C **********
C
C THIS PROGRAM TESTS CODES FOR THE SOLUTION OF N NONLINEAR
C EQUATIONS IN N VARIABLES. IT CONSISTS OF A DRIVER AND AN
C INTERFACE SUBROUTINE FCN. THE DRIVER READS IN DATA, CALLS THE
C NONLINEAR EQUATION SOLVER, AND FINALLY PRINTS OUT INFORMATION
C ON THE PERFORMANCE OF THE SOLVER. THIS IS ONLY A SAMPLE DRIVER,
C MANY OTHER DRIVERS ARE POSSIBLE. THE INTERFACE SUBROUTINE FCN
C IS NECESSARY TO TAKE INTO ACCOUNT THE FORMS OF CALLING
C SEQUENCES USED BY THE FUNCTION SUBROUTINES IN THE VARIOUS
C NONLINEAR EQUATION SOLVERS.
C
C SUBPROGRAMS CALLED
C
C USER-SUPPLIED ...... FCN
C
C MINPACK-SUPPLIED ... DPMPAR,ENORM,HYBRD1,INITPT,VECFCN
C
C FORTRAN-SUPPLIED ... DSQRT
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. MARCH 1980.
C BURTON S. GARBOW, KENNETH E. HILLSTROM, JORGE J. MORE
C
C **********
INTEGER I,IC,INFO,K,LWA,N,NFEV,NPROB,NREAD,NTRIES,NWRITE
INTEGER NA(60),NF(60),NP(60),NX(60)
DOUBLE PRECISION FACTOR,FNORM1,FNORM2,ONE,TEN,TOL
DOUBLE PRECISION FNM(60),FVEC(40),WA(2660),X(40)
DOUBLE PRECISION DPMPAR,ENORM
EXTERNAL FCN
COMMON /REFNUM/ NPROB,NFEV
C
C LOGICAL INPUT UNIT IS ASSUMED TO BE NUMBER 5.
C LOGICAL OUTPUT UNIT IS ASSUMED TO BE NUMBER 6.
C
DATA NREAD,NWRITE /5,6/
C
DATA ONE,TEN /1.0D0,1.0D1/
TOL = DSQRT(DPMPAR(1))
LWA = 2660
IC = 0
10 CONTINUE
READ (NREAD,50) NPROB,N,NTRIES
IF (NPROB .LE. 0) GO TO 30
FACTOR = ONE
DO 20 K = 1, NTRIES
IC = IC + 1
CALL INITPT(N,X,NPROB,FACTOR)
CALL VECFCN(N,X,FVEC,NPROB)
FNORM1 = ENORM(N,FVEC)
WRITE (NWRITE,60) NPROB,N
NFEV = 0
CALL HYBRD1(FCN,N,X,FVEC,TOL,INFO,WA,LWA)
FNORM2 = ENORM(N,FVEC)
NP(IC) = NPROB
NA(IC) = N
NF(IC) = NFEV
NX(IC) = INFO
FNM(IC) = FNORM2
WRITE (NWRITE,70) FNORM1,FNORM2,NFEV,INFO,(X(I), I = 1, N)
FACTOR = TEN*FACTOR
20 CONTINUE
GO TO 10
30 CONTINUE
WRITE (NWRITE,80) IC
WRITE (NWRITE,90)
DO 40 I = 1, IC
WRITE (NWRITE,100) NP(I),NA(I),NF(I),NX(I),FNM(I)
40 CONTINUE
STOP
50 FORMAT (3I5)
60 FORMAT ( //// 5X, 8H PROBLEM, I5, 5X, 10H DIMENSION, I5, 5X //)
70 FORMAT (5X, 33H INITIAL L2 NORM OF THE RESIDUALS, D15.7 // 5X,
* 33H FINAL L2 NORM OF THE RESIDUALS , D15.7 // 5X,
* 33H NUMBER OF FUNCTION EVALUATIONS , I10 // 5X,
* 15H EXIT PARAMETER, 18X, I10 // 5X,
* 27H FINAL APPROXIMATE SOLUTION // (5X, 5D15.7))
80 FORMAT (12H1SUMMARY OF , I3, 16H CALLS TO HYBRD1 /)
90 FORMAT (39H NPROB N NFEV INFO FINAL L2 NORM /)
100 FORMAT (I4, I6, I7, I6, 1X, D15.7)
C
C LAST CARD OF DRIVER.
