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|
C ----------------------------------------------------------------------
C This file contains the LBFGS algorithm and supporting routines
C
C ****************
C LBFGS SUBROUTINE
C ****************
C
SUBROUTINE LBFGS(N,M,X,F,G,DIAGCO,DIAG,IPRINT,EPS,XTOL,W,IFLAG)
C
INTEGER N,M,IPRINT(2),IFLAG
DOUBLE PRECISION X(N),G(N),DIAG(N),W(N*(2*M+1)+2*M)
DOUBLE PRECISION F,EPS,XTOL
LOGICAL DIAGCO
C
C LIMITED MEMORY BFGS METHOD FOR LARGE SCALE OPTIMIZATION
C JORGE NOCEDAL
C *** July 1990 ***
C
C
C This subroutine solves the unconstrained minimization problem
C
C min F(x), x= (x1,x2,...,xN),
C
C using the limited memory BFGS method. The routine is especially
C effective on problems involving a large number of variables. In
C a typical iteration of this method an approximation Hk to the
C inverse of the Hessian is obtained by applying M BFGS updates to
C a diagonal matrix Hk0, using information from the previous M steps.
C The user specifies the number M, which determines the amount of
C storage required by the routine. The user may also provide the
C diagonal matrices Hk0 if not satisfied with the default choice.
C The algorithm is described in "On the limited memory BFGS method
C for large scale optimization", by D. Liu and J. Nocedal,
C Mathematical Programming B 45 (1989) 503-528.
C
C The user is required to calculate the function value F and its
C gradient G. In order to allow the user complete control over
C these computations, reverse communication is used. The routine
C must be called repeatedly under the control of the parameter
C IFLAG.
C
C The steplength is determined at each iteration by means of the
C line search routine MCVSRCH, which is a slight modification of
C the routine CSRCH written by More' and Thuente.
C
C The calling statement is
C
C CALL LBFGS(N,M,X,F,G,DIAGCO,DIAG,IPRINT,EPS,XTOL,W,IFLAG)
C
C where
C
C N is an INTEGER variable that must be set by the user to the
C number of variables. It is not altered by the routine.
C Restriction: N>0.
C
C M is an INTEGER variable that must be set by the user to
C the number of corrections used in the BFGS update. It
C is not altered by the routine. Values of M less than 3 are
C not recommended; large values of M will result in excessive
C computing time. 3<= M <=7 is recommended. Restriction: M>0.
C
C X is a DOUBLE PRECISION array of length N. On initial entry
C it must be set by the user to the values of the initial
C estimate of the solution vector. On exit with IFLAG=0, it
C contains the values of the variables at the best point
C found (usually a solution).
C
C F is a DOUBLE PRECISION variable. Before initial entry and on
C a re-entry with IFLAG=1, it must be set by the user to
C contain the value of the function F at the point X.
C
C G is a DOUBLE PRECISION array of length N. Before initial
C entry and on a re-entry with IFLAG=1, it must be set by
C the user to contain the components of the gradient G at
C the point X.
C
C DIAGCO is a LOGICAL variable that must be set to .TRUE. if the
C user wishes to provide the diagonal matrix Hk0 at each
C iteration. Otherwise it should be set to .FALSE., in which
C case LBFGS will use a default value described below. If
C DIAGCO is set to .TRUE. the routine will return at each
C iteration of the algorithm with IFLAG=2, and the diagonal
C matrix Hk0 must be provided in the array DIAG.
C
C
C DIAG is a DOUBLE PRECISION array of length N. If DIAGCO=.TRUE.,
C then on initial entry or on re-entry with IFLAG=2, DIAG
C it must be set by the user to contain the values of the
C diagonal matrix Hk0. Restriction: all elements of DIAG
C must be positive.
C
C IPRINT is an INTEGER array of length two which must be set by the
C user.
C
C IPRINT(1) specifies the frequency of the output:
C IPRINT(1) < 0 : no output is generated,
C IPRINT(1) = 0 : output only at first and last iteration,
C IPRINT(1) > 0 : output every IPRINT(1) iterations.
C
C IPRINT(2) specifies the type of output generated:
C IPRINT(2) = 0 : iteration count, number of function
C evaluations, function value, norm of the
C gradient, and steplength,
C IPRINT(2) = 1 : same as IPRINT(2)=0, plus vector of
C variables and gradient vector at the
C initial point,
C IPRINT(2) = 2 : same as IPRINT(2)=1, plus vector of
C variables,
C IPRINT(2) = 3 : same as IPRINT(2)=2, plus gradient vector.
C
C
C EPS is a positive DOUBLE PRECISION variable that must be set by
C the user, and determines the accuracy with which the solution
C is to be found. The subroutine terminates when
C
C ||G|| < EPS max(1,||X||),
C
C where ||.|| denotes the Euclidean norm.
C
C XTOL is a positive DOUBLE PRECISION variable that must be set by
C the user to an estimate of the machine precision (e.g.
C 10**(-16) on a SUN station 3/60). The line search routine will
C terminate if the relative width of the interval of uncertainty
C is less than XTOL.
C
C W is a DOUBLE PRECISION array of length N(2M+1)+2M used as
C workspace for LBFGS. This array must not be altered by the
C user.
C
C IFLAG is an INTEGER variable that must be set to 0 on initial entry
C to the subroutine. A return with IFLAG<0 indicates an error,
C and IFLAG=0 indicates that the routine has terminated without
C detecting errors. On a return with IFLAG=1, the user must
C evaluate the function F and gradient G. On a return with
C IFLAG=2, the user must provide the diagonal matrix Hk0.
C
C The following negative values of IFLAG, detecting an error,
C are possible:
C
C IFLAG=-1 The line search routine MCSRCH failed. The
C parameter INFO provides more detailed information
C (see also the documentation of MCSRCH):
C
C INFO = 0 IMPROPER INPUT PARAMETERS.
