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|
C **********
C
C THIS PROGRAM TESTS CODES FOR THE LEAST-SQUARES SOLUTION OF
C M NONLINEAR EQUATIONS IN N VARIABLES. IT CONSISTS OF A DRIVER
C AND AN INTERFACE SUBROUTINE FCN. THE DRIVER READS IN DATA,
C CALLS THE NONLINEAR LEAST-SQUARES SOLVER, AND FINALLY PRINTS
C OUT INFORMATION ON THE PERFORMANCE OF THE SOLVER. THIS IS
C ONLY A SAMPLE DRIVER, MANY OTHER DRIVERS ARE POSSIBLE. THE
C INTERFACE SUBROUTINE FCN IS NECESSARY TO TAKE INTO ACCOUNT THE
C FORMS OF CALLING SEQUENCES USED BY THE FUNCTION AND JACOBIAN
C SUBROUTINES IN THE VARIOUS NONLINEAR LEAST-SQUARES SOLVERS.
C
C SUBPROGRAMS CALLED
C
C USER-SUPPLIED ...... FCN
C
C MINPACK-SUPPLIED ... SPMPAR,ENORM,INITPT,LMSTR1,SSQFCN
C
C FORTRAN-SUPPLIED ... SQRT
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,LDFJAC,LWA,M,N,NFEV,NJEV,NPROB,NREAD,NTRIES,
* NWRITE
INTEGER IWA(40),MA(60),NA(60),NF(60),NJ(60),NP(60),NX(60)
REAL FACTOR,FNORM1,FNORM2,ONE,TEN,TOL
REAL FJAC(40,40),FNM(60),FVEC(65),WA(265),X(40)
REAL SPMPAR,ENORM
EXTERNAL FCN
COMMON /REFNUM/ NPROB,NFEV,NJEV
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 /1.0E0,1.0E1/
TOL = SQRT(SPMPAR(1))
LDFJAC = 40
LWA = 265
IC = 0
10 CONTINUE
READ (NREAD,50) NPROB,N,M,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 SSQFCN(M,N,X,FVEC,NPROB)
FNORM1 = ENORM(M,FVEC)
WRITE (NWRITE,60) NPROB,N,M
NFEV = 0
NJEV = 0
CALL LMSTR1(FCN,M,N,X,FVEC,FJAC,LDFJAC,TOL,INFO,IWA,WA,
* LWA)
CALL SSQFCN(M,N,X,FVEC,NPROB)
FNORM2 = ENORM(M,FVEC)
NP(IC) = NPROB
NA(IC) = N
MA(IC) = M
NF(IC) = NFEV
NJ(IC) = NJEV
NX(IC) = INFO
FNM(IC) = FNORM2
WRITE (NWRITE,70)
* FNORM1,FNORM2,NFEV,NJEV,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),MA(I),NF(I),NJ(I),NX(I),FNM(I)
40 CONTINUE
STOP
50 FORMAT (4I5)
60 FORMAT ( //// 5X, 8H PROBLEM, I5, 5X, 11H DIMENSIONS, 2I5, 5X //
* )
70 FORMAT (5X, 33H INITIAL L2 NORM OF THE RESIDUALS, E15.7 // 5X,
* 33H FINAL L2 NORM OF THE RESIDUALS , E15.7 // 5X,
* 33H NUMBER OF FUNCTION EVALUATIONS , I10 // 5X,
* 33H NUMBER OF JACOBIAN EVALUATIONS , I10 // 5X,
* 15H EXIT PARAMETER, 18X, I10 // 5X,
* 27H FINAL APPROXIMATE SOLUTION // (5X, 5E15.7))
80 FORMAT (12H1SUMMARY OF , I3, 16H CALLS TO LMSTR1 /)
90 FORMAT (49H NPROB N M NFEV NJEV INFO FINAL L2 NORM /)
100 FORMAT (3I5, 3I6, 2X, E15.7)
C
C LAST CARD OF DRIVER.