C
END
SUBROUTINE FCN(N,X,FVEC,IFLAG)
INTEGER N,IFLAG
DOUBLE PRECISION X(N),FVEC(N)
C **********
C
C THE CALLING SEQUENCE OF FCN SHOULD BE IDENTICAL TO THE
C CALLING SEQUENCE OF THE FUNCTION SUBROUTINE IN THE NONLINEAR
C EQUATION SOLVER. FCN SHOULD ONLY CALL THE TESTING FUNCTION
C SUBROUTINE VECFCN WITH THE APPROPRIATE VALUE OF PROBLEM
C NUMBER (NPROB).
C
C SUBPROGRAMS CALLED
C
C MINPACK-SUPPLIED ... VECFCN
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. MARCH 1980.
C BURTON S. GARBOW, KENNETH E. HILLSTROM, JORGE J. MORE
C
C **********
INTEGER NPROB,NFEV
COMMON /REFNUM/ NPROB,NFEV
CALL VECFCN(N,X,FVEC,NPROB)
NFEV = NFEV + 1
RETURN
C
C LAST CARD OF INTERFACE SUBROUTINE FCN.
C
END
SUBROUTINE VECFCN(N,X,FVEC,NPROB)
INTEGER N,NPROB
DOUBLE PRECISION X(N),FVEC(N)
C **********
C
C SUBROUTINE VECFCN
C
C THIS SUBROUTINE DEFINES FOURTEEN TEST FUNCTIONS. THE FIRST
C FIVE TEST FUNCTIONS ARE OF DIMENSIONS 2,4,2,4,3, RESPECTIVELY,
C WHILE THE REMAINING TEST FUNCTIONS ARE OF VARIABLE DIMENSION
C N FOR ANY N GREATER THAN OR EQUAL TO 1 (PROBLEM 6 IS AN
C EXCEPTION TO THIS, SINCE IT DOES NOT ALLOW N = 1).
C
C THE SUBROUTINE STATEMENT IS
C
C SUBROUTINE VECFCN(N,X,FVEC,NPROB)
C
C WHERE
C
C N IS A POSITIVE INTEGER INPUT VARIABLE.
C
C X IS AN INPUT ARRAY OF LENGTH N.
C
C FVEC IS AN OUTPUT ARRAY OF LENGTH N WHICH CONTAINS THE NPROB
C FUNCTION VECTOR EVALUATED AT X.
C
C NPROB IS A POSITIVE INTEGER INPUT VARIABLE WHICH DEFINES THE
C NUMBER OF THE PROBLEM. NPROB MUST NOT EXCEED 14.
C
C SUBPROGRAMS CALLED
C
C FORTRAN-SUPPLIED ... DATAN,DCOS,DEXP,DSIGN,DSIN,DSQRT,
C MAX0,MIN0
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. MARCH 1980.
C BURTON S. GARBOW, KENNETH E. HILLSTROM, JORGE J. MORE
C
C **********
INTEGER I,IEV,IVAR,J,K,K1,K2,KP1,ML,MU
DOUBLE PRECISION C1,C2,C3,C4,C5,C6,C7,C8,C9,EIGHT,FIVE,H,ONE,
* PROD,SUM,SUM1,SUM2,TEMP,TEMP1,TEMP2,TEN,THREE,
* TI,TJ,TK,TPI,TWO,ZERO
DOUBLE PRECISION DFLOAT
DATA ZERO,ONE,TWO,THREE,FIVE,EIGHT,TEN
* /0.0D0,1.0D0,2.0D0,3.0D0,5.0D0,8.0D0,1.0D1/
DATA C1,C2,C3,C4,C5,C6,C7,C8,C9
* /1.0D4,1.0001D0,2.0D2,2.02D1,1.98D1,1.8D2,2.5D-1,5.0D-1,
* 2.9D1/
DFLOAT(IVAR) = IVAR
C
C PROBLEM SELECTOR.
C
GO TO (10,20,30,40,50,60,120,170,200,220,270,300,330,350), NPROB
C
C ROSENBROCK FUNCTION.
C
10 CONTINUE
FVEC(1) = ONE - X(1)
FVEC(2) = TEN*(X(2) - X(1)**2)
GO TO 380
C
C POWELL SINGULAR FUNCTION.