C
C INFO = 2 RELATIVE WIDTH OF THE INTERVAL OF
C UNCERTAINTY IS AT MOST XTOL.
C
C INFO = 3 MORE THAN 20 FUNCTION EVALUATIONS WERE
C REQUIRED AT THE PRESENT ITERATION.
C
C INFO = 4 THE STEP IS TOO SMALL.
C
C INFO = 5 THE STEP IS TOO LARGE.
C
C INFO = 6 ROUNDING ERRORS PREVENT FURTHER PROGRESS.
C THERE MAY NOT BE A STEP WHICH SATISFIES
C THE SUFFICIENT DECREASE AND CURVATURE
C CONDITIONS. TOLERANCES MAY BE TOO SMALL.
C
C
C IFLAG=-2 The i-th diagonal element of the diagonal inverse
C Hessian approximation, given in DIAG, is not
C positive.
C
C IFLAG=-3 Improper input parameters for LBFGS (N or M are
C not positive).
C
C
C
C ON THE DRIVER:
C
C The program that calls LBFGS must contain the declaration:
C
C EXTERNAL LB2
C
C LB2 is a BLOCK DATA that defines the default values of several
C parameters described in the COMMON section.
C
C
C
C COMMON:
C
C The subroutine contains one common area, which the user may wish to
C reference:
C
COMMON /LB3/MP,LP,GTOL,STPMIN,STPMAX
C
C MP is an INTEGER variable with default value 6. It is used as the
C unit number for the printing of the monitoring information
C controlled by IPRINT.
C
C LP is an INTEGER variable with default value 6. It is used as the
C unit number for the printing of error messages. This printing
C may be suppressed by setting LP to a non-positive value.
C
C GTOL is a DOUBLE PRECISION variable with default value 0.9, which
C controls the accuracy of the line search routine MCSRCH. If the
C function and gradient evaluations are inexpensive with respect
C to the cost of the iteration (which is sometimes the case when
C solving very large problems) it may be advantageous to set GTOL
C to a small value. A typical small value is 0.1. Restriction:
C GTOL should be greater than 1.D-04.
C
C STPMIN and STPMAX are non-negative DOUBLE PRECISION variables which
C specify lower and uper bounds for the step in the line search.
C Their default values are 1.D-20 and 1.D+20, respectively. These
C values need not be modified unless the exponents are too large
C for the machine being used, or unless the problem is extremely
C badly scaled (in which case the exponents should be increased).
C
C
C MACHINE DEPENDENCIES
C
C The only variables that are machine-dependent are XTOL,
C STPMIN and STPMAX.
C
C
C GENERAL INFORMATION
C
C Other routines called directly: DAXPY, DDOT, LB1, MCSRCH
C
C Input/Output : No input; diagnostic messages on unit MP and
C error messages on unit LP.
C
C
C - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
C
DOUBLE PRECISION GTOL,ONE,ZERO,GNORM,DDOT,STP1,FTOL,STPMIN,
. STPMAX,STP,YS,YY,SQ,YR,BETA,XNORM
INTEGER MP,LP,ITER,NFUN,POINT,ISPT,IYPT,MAXFEV,INFO,
. BOUND,NPT,CP,I,NFEV,INMC,IYCN,ISCN
LOGICAL FINISH
C
SAVE
DATA ONE,ZERO/1.0D+0,0.0D+0/
C
C INITIALIZE
C ----------
C
IF(IFLAG.EQ.0) GO TO 10
GO TO (172,100) IFLAG
10 ITER= 0
IF(N.LE.0.OR.M.LE.0) GO TO 196
IF(GTOL.LE.1.D-04) THEN
IF(LP.GT.0) WRITE(LP,245)
GTOL=9.D-01
ENDIF
NFUN= 1
POINT= 0
FINISH= .FALSE.
IF(DIAGCO) THEN
DO 30 I=1,N
30 IF (DIAG(I).LE.ZERO) GO TO 195
ELSE
DO 40 I=1,N
40 DIAG(I)= 1.0D0
ENDIF
C
C THE WORK VECTOR W IS DIVIDED AS FOLLOWS:
C ---------------------------------------
C THE FIRST N LOCATIONS ARE USED TO STORE THE GRADIENT AND
C OTHER TEMPORARY INFORMATION.
C LOCATIONS (N+1)...(N+M) STORE THE SCALARS RHO.
C LOCATIONS (N+M+1)...(N+2M) STORE THE NUMBERS ALPHA USED
C IN THE FORMULA THAT COMPUTES H*G.
C LOCATIONS (N+2M+1)...(N+2M+NM) STORE THE LAST M SEARCH
C STEPS.
C LOCATIONS (N+2M+NM+1)...(N+2M+2NM) STORE THE LAST M
C GRADIENT DIFFERENCES.
C
C THE SEARCH STEPS AND GRADIENT DIFFERENCES ARE STORED IN A
C CIRCULAR ORDER CONTROLLED BY THE PARAMETER POINT.
C
ISPT= N+2*M
IYPT= ISPT+N*M
DO 50 I=1,N
50 W(ISPT+I)= -G(I)*DIAG(I)
GNORM= DSQRT(DDOT(N,G,1,G,1))
STP1= ONE/GNORM
C
C PARAMETERS FOR LINE SEARCH ROUTINE
C
FTOL= 1.0D-4
MAXFEV= 20
C
IF(IPRINT(1).GE.0) CALL LB1(IPRINT,ITER,NFUN,
* GNORM,N,M,X,F,G,STP,FINISH)
C
C --------------------
C MAIN ITERATION LOOP
C --------------------
C
80 ITER= ITER+1
INFO=0
BOUND=ITER-1
IF(ITER.EQ.1) GO TO 165
IF (ITER .GT. M)BOUND=M
C
YS= DDOT(N,W(IYPT+NPT+1),1,W(ISPT+NPT+1),1)
IF(.NOT.DIAGCO) THEN
YY= DDOT(N,W(IYPT+NPT+1),1,W(IYPT+NPT+1),1)
DO 90 I=1,N
90 DIAG(I)= YS/YY
ELSE
IFLAG=2
RETURN
ENDIF
100 CONTINUE
IF(DIAGCO) THEN
DO 110 I=1,N
110 IF (DIAG(I).LE.ZERO) GO TO 195
ENDIF
C
C COMPUTE -H*G USING THE FORMULA GIVEN IN: Nocedal, J. 1980,
C "Updating quasi-Newton matrices with limited storage",
C Mathematics of Computation, Vol.24, No.151, pp. 773-782.