C
END
SUBROUTINE FCN(M,N,X,FVEC,FJROW,IFLAG)
INTEGER M,N,IFLAG
REAL X(N),FVEC(M),FJROW(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 LEAST SQUARES SOLVER. IF IFLAG = 1, FCN SHOULD ONLY CALL THE
C TESTING FUNCTION SUBROUTINE SSQFCN. IF IFLAG = I, I .GE. 2,
C FCN SHOULD ONLY CALL SUBROUTINE SSQJAC TO CALCULATE THE
C (I-1)-ST ROW OF THE JACOBIAN. (THE SSQJAC SUBROUTINE PROVIDED
C HERE FOR TESTING PURPOSES CALCULATES THE ENTIRE JACOBIAN
C MATRIX AND IS THEREFORE CALLED ONLY WHEN IFLAG = 2.) EACH
C CALL TO SSQFCN OR SSQJAC SHOULD SPECIFY THE APPROPRIATE
C VALUE OF PROBLEM NUMBER (NPROB).
C
C SUBPROGRAMS CALLED
C
C MINPACK-SUPPLIED ... SSQFCN,SSQJAC
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. MARCH 1980.
C BURTON S. GARBOW, KENNETH E. HILLSTROM, JORGE J. MORE
C
C **********
INTEGER NPROB,NFEV,NJEV,J
REAL TEMP(65,40)
COMMON /REFNUM/ NPROB,NFEV,NJEV
IF (IFLAG .EQ. 1) CALL SSQFCN(M,N,X,FVEC,NPROB)
IF (IFLAG .EQ. 2) CALL SSQJAC(M,N,X,TEMP,65,NPROB)
IF (IFLAG .EQ. 1) NFEV = NFEV + 1
IF (IFLAG .EQ. 2) NJEV = NJEV + 1
IF (IFLAG .EQ. 1) GO TO 120
DO 110 J = 1, N
FJROW(J) = TEMP(IFLAG-1,J)
110 CONTINUE
120 CONTINUE
RETURN
C
C LAST CARD OF INTERFACE SUBROUTINE FCN.
C
END
SUBROUTINE SSQJAC(M,N,X,FJAC,LDFJAC,NPROB)
INTEGER M,N,LDFJAC,NPROB
REAL X(N),FJAC(LDFJAC,N)
C **********
C
C SUBROUTINE SSQJAC
C
C THIS SUBROUTINE DEFINES THE JACOBIAN MATRICES OF EIGHTEEN
C NONLINEAR LEAST SQUARES PROBLEMS. THE PROBLEM DIMENSIONS ARE
C AS DESCRIBED IN THE PROLOGUE COMMENTS OF SSQFCN.
C
C THE SUBROUTINE STATEMENT IS
C
C SUBROUTINE SSQJAC(M,N,X,FJAC,LDFJAC,NPROB)
C
C WHERE
C
C M AND N ARE POSITIVE INTEGER INPUT VARIABLES. N MUST NOT
C EXCEED M.
C
C X IS AN INPUT ARRAY OF LENGTH N.
C
C FJAC IS AN M BY N OUTPUT ARRAY WHICH CONTAINS THE JACOBIAN
C MATRIX OF THE NPROB FUNCTION EVALUATED AT X.
C
C LDFJAC IS A POSITIVE INTEGER INPUT VARIABLE NOT LESS THAN M
C WHICH SPECIFIES THE LEADING DIMENSION OF THE ARRAY FJAC.
C
C NPROB IS A POSITIVE INTEGER VARIABLE WHICH DEFINES THE
C NUMBER OF THE PROBLEM. NPROB MUST NOT EXCEED 18.
C
C SUBPROGRAMS CALLED
C
C FORTRAN-SUPPLIED ... ATAN,COS,EXP,SIN,SQRT
C
C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. MARCH 1980.
C BURTON S. GARBOW, KENNETH E. HILLSTROM, JORGE J. MORE
C
C **********
INTEGER I,IVAR,J,K,MM1,NM1
REAL C14,C20,C29,C45,C100,DIV,DX,EIGHT,FIVE,FOUR,ONE,PROD,S2,
* TEMP,TEN,THREE,TI,TMP1,TMP2,TMP3,TMP4,TPI,TWO,ZERO
REAL V(11)
REAL FLOAT
DATA ZERO,ONE,TWO,THREE,FOUR,FIVE,EIGHT,TEN,C14,C20,C29,C45,C100
* /0.0E0,1.0E0,2.0E0,3.0E0,4.0E0,5.0E0,8.0E0,1.0E1,1.4E1,
* 2.0E1,2.9E1,4.5E1,1.0E2/
DATA V(1),V(2),V(3),V(4),V(5),V(6),V(7),V(8),V(9),V(10),V(11)
* /4.0E0,2.0E0,1.0E0,5.0E-1,2.5E-1,1.67E-1,1.25E-1,1.0E-1,
* 8.33E-2,7.14E-2,6.25E-2/
FLOAT(IVAR) = IVAR
C
C JACOBIAN ROUTINE SELECTOR.