C
20 CONTINUE
FVEC(1) = X(1) + TEN*X(2)
FVEC(2) = DSQRT(FIVE)*(X(3) - X(4))
FVEC(3) = (X(2) - TWO*X(3))**2
FVEC(4) = DSQRT(TEN)*(X(1) - X(4))**2
GO TO 380
C
C POWELL BADLY SCALED FUNCTION.
C
30 CONTINUE
FVEC(1) = C1*X(1)*X(2) - ONE
FVEC(2) = DEXP(-X(1)) + DEXP(-X(2)) - C2
GO TO 380
C
C WOOD FUNCTION.
C
40 CONTINUE
TEMP1 = X(2) - X(1)**2
TEMP2 = X(4) - X(3)**2
FVEC(1) = -C3*X(1)*TEMP1 - (ONE - X(1))
FVEC(2) = C3*TEMP1 + C4*(X(2) - ONE) + C5*(X(4) - ONE)
FVEC(3) = -C6*X(3)*TEMP2 - (ONE - X(3))
FVEC(4) = C6*TEMP2 + C4*(X(4) - ONE) + C5*(X(2) - ONE)
GO TO 380
C
C HELICAL VALLEY FUNCTION.
C
50 CONTINUE
TPI = EIGHT*DATAN(ONE)
TEMP1 = DSIGN(C7,X(2))
IF (X(1) .GT. ZERO) TEMP1 = DATAN(X(2)/X(1))/TPI
IF (X(1) .LT. ZERO) TEMP1 = DATAN(X(2)/X(1))/TPI + C8
TEMP2 = DSQRT(X(1)**2+X(2)**2)
FVEC(1) = TEN*(X(3) - TEN*TEMP1)
FVEC(2) = TEN*(TEMP2 - ONE)
FVEC(3) = X(3)
GO TO 380
C
C WATSON FUNCTION.
C
60 CONTINUE
DO 70 K = 1, N
FVEC(K) = ZERO
70 CONTINUE
DO 110 I = 1, 29
TI = DFLOAT(I)/C9
SUM1 = ZERO
TEMP = ONE
DO 80 J = 2, N
SUM1 = SUM1 + DFLOAT(J-1)*TEMP*X(J)
TEMP = TI*TEMP
80 CONTINUE
SUM2 = ZERO
TEMP = ONE
DO 90 J = 1, N
SUM2 = SUM2 + TEMP*X(J)
TEMP = TI*TEMP
90 CONTINUE
TEMP1 = SUM1 - SUM2**2 - ONE
TEMP2 = TWO*TI*SUM2
TEMP = ONE/TI
DO 100 K = 1, N
FVEC(K) = FVEC(K) + TEMP*(DFLOAT(K-1) - TEMP2)*TEMP1
TEMP = TI*TEMP
100 CONTINUE
110 CONTINUE
TEMP = X(2) - X(1)**2 - ONE
FVEC(1) = FVEC(1) + X(1)*(ONE - TWO*TEMP)
FVEC(2) = FVEC(2) + TEMP
GO TO 380
C
C CHEBYQUAD FUNCTION.
C
120 CONTINUE
DO 130 K = 1, N
FVEC(K) = ZERO
130 CONTINUE
DO 150 J = 1, N
TEMP1 = ONE
TEMP2 = TWO*X(J) - ONE
TEMP = TWO*TEMP2
DO 140 I = 1, N
FVEC(I) = FVEC(I) + TEMP2
TI = TEMP*TEMP2 - TEMP1
TEMP1 = TEMP2
TEMP2 = TI
140 CONTINUE
150 CONTINUE
TK = ONE/DFLOAT(N)
IEV = -1
DO 160 K = 1, N
FVEC(K) = TK*FVEC(K)
IF (IEV .GT. 0) FVEC(K) = FVEC(K) + ONE/(DFLOAT(K)**2 - ONE)
IEV = -IEV
160 CONTINUE
GO TO 380
C
C BROWN ALMOST-LINEAR FUNCTION.
C
170 CONTINUE
SUM = -DFLOAT(N+1)
PROD = ONE
DO 180 J = 1, N
SUM = SUM + X(J)
PROD = X(J)*PROD
180 CONTINUE
DO 190 K = 1, N
FVEC(K) = X(K) + SUM
190 CONTINUE
FVEC(N) = PROD - ONE
GO TO 380
C
C DISCRETE BOUNDARY VALUE FUNCTION.