C ---------------------------------------------------------
C
CP= POINT
IF (POINT.EQ.0) CP=M
W(N+CP)= ONE/YS
DO 112 I=1,N
112 W(I)= -G(I)
CP= POINT
DO 125 I= 1,BOUND
CP=CP-1
IF (CP.EQ. -1)CP=M-1
SQ= DDOT(N,W(ISPT+CP*N+1),1,W,1)
INMC=N+M+CP+1
IYCN=IYPT+CP*N
W(INMC)= W(N+CP+1)*SQ
CALL DAXPY(N,-W(INMC),W(IYCN+1),1,W,1)
125 CONTINUE
C
DO 130 I=1,N
130 W(I)=DIAG(I)*W(I)
C
DO 145 I=1,BOUND
YR= DDOT(N,W(IYPT+CP*N+1),1,W,1)
BETA= W(N+CP+1)*YR
INMC=N+M+CP+1
BETA= W(INMC)-BETA
ISCN=ISPT+CP*N
CALL DAXPY(N,BETA,W(ISCN+1),1,W,1)
CP=CP+1
IF (CP.EQ.M)CP=0
145 CONTINUE
C
C STORE THE NEW SEARCH DIRECTION
C ------------------------------
C
DO 160 I=1,N
160 W(ISPT+POINT*N+I)= W(I)
C
C OBTAIN THE ONE-DIMENSIONAL MINIMIZER OF THE FUNCTION
C BY USING THE LINE SEARCH ROUTINE MCSRCH
C ----------------------------------------------------
165 NFEV=0
STP=ONE
IF (ITER.EQ.1) STP=STP1
DO 170 I=1,N
170 W(I)=G(I)
172 CONTINUE
CALL MCSRCH(N,X,F,G,W(ISPT+POINT*N+1),STP,FTOL,
* XTOL,MAXFEV,INFO,NFEV,DIAG)
IF (INFO .EQ. -1) THEN
IFLAG=1
RETURN
ENDIF
IF (INFO .NE. 1) GO TO 190
NFUN= NFUN + NFEV
C
C COMPUTE THE NEW STEP AND GRADIENT CHANGE
C -----------------------------------------
C
NPT=POINT*N
DO 175 I=1,N
W(ISPT+NPT+I)= STP*W(ISPT+NPT+I)
175 W(IYPT+NPT+I)= G(I)-W(I)
POINT=POINT+1
IF (POINT.EQ.M)POINT=0
C
C TERMINATION TEST
C ----------------
C
GNORM= DSQRT(DDOT(N,G,1,G,1))
XNORM= DSQRT(DDOT(N,X,1,X,1))
XNORM= DMAX1(1.0D0,XNORM)
IF (GNORM/XNORM .LE. EPS) FINISH=.TRUE.
C
IF(IPRINT(1).GE.0) CALL LB1(IPRINT,ITER,NFUN,
* GNORM,N,M,X,F,G,STP,FINISH)
IF (FINISH) THEN
IFLAG=0
RETURN
ENDIF
GO TO 80
C
C ------------------------------------------------------------
C END OF MAIN ITERATION LOOP. ERROR EXITS.
C ------------------------------------------------------------
C
190 IFLAG=-1
IF(LP.GT.0) WRITE(LP,200) INFO
RETURN
195 IFLAG=-2
IF(LP.GT.0) WRITE(LP,235) I
RETURN
196 IFLAG= -3
IF(LP.GT.0) WRITE(LP,240)
C
C FORMATS
C -------
C
200 FORMAT(/' IFLAG= -1 ',/' LINE SEARCH FAILED. SEE'
. ' DOCUMENTATION OF ROUTINE MCSRCH',/' ERROR RETURN'
. ' OF LINE SEARCH: INFO= ',I2,/
. ' POSSIBLE CAUSES: FUNCTION OR GRADIENT ARE INCORRECT',/,
. ' OR INCORRECT TOLERANCES')
235 FORMAT(/' IFLAG= -2',/' THE',I5,'-TH DIAGONAL ELEMENT OF THE',/,
. ' INVERSE HESSIAN APPROXIMATION IS NOT POSITIVE')
240 FORMAT(/' IFLAG= -3',/' IMPROPER INPUT PARAMETERS (N OR M',
. ' ARE NOT POSITIVE)')
245 FORMAT(/' GTOL IS LESS THAN OR EQUAL TO 1.D-04',
. / ' IT HAS BEEN RESET TO 9.D-01')
RETURN
END
C
C LAST LINE OF SUBROUTINE LBFGS
C
C
SUBROUTINE LB1(IPRINT,ITER,NFUN,
* GNORM,N,M,X,F,G,STP,FINISH)
C
C -------------------------------------------------------------
C THIS ROUTINE PRINTS MONITORING INFORMATION. THE FREQUENCY AND
C AMOUNT OF OUTPUT ARE CONTROLLED BY IPRINT.
