS .EQ. -1) THEN
DELTA=DMIN1(TENTH*DELTA,HALF*DIST)
IF (DELTA .LE. 1.5D0*RHO) DELTA=RHO
END IF
NTRITS=0
ADELT=DMAX1(DMIN1(TENTH*DIST,DELTA),RHO)
DSQ=ADELT*ADELT
GOTO 90
END IF
IF (NTRITS .EQ. -1) GOTO 680
IF (RATIO .GT. ZERO) GOTO 60
IF (DMAX1(DELTA,DNORM) .GT. RHO) GOTO 60
C
C The calculations with the current value of RHO are complete. Pick the
C next values of RHO and DELTA.
C
680 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 690
690 FORMAT (5X)
PRINT 700, RHO,NF
700 FORMAT (/4X,'New RHO =',1PD11.4,5X,'Number of',
1 ' function values =',I6)
PRINT 710, FVAL(KOPT),(XBASE(I)+XOPT(I),I=1,N)
710 FORMAT (4X,'Least value of F =',1PD23.15,9X,
1 'The corresponding X is:'/(2X,5D15.6))
END IF
NTRITS=0
NFSAV=NF
GOTO 60
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 (NTRITS .EQ. -1) GOTO 360
720 IF (FVAL(KOPT) .LE. FSAVE) THEN
DO 730 I=1,N
X(I)=DMIN1(DMAX1(XL(I),XBASE(I)+XOPT(I)),XU(I))
IF (XOPT(I) .EQ. SL(I)) X(I)=XL(I)
IF (XOPT(I) .EQ. SU(I)) X(I)=XU(I)
730 CONTINUE
F=FVAL(KOPT)
END IF
IF (IPRINT .GE. 1) THEN
PRINT 740, NF
740 FORMAT (/4X,'At the return from BOBYQA',5X,
1 'Number of function values =',I6)
PRINT 710, F,(X(I),I=1,N)
END IF
RETURN
END
SUBROUTINE ALTMOV (N,NPT,XPT,XOPT,BMAT,ZMAT,NDIM,SL,SU,KOPT,
1 KNEW,ADELT,XNEW,XALT,ALPHA,CAUCHY,GLAG,HCOL,W)
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION XPT(NPT,*),XOPT(*),BMAT(NDIM,*),ZMAT(NPT,*),SL(*),
1 SU(*),XNEW(*),XALT(*),GLAG(*),HCOL(*),W(*)
C
C The arguments N, NPT, XPT, XOPT, BMAT, ZMAT, NDIM, SL and SU all have
C the same meanings as the corresponding arguments of BOBYQB.
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 ADELT is the current trust region bound.
C XNEW will be set to a suitable new position for the interpolation point
C XPT(KNEW,.). Specifically, it satisfies the SL, SU and trust region
C bounds and it should provide a large denominator in the next call of
C UPDATE. The step XNEW-XOPT from XOPT is restricted to moves along the
C straight lines through XOPT and another interpolation point.
C XALT also provides a large value of the modulus of the KNEW-th Lagrange
C function subject to the constraints that have been mentioned, its main
C difference from XNEW being that XALT-XOPT is a constrained version of
C the Cauchy step within the trust region. An exception is that XALT is
C not calculated if all components of GLAG (see below) are zero.
C ALPHA will be set to the KNEW-th diagonal element of the H matrix.
C CAUCHY will be set to the square of the KNEW-th Lagrange function at
C the step XALT-XOPT from XOPT for the vector XALT that is returned,
C except that CAUCHY is set to zero if XALT is not calculated.
C GLAG is a working space vector of length N for the gradient of the
C KNEW-th Lagrange function at XOPT.
C HCOL is a working space vector of length NPT for the second derivative
C coefficients of the KNEW-th Lagrange function.
C W is a working space vector of length 2N that is going to hold the
C constrained Cauchy step from XOPT of the Lagrange function, followed
C by the downhill version of XALT when the uphill step is calculated.
C
C Set the first NPT components of W to the leading elements of the
C KNEW-th column of the H matrix.
C
HALF=0.5D0
ONE=1.0D0
ZERO=0.0D0
CONST=ONE+DSQRT(2.0D0)
DO 10 K=1,NPT
10 HCOL(K)=ZERO
DO 20 J=1,NPT-N-1
TEMP=ZMAT(KNEW,J)
DO 20 K=1,NPT
20 HCOL(K)=HCOL(K)+TEMP*ZMAT(K,J)
ALPHA=HCOL(KNEW)
HA=HALF*ALPHA
C
C Calculate the gradient of the KNEW-th Lagrange function at XOPT.
C
DO 30 I=1,N
30 GLAG(I)=BMAT(KNEW,I)
DO 50 K=1,NPT
TEMP=ZERO
DO 40 J=1,N
40 TEMP=TEMP+XPT(K,J)*XOPT(J)
TEMP=HCOL(K)*TEMP
DO 50 I=1,N
50 GLAG(I)=GLAG(I)+TEMP*XPT(K,I)
C
C Search for a large denominator along the straight lines through XOPT
C and another interpolation point. SLBD and SUBD will be lower and upper
C bounds on the step along each of these lines in turn. PREDSQ will be
C set to the square of the predicted denominator for each line. PRESAV
C will be set to the largest admissible value of PREDSQ that occurs.