C
200 CONTINUE
H = ONE/DFLOAT(N+1)
DO 210 K = 1, N
TEMP = (X(K) + DFLOAT(K)*H + ONE)**3
TEMP1 = ZERO
IF (K .NE. 1) TEMP1 = X(K-1)
TEMP2 = ZERO
IF (K .NE. N) TEMP2 = X(K+1)
FVEC(K) = TWO*X(K) - TEMP1 - TEMP2 + TEMP*H**2/TWO
210 CONTINUE
GO TO 380
C
C DISCRETE INTEGRAL EQUATION FUNCTION.
C
220 CONTINUE
H = ONE/DFLOAT(N+1)
DO 260 K = 1, N
TK = DFLOAT(K)*H
SUM1 = ZERO
DO 230 J = 1, K
TJ = DFLOAT(J)*H
TEMP = (X(J) + TJ + ONE)**3
SUM1 = SUM1 + TJ*TEMP
230 CONTINUE
SUM2 = ZERO
KP1 = K + 1
IF (N .LT. KP1) GO TO 250
DO 240 J = KP1, N
TJ = DFLOAT(J)*H
TEMP = (X(J) + TJ + ONE)**3
SUM2 = SUM2 + (ONE - TJ)*TEMP
240 CONTINUE
250 CONTINUE
FVEC(K) = X(K) + H*((ONE - TK)*SUM1 + TK*SUM2)/TWO
260 CONTINUE
GO TO 380
C
C TRIGONOMETRIC FUNCTION.
C
270 CONTINUE
SUM = ZERO
DO 280 J = 1, N
FVEC(J) = DCOS(X(J))
SUM = SUM + FVEC(J)
280 CONTINUE
DO 290 K = 1, N
FVEC(K) = DFLOAT(N+K) - DSIN(X(K)) - SUM - DFLOAT(K)*FVEC(K)
290 CONTINUE
GO TO 380
C
C VARIABLY DIMENSIONED FUNCTION.
C
300 CONTINUE
SUM = ZERO
DO 310 J = 1, N
SUM = SUM + DFLOAT(J)*(X(J) - ONE)
310 CONTINUE
TEMP = SUM*(ONE + TWO*SUM**2)
DO 320 K = 1, N
FVEC(K) = X(K) - ONE + DFLOAT(K)*TEMP
320 CONTINUE
GO TO 380
C
C BROYDEN TRIDIAGONAL FUNCTION.
C
330 CONTINUE
DO 340 K = 1, N
TEMP = (THREE - TWO*X(K))*X(K)
TEMP1 = ZERO
IF (K .NE. 1) TEMP1 = X(K-1)
TEMP2 = ZERO
IF (K .NE. N) TEMP2 = X(K+1)
FVEC(K) = TEMP - TEMP1 - TWO*TEMP2 + ONE
340 CONTINUE
GO TO 380
C
C BROYDEN BANDED FUNCTION.
C
350 CONTINUE
ML = 5
MU = 1
DO 370 K = 1, N
K1 = MAX0(1,K-ML)
K2 = MIN0(K+MU,N)
TEMP = ZERO
DO 360 J = K1, K2
IF (J .NE. K) TEMP = TEMP + X(J)*(ONE + X(J))
360 CONTINUE
FVEC(K) = X(K)*(TWO + FIVE*X(K)**2) + ONE - TEMP
370 CONTINUE
380 CONTINUE
RETURN
C
C LAST CARD OF SUBROUTINE VECFCN.
C
END
SUBROUTINE INITPT(N,X,NPROB,FACTOR)
INTEGER N,NPROB
DOUBLE PRECISION FACTOR
DOUBLE PRECISION X(N)
C **********
C
C SUBROUTINE INITPT
C
C THIS SUBROUTINE SPECIFIES THE STANDARD STARTING POINTS FOR
C THE FUNCTIONS DEFINED BY SUBROUTINE VECFCN. THE SUBROUTINE
C RETURNS IN X A MULTIPLE (FACTOR) OF THE STANDARD STARTING
C POINT. FOR THE SIXTH FUNCTION THE STANDARD STARTING POINT IS
C ZERO, SO IN THIS CASE, IF FACTOR IS NOT UNITY, THEN THE
C SUBROUTINE RETURNS THE VECTOR X(J) = FACTOR, J=1,...,N.
C
C THE SUBROUTINE STATEMENT IS
C
C SUBROUTINE INITPT(N,X,NPROB,FACTOR)
C
C WHERE
C
C N IS A POSITIVE INTEGER INPUT VARIABLE.