C -------------------------------------------------------------
C
INTEGER IPRINT(2),ITER,NFUN,LP,MP,N,M
DOUBLE PRECISION X(N),G(N),F,GNORM,STP,GTOL,STPMIN,STPMAX
LOGICAL FINISH
COMMON /LB3/MP,LP,GTOL,STPMIN,STPMAX
C
IF (ITER.EQ.0)THEN
WRITE(MP,10)
WRITE(MP,20) N,M
WRITE(MP,30)F,GNORM
IF (IPRINT(2).GE.1)THEN
WRITE(MP,40)
WRITE(MP,50) (X(I),I=1,N)
WRITE(MP,60)
WRITE(MP,50) (G(I),I=1,N)
ENDIF
WRITE(MP,10)
WRITE(MP,70)
ELSE
IF ((IPRINT(1).EQ.0).AND.(ITER.NE.1.AND..NOT.FINISH))RETURN
IF (IPRINT(1).NE.0)THEN
IF(MOD(ITER-1,IPRINT(1)).EQ.0.OR.FINISH)THEN
IF(IPRINT(2).GT.1.AND.ITER.GT.1) WRITE(MP,70)
WRITE(MP,80)ITER,NFUN,F,GNORM,STP
ELSE
RETURN
ENDIF
ELSE
IF( IPRINT(2).GT.1.AND.FINISH) WRITE(MP,70)
WRITE(MP,80)ITER,NFUN,F,GNORM,STP
ENDIF
IF (IPRINT(2).EQ.2.OR.IPRINT(2).EQ.3)THEN
IF (FINISH)THEN
WRITE(MP,90)
ELSE
WRITE(MP,40)
ENDIF
WRITE(MP,50)(X(I),I=1,N)
IF (IPRINT(2).EQ.3)THEN
WRITE(MP,60)
WRITE(MP,50)(G(I),I=1,N)
ENDIF
ENDIF
IF (FINISH) WRITE(MP,100)
ENDIF
C
10 FORMAT('*************************************************')
20 FORMAT(' N=',I5,' NUMBER OF CORRECTIONS=',I2,
. /, ' INITIAL VALUES')
30 FORMAT(' F= ',1PD10.3,' GNORM= ',1PD10.3)
40 FORMAT(' VECTOR X= ')
50 FORMAT(6(2X,1PD10.3))
60 FORMAT(' GRADIENT VECTOR G= ')
70 FORMAT(/' I NFN',4X,'FUNC',8X,'GNORM',7X,'STEPLENGTH'/)
80 FORMAT(2(I4,1X),3X,3(1PD10.3,2X))
90 FORMAT(' FINAL POINT X= ')
100 FORMAT(/' THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.',
. /' IFLAG = 0')
C
RETURN
END
C ******
C
C
C ----------------------------------------------------------
C DATA
C ----------------------------------------------------------
C
BLOCK DATA LB2
INTEGER LP,MP
DOUBLE PRECISION GTOL,STPMIN,STPMAX
COMMON /LB3/MP,LP,GTOL,STPMIN,STPMAX
DATA MP,LP,GTOL,STPMIN,STPMAX/6,6,9.0D-01,1.0D-20,1.0D+20/
END
C ------------------------------------------------------------------
C
C **************************
C LINE SEARCH ROUTINE MCSRCH
C **************************
C
SUBROUTINE MCSRCH(N,X,F,G,S,STP,FTOL,XTOL,MAXFEV,INFO,NFEV,WA)
INTEGER N,MAXFEV,INFO,NFEV
DOUBLE PRECISION F,STP,FTOL,GTOL,XTOL,STPMIN,STPMAX
DOUBLE PRECISION X(N),G(N),S(N),WA(N)
COMMON /LB3/MP,LP,GTOL,STPMIN,STPMAX
SAVE
C
C SUBROUTINE MCSRCH
C
C A slight modification of the subroutine CSRCH of More' and Thuente.
C The changes are to allow reverse communication, and do not affect
C the performance of the routine.
C
C THE PURPOSE OF MCSRCH IS TO FIND A STEP WHICH SATISFIES
C A SUFFICIENT DECREASE CONDITION AND A CURVATURE CONDITION.
C
C AT EACH STAGE THE SUBROUTINE UPDATES AN INTERVAL OF
C UNCERTAINTY WITH ENDPOINTS STX AND STY. THE INTERVAL OF
C UNCERTAINTY IS INITIALLY CHOSEN SO THAT IT CONTAINS A
C MINIMIZER OF THE MODIFIED FUNCTION
C
C F(X+STP*S) - F(X) - FTOL*STP*(GRADF(X)'S).
C
C IF A STEP IS OBTAINED FOR WHICH THE MODIFIED FUNCTION
C HAS A NONPOSITIVE FUNCTION VALUE AND NONNEGATIVE DERIVATIVE,
C THEN THE INTERVAL OF UNCERTAINTY IS CHOSEN SO THAT IT
C CONTAINS A MINIMIZER OF F(X+STP*S).
C
C THE ALGORITHM IS DESIGNED TO FIND A STEP WHICH SATISFIES
C THE SUFFICIENT DECREASE CONDITION
C
C F(X+STP*S) .LE. F(X) + FTOL*STP*(GRADF(X)'S),
C
C AND THE CURVATURE CONDITION
C
C ABS(GRADF(X+STP*S)'S)) .LE. GTOL*ABS(GRADF(X)'S).
C
C IF FTOL IS LESS THAN GTOL AND IF, FOR EXAMPLE, THE FUNCTION
C IS BOUNDED BELOW, THEN THERE IS ALWAYS A STEP WHICH SATISFIES
C BOTH CONDITIONS. IF NO STEP CAN BE FOUND WHICH SATISFIES BOTH
C CONDITIONS, THEN THE ALGORITHM USUALLY STOPS WHEN ROUNDING
C ERRORS PREVENT FURTHER PROGRESS. IN THIS CASE STP ONLY
C SATISFIES THE SUFFICIENT DECREASE CONDITION.