C
PRESAV=ZERO
DO 80 K=1,NPT
IF (K .EQ. KOPT) GOTO 80
DDERIV=ZERO
DISTSQ=ZERO
DO 60 I=1,N
TEMP=XPT(K,I)-XOPT(I)
DDERIV=DDERIV+GLAG(I)*TEMP
60 DISTSQ=DISTSQ+TEMP*TEMP
SUBD=ADELT/DSQRT(DISTSQ)
SLBD=-SUBD
ILBD=0
IUBD=0
SUMIN=DMIN1(ONE,SUBD)
C
C Revise SLBD and SUBD if necessary because of the bounds in SL and SU.
C
DO 70 I=1,N
TEMP=XPT(K,I)-XOPT(I)
IF (TEMP .GT. ZERO) THEN
IF (SLBD*TEMP .LT. SL(I)-XOPT(I)) THEN
SLBD=(SL(I)-XOPT(I))/TEMP
ILBD=-I
END IF
IF (SUBD*TEMP .GT. SU(I)-XOPT(I)) THEN
SUBD=DMAX1(SUMIN,(SU(I)-XOPT(I))/TEMP)
IUBD=I
END IF
ELSE IF (TEMP .LT. ZERO) THEN
IF (SLBD*TEMP .GT. SU(I)-XOPT(I)) THEN
SLBD=(SU(I)-XOPT(I))/TEMP
ILBD=I
END IF
IF (SUBD*TEMP .LT. SL(I)-XOPT(I)) THEN
SUBD=DMAX1(SUMIN,(SL(I)-XOPT(I))/TEMP)
IUBD=-I
END IF
END IF
70 CONTINUE
C
C Seek a large modulus of the KNEW-th Lagrange function when the index
C of the other interpolation point on the line through XOPT is KNEW.
C
IF (K .EQ. KNEW) THEN
DIFF=DDERIV-ONE
STEP=SLBD
VLAG=SLBD*(DDERIV-SLBD*DIFF)
ISBD=ILBD
TEMP=SUBD*(DDERIV-SUBD*DIFF)
IF (DABS(TEMP) .GT. DABS(VLAG)) THEN
STEP=SUBD
VLAG=TEMP
ISBD=IUBD
END IF
TEMPD=HALF*DDERIV
TEMPA=TEMPD-DIFF*SLBD
TEMPB=TEMPD-DIFF*SUBD
IF (TEMPA*TEMPB .LT. ZERO) THEN
TEMP=TEMPD*TEMPD/DIFF
IF (DABS(TEMP) .GT. DABS(VLAG)) THEN
STEP=TEMPD/DIFF
VLAG=TEMP
ISBD=0
END IF
END IF
C
C Search along each of the other lines through XOPT and another point.
C
ELSE
STEP=SLBD
VLAG=SLBD*(ONE-SLBD)
ISBD=ILBD
TEMP=SUBD*(ONE-SUBD)
IF (DABS(TEMP) .GT. DABS(VLAG)) THEN
STEP=SUBD
VLAG=TEMP
ISBD=IUBD
END IF
IF (SUBD .GT. HALF) THEN
IF (DABS(VLAG) .LT. 0.25D0) THEN
STEP=HALF
VLAG=0.25D0
ISBD=0
END IF
END IF
VLAG=VLAG*DDERIV
END IF
C
C Calculate PREDSQ for the current line search and maintain PRESAV.
C
TEMP=STEP*(ONE-STEP)*DISTSQ
PREDSQ=VLAG*VLAG*(VLAG*VLAG+HA*TEMP*TEMP)
IF (PREDSQ .GT. PRESAV) THEN
PRESAV=PREDSQ
KSAV=K
STPSAV=STEP
IBDSAV=ISBD
END IF
80 CONTINUE
C
C Construct XNEW in a way that satisfies the bound constraints exactly.
C
DO 90 I=1,N
TEMP=XOPT(I)+STPSAV*(XPT(KSAV,I)-XOPT(I))
90 XNEW(I)=DMAX1(SL(I),DMIN1(SU(I),TEMP))
IF (IBDSAV .LT. 0) XNEW(-IBDSAV)=SL(-IBDSAV)
IF (IBDSAV .GT. 0) XNEW(IBDSAV)=SU(IBDSAV)
C
C Prepare for the iterative method that assembles the constrained Cauchy
C step in W. The sum of squares of the fixed components of W is formed in
C WFIXSQ, and the free components of W are set to BIGSTP.
C
BIGSTP=ADELT+ADELT
IFLAG=0
100 WFIXSQ=ZERO
GGFREE=ZERO
DO 110 I=1,N
W(I)=ZERO
TEMPA=DMIN1(XOPT(I)-SL(I),GLAG(I))
TEMPB=DMAX1(XOPT(I)-SU(I),GLAG(I))