C
C X IS AN OUTPUT ARRAY OF LENGTH N WHICH CONTAINS THE STANDARD
C STARTING POINT FOR PROBLEM NPROB MULTIPLIED BY FACTOR.
C
C NPROB IS A POSITIVE INTEGER INPUT VARIABLE WHICH DEFINES THE
C NUMBER OF THE PROBLEM. NPROB MUST NOT EXCEED 14.
C
C FACTOR IS AN INPUT VARIABLE WHICH SPECIFIES THE MULTIPLE OF
C THE STANDARD STARTING POINT. IF FACTOR IS UNITY, NO
C MULTIPLICATION IS PERFORMED.
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. MARCH 1980.
C BURTON S. GARBOW, KENNETH E. HILLSTROM, JORGE J. MORE
C
C **********
INTEGER IVAR,J
DOUBLE PRECISION C1,H,HALF,ONE,THREE,TJ,ZERO
DOUBLE PRECISION DFLOAT
DATA ZERO,HALF,ONE,THREE,C1 /0.0D0,5.0D-1,1.0D0,3.0D0,1.2D0/
DFLOAT(IVAR) = IVAR
C
C SELECTION OF INITIAL POINT.
C
GO TO (10,20,30,40,50,60,80,100,120,120,140,160,180,180), NPROB
C
C ROSENBROCK FUNCTION.
C
10 CONTINUE
X(1) = -C1
X(2) = ONE
GO TO 200
C
C POWELL SINGULAR FUNCTION.
C
20 CONTINUE
X(1) = THREE
X(2) = -ONE
X(3) = ZERO
X(4) = ONE
GO TO 200
C
C POWELL BADLY SCALED FUNCTION.
C
30 CONTINUE
X(1) = ZERO
X(2) = ONE
GO TO 200
C
C WOOD FUNCTION.
C
40 CONTINUE
X(1) = -THREE
X(2) = -ONE
X(3) = -THREE
X(4) = -ONE
GO TO 200
C
C HELICAL VALLEY FUNCTION.
C
50 CONTINUE
X(1) = -ONE
X(2) = ZERO
X(3) = ZERO
GO TO 200
C
C WATSON FUNCTION.
C
60 CONTINUE
DO 70 J = 1, N
X(J) = ZERO
70 CONTINUE
GO TO 200
C
C CHEBYQUAD FUNCTION.
C
80 CONTINUE
H = ONE/DFLOAT(N+1)
DO 90 J = 1, N
X(J) = DFLOAT(J)*H
90 CONTINUE
GO TO 200
C
C BROWN ALMOST-LINEAR FUNCTION.
C
100 CONTINUE
DO 110 J = 1, N
X(J) = HALF
110 CONTINUE
GO TO 200
C
C DISCRETE BOUNDARY VALUE AND INTEGRAL EQUATION FUNCTIONS.
C
120 CONTINUE
H = ONE/DFLOAT(N+1)
DO 130 J = 1, N
TJ = DFLOAT(J)*H
X(J) = TJ*(TJ - ONE)
130 CONTINUE
GO TO 200
C
C TRIGONOMETRIC FUNCTION.
C
140 CONTINUE
H = ONE/DFLOAT(N)
DO 150 J = 1, N
X(J) = H
150 CONTINUE
GO TO 200
C
C VARIABLY DIMENSIONED FUNCTION.
C
160 CONTINUE
H = ONE/DFLOAT(N)
DO 170 J = 1, N
X(J) = ONE - DFLOAT(J)*H
170 CONTINUE
GO TO 200
C
C BROYDEN TRIDIAGONAL AND BANDED FUNCTIONS.
C
180 CONTINUE
DO 190 J = 1, N
X(J) = -ONE
190 CONTINUE
200 CONTINUE
C
C COMPUTE MULTIPLE OF INITIAL POINT.
C
IF (FACTOR .EQ. ONE) GO TO 250
IF (NPROB .EQ. 6) GO TO 220
DO 210 J = 1, N
X(J) = FACTOR*X(J)
210 CONTINUE
GO TO 240
220 CONTINUE
DO 230 J = 1, N
X(J) = FACTOR
230 CONTINUE
240 CONTINUE
250 CONTINUE
RETURN
C
C LAST CARD OF SUBROUTINE INITPT.
C
END