C
C THE SUBROUTINE STATEMENT IS
C
C SUBROUTINE MCSRCH(N,X,F,G,S,STP,FTOL,XTOL, MAXFEV,INFO,NFEV,WA)
C WHERE
C
C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER
C OF VARIABLES.
C
C X IS AN ARRAY OF LENGTH N. ON INPUT IT MUST CONTAIN THE
C BASE POINT FOR THE LINE SEARCH. ON OUTPUT IT CONTAINS
C X + STP*S.
C
C F IS A VARIABLE. ON INPUT IT MUST CONTAIN THE VALUE OF F
C AT X. ON OUTPUT IT CONTAINS THE VALUE OF F AT X + STP*S.
C
C G IS AN ARRAY OF LENGTH N. ON INPUT IT MUST CONTAIN THE
C GRADIENT OF F AT X. ON OUTPUT IT CONTAINS THE GRADIENT
C OF F AT X + STP*S.
C
C S IS AN INPUT ARRAY OF LENGTH N WHICH SPECIFIES THE
C SEARCH DIRECTION.
C
C STP IS A NONNEGATIVE VARIABLE. ON INPUT STP CONTAINS AN
C INITIAL ESTIMATE OF A SATISFACTORY STEP. ON OUTPUT
C STP CONTAINS THE FINAL ESTIMATE.
C
C FTOL AND GTOL ARE NONNEGATIVE INPUT VARIABLES. (In this reverse
C communication implementation GTOL is defined in a COMMON
C statement.) TERMINATION OCCURS WHEN THE SUFFICIENT DECREASE
C CONDITION AND THE DIRECTIONAL DERIVATIVE CONDITION ARE
C SATISFIED.
C
C XTOL IS A NONNEGATIVE INPUT VARIABLE. TERMINATION OCCURS
C WHEN THE RELATIVE WIDTH OF THE INTERVAL OF UNCERTAINTY
C IS AT MOST XTOL.
C
C STPMIN AND STPMAX ARE NONNEGATIVE INPUT VARIABLES WHICH
C SPECIFY LOWER AND UPPER BOUNDS FOR THE STEP. (In this reverse
C communication implementatin they are defined in a COMMON
C statement).
C
C MAXFEV IS A POSITIVE INTEGER INPUT VARIABLE. TERMINATION
C OCCURS WHEN THE NUMBER OF CALLS TO FCN IS AT LEAST
C MAXFEV BY THE END OF AN ITERATION.
C
C INFO IS AN INTEGER OUTPUT VARIABLE SET AS FOLLOWS:
C
C INFO = 0 IMPROPER INPUT PARAMETERS.
C
C INFO =-1 A RETURN IS MADE TO COMPUTE THE FUNCTION AND GRADIENT.
C
C INFO = 1 THE SUFFICIENT DECREASE CONDITION AND THE
C DIRECTIONAL DERIVATIVE CONDITION HOLD.
C
C INFO = 2 RELATIVE WIDTH OF THE INTERVAL OF UNCERTAINTY
C IS AT MOST XTOL.
C
C INFO = 3 NUMBER OF CALLS TO FCN HAS REACHED MAXFEV.
C
C INFO = 4 THE STEP IS AT THE LOWER BOUND STPMIN.
C
C INFO = 5 THE STEP IS AT THE UPPER BOUND STPMAX.
C
C INFO = 6 ROUNDING ERRORS PREVENT FURTHER PROGRESS.
C THERE MAY NOT BE A STEP WHICH SATISFIES THE
C SUFFICIENT DECREASE AND CURVATURE CONDITIONS.
C TOLERANCES MAY BE TOO SMALL.
C
C NFEV IS AN INTEGER OUTPUT VARIABLE SET TO THE NUMBER OF
C CALLS TO FCN.
C
C WA IS A WORK ARRAY OF LENGTH N.
C
C SUBPROGRAMS CALLED
C
C MCSTEP
C
C FORTRAN-SUPPLIED...ABS,MAX,MIN
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JUNE 1983
C JORGE J. MORE', DAVID J. THUENTE
C
C **********
INTEGER INFOC,J
LOGICAL BRACKT,STAGE1
DOUBLE PRECISION DG,DGM,DGINIT,DGTEST,DGX,DGXM,DGY,DGYM,
* FINIT,FTEST1,FM,FX,FXM,FY,FYM,P5,P66,STX,STY,
* STMIN,STMAX,WIDTH,WIDTH1,XTRAPF,ZERO
DATA P5,P66,XTRAPF,ZERO /0.5D0,0.66D0,4.0D0,0.0D0/
IF(INFO.EQ.-1) GO TO 45
INFOC = 1
C
C CHECK THE INPUT PARAMETERS FOR ERRORS.
C
IF (N .LE. 0 .OR. STP .LE. ZERO .OR. FTOL .LT. ZERO .OR.
* GTOL .LT. ZERO .OR. XTOL .LT. ZERO .OR. STPMIN .LT. ZERO
* .OR. STPMAX .LT. STPMIN .OR. MAXFEV .LE. 0) RETURN
C
C COMPUTE THE INITIAL GRADIENT IN THE SEARCH DIRECTION
C AND CHECK THAT S IS A DESCENT DIRECTION.
C
DGINIT = ZERO
DO 10 J = 1, N
DGINIT = DGINIT + G(J)*S(J)
10 CONTINUE
IF (DGINIT .GE. ZERO) then
write(LP,15)
15 FORMAT(/' THE SEARCH DIRECTION IS NOT A DESCENT DIRECTION')
RETURN
ENDIF
C
C INITIALIZE LOCAL VARIABLES.
C
BRACKT = .FALSE.
STAGE1 = .TRUE.