C
GO TO (10,40,70,130,140,150,180,190,210,230,250,310,330,350,370,
* 400,460,480), NPROB
C
C LINEAR FUNCTION - FULL RANK.
C
10 CONTINUE
TEMP = TWO/FLOAT(M)
DO 30 J = 1, N
DO 20 I = 1, M
FJAC(I,J) = -TEMP
20 CONTINUE
FJAC(J,J) = FJAC(J,J) + ONE
30 CONTINUE
GO TO 500
C
C LINEAR FUNCTION - RANK 1.
C
40 CONTINUE
DO 60 J = 1, N
DO 50 I = 1, M
FJAC(I,J) = FLOAT(I)*FLOAT(J)
50 CONTINUE
60 CONTINUE
GO TO 500
C
C LINEAR FUNCTION - RANK 1 WITH ZERO COLUMNS AND ROWS.
C
70 CONTINUE
DO 90 J = 1, N
DO 80 I = 1, M
FJAC(I,J) = ZERO
80 CONTINUE
90 CONTINUE
NM1 = N - 1
MM1 = M - 1
IF (NM1 .LT. 2) GO TO 120
DO 110 J = 2, NM1
DO 100 I = 2, MM1
FJAC(I,J) = FLOAT(I-1)*FLOAT(J)
100 CONTINUE
110 CONTINUE
120 CONTINUE
GO TO 500
C
C ROSENBROCK FUNCTION.
C
130 CONTINUE
FJAC(1,1) = -C20*X(1)
FJAC(1,2) = TEN
FJAC(2,1) = -ONE
FJAC(2,2) = ZERO
GO TO 500
C
C HELICAL VALLEY FUNCTION.
C
140 CONTINUE
TPI = EIGHT*ATAN(ONE)
TEMP = X(1)**2 + X(2)**2
TMP1 = TPI*TEMP
TMP2 = SQRT(TEMP)
FJAC(1,1) = C100*X(2)/TMP1
FJAC(1,2) = -C100*X(1)/TMP1
FJAC(1,3) = TEN
FJAC(2,1) = TEN*X(1)/TMP2
FJAC(2,2) = TEN*X(2)/TMP2
FJAC(2,3) = ZERO
FJAC(3,1) = ZERO
FJAC(3,2) = ZERO
FJAC(3,3) = ONE
GO TO 500
C
C POWELL SINGULAR FUNCTION.
C
150 CONTINUE
DO 170 J = 1, 4
DO 160 I = 1, 4
FJAC(I,J) = ZERO
160 CONTINUE
170 CONTINUE
FJAC(1,1) = ONE
FJAC(1,2) = TEN
FJAC(2,3) = SQRT(FIVE)
FJAC(2,4) = -FJAC(2,3)
FJAC(3,2) = TWO*(X(2) - TWO*X(3))
FJAC(3,3) = -TWO*FJAC(3,2)
FJAC(4,1) = TWO*SQRT(TEN)*(X(1) - X(4))
FJAC(4,4) = -FJAC(4,1)
GO TO 500
C
C FREUDENSTEIN AND ROTH FUNCTION.
C
180 CONTINUE
FJAC(1,1) = ONE
FJAC(1,2) = X(2)*(TEN - THREE*X(2)) - TWO
FJAC(2,1) = ONE
FJAC(2,2) = X(2)*(TWO + THREE*X(2)) - C14
GO TO 500
C
C BARD FUNCTION.
C
190 CONTINUE
DO 200 I = 1, 15
TMP1 = FLOAT(I)
TMP2 = FLOAT(16-I)
TMP3 = TMP1
IF (I .GT. 8) TMP3 = TMP2
TMP4 = (X(2)*TMP2 + X(3)*TMP3)**2
FJAC(I,1) = -ONE
FJAC(I,2) = TMP1*TMP2/TMP4
FJAC(I,3) = TMP1*TMP3/TMP4
200 CONTINUE
GO TO 500
C
C KOWALIK AND OSBORNE FUNCTION.