NFEV = 0
FINIT = F
DGTEST = FTOL*DGINIT
WIDTH = STPMAX - STPMIN
WIDTH1 = WIDTH/P5
DO 20 J = 1, N
WA(J) = X(J)
20 CONTINUE
C
C THE VARIABLES STX, FX, DGX CONTAIN THE VALUES OF THE STEP,
C FUNCTION, AND DIRECTIONAL DERIVATIVE AT THE BEST STEP.
C THE VARIABLES STY, FY, DGY CONTAIN THE VALUE OF THE STEP,
C FUNCTION, AND DERIVATIVE AT THE OTHER ENDPOINT OF
C THE INTERVAL OF UNCERTAINTY.
C THE VARIABLES STP, F, DG CONTAIN THE VALUES OF THE STEP,
C FUNCTION, AND DERIVATIVE AT THE CURRENT STEP.
C
STX = ZERO
FX = FINIT
DGX = DGINIT
STY = ZERO
FY = FINIT
DGY = DGINIT
C
C START OF ITERATION.
C
30 CONTINUE
C
C SET THE MINIMUM AND MAXIMUM STEPS TO CORRESPOND
C TO THE PRESENT INTERVAL OF UNCERTAINTY.
C
IF (BRACKT) THEN
STMIN = MIN(STX,STY)
STMAX = MAX(STX,STY)
ELSE
STMIN = STX
STMAX = STP + XTRAPF*(STP - STX)
END IF
C
C FORCE THE STEP TO BE WITHIN THE BOUNDS STPMAX AND STPMIN.
C
STP = MAX(STP,STPMIN)
STP = MIN(STP,STPMAX)
C
C IF AN UNUSUAL TERMINATION IS TO OCCUR THEN LET
C STP BE THE LOWEST POINT OBTAINED SO FAR.
C
IF ((BRACKT .AND. (STP .LE. STMIN .OR. STP .GE. STMAX))
* .OR. NFEV .GE. MAXFEV-1 .OR. INFOC .EQ. 0
* .OR. (BRACKT .AND. STMAX-STMIN .LE. XTOL*STMAX)) STP = STX
C
C EVALUATE THE FUNCTION AND GRADIENT AT STP
C AND COMPUTE THE DIRECTIONAL DERIVATIVE.
C We return to main program to obtain F and G.
C
DO 40 J = 1, N
X(J) = WA(J) + STP*S(J)
40 CONTINUE
INFO=-1
RETURN
C
45 INFO=0
NFEV = NFEV + 1
DG = ZERO
DO 50 J = 1, N
DG = DG + G(J)*S(J)
50 CONTINUE
FTEST1 = FINIT + STP*DGTEST
C
C TEST FOR CONVERGENCE.
C
IF ((BRACKT .AND. (STP .LE. STMIN .OR. STP .GE. STMAX))
* .OR. INFOC .EQ. 0) INFO = 6
IF (STP .EQ. STPMAX .AND.
* F .LE. FTEST1 .AND. DG .LE. DGTEST) INFO = 5
IF (STP .EQ. STPMIN .AND.
* (F .GT. FTEST1 .OR. DG .GE. DGTEST)) INFO = 4
IF (NFEV .GE. MAXFEV) INFO = 3
IF (BRACKT .AND. STMAX-STMIN .LE. XTOL*STMAX) INFO = 2
IF (F .LE. FTEST1 .AND. ABS(DG) .LE. GTOL*(-DGINIT)) INFO = 1
C
C CHECK FOR TERMINATION.
C
IF (INFO .NE. 0) RETURN
C
C IN THE FIRST STAGE WE SEEK A STEP FOR WHICH THE MODIFIED
C FUNCTION HAS A NONPOSITIVE VALUE AND NONNEGATIVE DERIVATIVE.
C
IF (STAGE1 .AND. F .LE. FTEST1 .AND.
* DG .GE. MIN(FTOL,GTOL)*DGINIT) STAGE1 = .FALSE.
C
C A MODIFIED FUNCTION IS USED TO PREDICT THE STEP ONLY IF
C WE HAVE NOT OBTAINED A STEP FOR WHICH THE MODIFIED
C FUNCTION HAS A NONPOSITIVE FUNCTION VALUE AND NONNEGATIVE
C DERIVATIVE, AND IF A LOWER FUNCTION VALUE HAS BEEN
C OBTAINED BUT THE DECREASE IS NOT SUFFICIENT.
C
IF (STAGE1 .AND. F .LE. FX .AND. F .GT. FTEST1) THEN
C
C DEFINE THE MODIFIED FUNCTION AND DERIVATIVE VALUES.
C
FM = F - STP*DGTEST
FXM = FX - STX*DGTEST
FYM = FY - STY*DGTEST
DGM = DG - DGTEST
DGXM = DGX - DGTEST
DGYM = DGY - DGTEST
C
C CALL CSTEP TO UPDATE THE INTERVAL OF UNCERTAINTY
C AND TO COMPUTE THE NEW STEP.
C
CALL MCSTEP(STX,FXM,DGXM,STY,FYM,DGYM,STP,FM,DGM,
* BRACKT,STMIN,STMAX,INFOC)
C
C RESET THE FUNCTION AND GRADIENT VALUES FOR F.
C
FX = FXM + STX*DGTEST
FY = FYM + STY*DGTEST
DGX = DGXM + DGTEST
DGY = DGYM + DGTEST
ELSE
C
C CALL MCSTEP TO UPDATE THE INTERVAL OF UNCERTAINTY
C AND TO COMPUTE THE NEW STEP.
C
CALL MCSTEP(STX,FX,DGX,STY,FY,DGY,STP,F,DG,
* BRACKT,STMIN,STMAX,INFOC)
END IF
C
C FORCE A SUFFICIENT DECREASE IN THE SIZE OF THE
C INTERVAL OF UNCERTAINTY.
C
IF (BRACKT) THEN
IF (ABS(STY-STX) .GE. P66*WIDTH1)
* STP = STX + P5*(STY - STX)
WIDTH1 = WIDTH
WIDTH = ABS(STY-STX)
END IF
C
C END OF ITERATION.
C
GO TO 30
C
C LAST LINE OF SUBROUTINE MCSRCH.
C
END
SUBROUTINE MCSTEP(STX,FX,DX,STY,FY,DY,STP,FP,DP,BRACKT,
* STPMIN,STPMAX,INFO)
INTEGER INFO
DOUBLE PRECISION STX,FX,DX,STY,FY,DY,STP,FP,DP,STPMIN,STPMAX
LOGICAL BRACKT,BOUND
C
C SUBROUTINE MCSTEP
C
C THE PURPOSE OF MCSTEP IS TO COMPUTE A SAFEGUARDED STEP FOR
C A LINESEARCH AND TO UPDATE AN INTERVAL OF UNCERTAINTY FOR
C A MINIMIZER OF THE FUNCTION.
C
C THE PARAMETER STX CONTAINS THE STEP WITH THE LEAST FUNCTION
C VALUE. THE PARAMETER STP CONTAINS THE CURRENT STEP. IT IS
C ASSUMED THAT THE DERIVATIVE AT STX IS NEGATIVE IN THE
C DIRECTION OF THE STEP. IF BRACKT IS SET TRUE THEN A
C MINIMIZER HAS BEEN BRACKETED IN AN INTERVAL OF UNCERTAINTY
C WITH ENDPOINTS STX AND STY.
C
C THE SUBROUTINE STATEMENT IS
C
C SUBROUTINE MCSTEP(STX,FX,DX,STY,FY,DY,STP,FP,DP,BRACKT,
C STPMIN,STPMAX,INFO)
C
C WHERE
C
C STX, FX, AND DX ARE VARIABLES WHICH SPECIFY THE STEP,
C THE FUNCTION, AND THE DERIVATIVE AT THE BEST STEP OBTAINED
C SO FAR. THE DERIVATIVE MUST BE NEGATIVE IN THE DIRECTION
C OF THE STEP, THAT IS, DX AND STP-STX MUST HAVE OPPOSITE
C SIGNS. ON OUTPUT THESE PARAMETERS ARE UPDATED APPROPRIATELY.
C
C STY, FY, AND DY ARE VARIABLES WHICH SPECIFY THE STEP,
C THE FUNCTION, AND THE DERIVATIVE AT THE OTHER ENDPOINT OF
C THE INTERVAL OF UNCERTAINTY. ON OUTPUT THESE PARAMETERS ARE
C UPDATED APPROPRIATELY.
C
C STP, FP, AND DP ARE VARIABLES WHICH SPECIFY THE STEP,
C THE FUNCTION, AND THE DERIVATIVE AT THE CURRENT STEP.
C IF BRACKT IS SET TRUE THEN ON INPUT STP MUST BE
C BETWEEN STX AND STY. ON OUTPUT STP IS SET TO THE NEW STEP.
C
C BRACKT IS A LOGICAL VARIABLE WHICH SPECIFIES IF A MINIMIZER
C HAS BEEN BRACKETED. IF THE MINIMIZER HAS NOT BEEN BRACKETED
C THEN ON INPUT BRACKT MUST BE SET FALSE. IF THE MINIMIZER
C IS BRACKETED THEN ON OUTPUT BRACKT IS SET TRUE.
C
C STPMIN AND STPMAX ARE INPUT VARIABLES WHICH SPECIFY LOWER
C AND UPPER BOUNDS FOR THE STEP.
C
C INFO IS AN INTEGER OUTPUT VARIABLE SET AS FOLLOWS:
C IF INFO = 1,2,3,4,5, THEN THE STEP HAS BEEN COMPUTED
C ACCORDING TO ONE OF THE FIVE CASES BELOW. OTHERWISE
C INFO = 0, AND THIS INDICATES IMPROPER INPUT PARAMETERS.
C
C SUBPROGRAMS CALLED
C
C FORTRAN-SUPPLIED ... ABS,MAX,MIN,SQRT
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JUNE 1983
C JORGE J. MORE', DAVID J. THUENTE
C
DOUBLE PRECISION GAMMA,P,Q,R,S,SGND,STPC,STPF,STPQ,THETA
INFO = 0
C
C CHECK THE INPUT PARAMETERS FOR ERRORS.
C
IF ((BRACKT .AND. (STP .LE. MIN(STX,STY) .OR.
* STP .GE. MAX(STX,STY))) .OR.
* DX*(STP-STX) .GE. 0.0 .OR. STPMAX .LT. STPMIN) RETURN
C
C DETERMINE IF THE DERIVATIVES HAVE OPPOSITE SIGN.
C
SGND = DP*(DX/ABS(DX))
C
C FIRST CASE. A HIGHER FUNCTION VALUE.
C THE MINIMUM IS BRACKETED. IF THE CUBIC STEP IS CLOSER
C TO STX THAN THE QUADRATIC STEP, THE CUBIC STEP IS TAKEN,
C ELSE THE AVERAGE OF THE CUBIC AND QUADRATIC STEPS IS TAKEN.
C
IF (FP .GT. FX) THEN
INFO = 1
BOUND = .TRUE.
THETA = 3*(FX - FP)/(STP - STX) + DX + DP
S = MAX(ABS(THETA),ABS(DX),ABS(DP))
GAMMA = S*SQRT((THETA/S)**2 - (DX/S)*(DP/S))
IF (STP .LT. STX) GAMMA = -GAMMA
P = (GAMMA - DX) + THETA
Q = ((GAMMA - DX) + GAMMA) + DP
R = P/Q
STPC = STX + R*(STP - STX)
STPQ = STX + ((DX/((FX-FP)/(STP-STX)+DX))/2)*(STP - STX)
IF (ABS(STPC-STX) .LT. ABS(STPQ-STX)) THEN
STPF = STPC
ELSE
STPF = STPC + (STPQ - STPC)/2
END IF
BRACKT = .TRUE.
C
C SECOND CASE. A LOWER FUNCTION VALUE AND DERIVATIVES OF
C OPPOSITE SIGN. THE MINIMUM IS BRACKETED. IF THE CUBIC
C STEP IS CLOSER TO STX THAN THE QUADRATIC (SECANT) STEP,
C THE CUBIC STEP IS TAKEN, ELSE THE QUADRATIC STEP IS TAKEN.
C
ELSE IF (SGND .LT. 0.0) THEN
INFO = 2
BOUND = .FALSE.
THETA = 3*(FX - FP)/(STP - STX) + DX + DP
S = MAX(ABS(THETA),ABS(DX),ABS(DP))
GAMMA = S*SQRT((THETA/S)**2 - (DX/S)*(DP/S))
IF (STP .GT. STX) GAMMA = -GAMMA
P = (GAMMA - DP) + THETA
Q = ((GAMMA - DP) + GAMMA) + DX
R = P/Q
STPC = STP + R*(STX - STP)
STPQ = STP + (DP/(DP-DX))*(STX - STP)
IF (ABS(STPC-STP) .GT. ABS(STPQ-STP)) THEN
STPF = STPC
ELSE
STPF = STPQ
END IF
BRACKT = .TRUE.
C
C THIRD CASE. A LOWER FUNCTION VALUE, DERIVATIVES OF THE
C SAME SIGN, AND THE MAGNITUDE OF THE DERIVATIVE DECREASES.
C THE CUBIC STEP IS ONLY USED IF THE CUBIC TENDS TO INFINITY
C IN THE DIRECTION OF THE STEP OR IF THE MINIMUM OF THE CUBIC
C IS BEYOND STP. OTHERWISE THE CUBIC STEP IS DEFINED TO BE
C EITHER STPMIN OR STPMAX. THE QUADRATIC (SECANT) STEP IS ALSO
C COMPUTED AND IF THE MINIMUM IS BRACKETED THEN THE THE STEP
C CLOSEST TO STX IS TAKEN, ELSE THE STEP FARTHEST AWAY IS TAKEN.
C
ELSE IF (ABS(DP) .LT. ABS(DX)) THEN
INFO = 3
BOUND = .TRUE.
THETA = 3*(FX - FP)/(STP - STX) + DX + DP
S = MAX(ABS(THETA),ABS(DX),ABS(DP))
C
C THE CASE GAMMA = 0 ONLY ARISES IF THE CUBIC DOES NOT TEND
C TO INFINITY IN THE DIRECTION OF THE STEP.
C
GAMMA = S*SQRT(MAX(0.0D0,(THETA/S)**2 - (DX/S)*(DP/S)))
IF (STP .GT. STX) GAMMA = -GAMMA
P = (GAMMA - DP) + THETA
Q = (GAMMA + (DX - DP)) + GAMMA
R = P/Q
IF (R .LT. 0.0 .AND. GAMMA .NE. 0.0) THEN
STPC = STP + R*(STX - STP)
ELSE IF (STP .GT. STX) THEN
STPC = STPMAX
ELSE
STPC = STPMIN
END IF
STPQ = STP + (DP/(DP-DX))*(STX - STP)
IF (BRACKT) THEN
IF (ABS(STP-STPC) .LT. ABS(STP-STPQ)) THEN
STPF = STPC
ELSE
STPF = STPQ
END IF
ELSE
IF (ABS(STP-STPC) .GT. ABS(STP-STPQ)) THEN
STPF = STPC
ELSE
STPF = STPQ
END IF
END IF
C
C FOURTH CASE. A LOWER FUNCTION VALUE, DERIVATIVES OF THE
C SAME SIGN, AND THE MAGNITUDE OF THE DERIVATIVE DOES
C NOT DECREASE. IF THE MINIMUM IS NOT BRACKETED, THE STEP
C IS EITHER STPMIN OR STPMAX, ELSE THE CUBIC STEP IS TAKEN.
C
ELSE
INFO = 4
BOUND = .FALSE.
IF (BRACKT) THEN
THETA = 3*(FP - FY)/(STY - STP) + DY + DP
S = MAX(ABS(THETA),ABS(DY),ABS(DP))
GAMMA = S*SQRT((THETA/S)**2 - (DY/S)*(DP/S))
IF (STP .GT. STY) GAMMA = -GAMMA
P = (GAMMA - DP) + THETA
Q = ((GAMMA - DP) + GAMMA) + DY
R = P/Q
STPC = STP + R*(STY - STP)
STPF = STPC
ELSE IF (STP .GT. STX) THEN
STPF = STPMAX
ELSE
STPF = STPMIN
END IF
END IF
C
C UPDATE THE INTERVAL OF UNCERTAINTY. THIS UPDATE DOES NOT
C DEPEND ON THE NEW STEP OR THE CASE ANALYSIS ABOVE.
C
IF (FP .GT. FX) THEN
STY = STP
FY = FP
DY = DP
ELSE
IF (SGND .LT. 0.0) THEN
STY = STX
FY = FX
DY = DX
END IF
STX = STP
FX = FP
DX = DP
END IF
C
C COMPUTE THE NEW STEP AND SAFEGUARD IT.
C
STPF = MIN(STPMAX,STPF)
STPF = MAX(STPMIN,STPF)
STP = STPF
IF (BRACKT .AND. BOUND) THEN
IF (STY .GT. STX) THEN
STP = MIN(STX+0.66*(STY-STX),STP)
ELSE
STP = MAX(STX+0.66*(STY-STX),STP)
END IF
END IF
RETURN
C
C LAST LINE OF SUBROUTINE MCSTEP.
C
END
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