C
210 CONTINUE
DO 220 I = 1, 11
TMP1 = V(I)*(V(I) + X(2))
TMP2 = V(I)*(V(I) + X(3)) + X(4)
FJAC(I,1) = -TMP1/TMP2
FJAC(I,2) = -V(I)*X(1)/TMP2
FJAC(I,3) = FJAC(I,1)*FJAC(I,2)
FJAC(I,4) = FJAC(I,3)/V(I)
220 CONTINUE
GO TO 500
C
C MEYER FUNCTION.
C
230 CONTINUE
DO 240 I = 1, 16
TEMP = FIVE*FLOAT(I) + C45 + X(3)
TMP1 = X(2)/TEMP
TMP2 = EXP(TMP1)
FJAC(I,1) = TMP2
FJAC(I,2) = X(1)*TMP2/TEMP
FJAC(I,3) = -TMP1*FJAC(I,2)
240 CONTINUE
GO TO 500
C
C WATSON FUNCTION.
C
250 CONTINUE
DO 280 I = 1, 29
DIV = FLOAT(I)/C29
S2 = ZERO
DX = ONE
DO 260 J = 1, N
S2 = S2 + DX*X(J)
DX = DIV*DX
260 CONTINUE
TEMP = TWO*DIV*S2
DX = ONE/DIV
DO 270 J = 1, N
FJAC(I,J) = DX*(FLOAT(J-1) - TEMP)
DX = DIV*DX
270 CONTINUE
280 CONTINUE
DO 300 J = 1, N
DO 290 I = 30, 31
FJAC(I,J) = ZERO
290 CONTINUE
300 CONTINUE
FJAC(30,1) = ONE
FJAC(31,1) = -TWO*X(1)
FJAC(31,2) = ONE
GO TO 500
C
C BOX 3-DIMENSIONAL FUNCTION.
C
310 CONTINUE
DO 320 I = 1, M
TEMP = FLOAT(I)
TMP1 = TEMP/TEN
FJAC(I,1) = -TMP1*EXP(-TMP1*X(1))
FJAC(I,2) = TMP1*EXP(-TMP1*X(2))
FJAC(I,3) = EXP(-TEMP) - EXP(-TMP1)
320 CONTINUE
GO TO 500
C
C JENNRICH AND SAMPSON FUNCTION.
C
330 CONTINUE
DO 340 I = 1, M
TEMP = FLOAT(I)
FJAC(I,1) = -TEMP*EXP(TEMP*X(1))
FJAC(I,2) = -TEMP*EXP(TEMP*X(2))
340 CONTINUE
GO TO 500
C
C BROWN AND DENNIS FUNCTION.
C
350 CONTINUE
DO 360 I = 1, M
TEMP = FLOAT(I)/FIVE
TI = SIN(TEMP)
TMP1 = X(1) + TEMP*X(2) - EXP(TEMP)
TMP2 = X(3) + TI*X(4) - COS(TEMP)
FJAC(I,1) = TWO*TMP1
FJAC(I,2) = TEMP*FJAC(I,1)
FJAC(I,3) = TWO*TMP2
FJAC(I,4) = TI*FJAC(I,3)
360 CONTINUE
GO TO 500
C
C CHEBYQUAD FUNCTION.
C
370 CONTINUE
DX = ONE/FLOAT(N)
DO 390 J = 1, N
TMP1 = ONE
TMP2 = TWO*X(J) - ONE
TEMP = TWO*TMP2
TMP3 = ZERO
TMP4 = TWO
DO 380 I = 1, M
FJAC(I,J) = DX*TMP4
TI = FOUR*TMP2 + TEMP*TMP4 - TMP3
TMP3 = TMP4
TMP4 = TI
TI = TEMP*TMP2 - TMP1
TMP1 = TMP2
TMP2 = TI
380 CONTINUE
390 CONTINUE
GO TO 500
C
C BROWN ALMOST-LINEAR FUNCTION.
C
400 CONTINUE
PROD = ONE
DO 420 J = 1, N
PROD = X(J)*PROD
DO 410 I = 1, N
FJAC(I,J) = ONE
410 CONTINUE
FJAC(J,J) = TWO
420 CONTINUE
DO 450 J = 1, N
TEMP = X(J)
IF (TEMP .NE. ZERO) GO TO 440
TEMP = ONE
PROD = ONE
DO 430 K = 1, N
IF (K .NE. J) PROD = X(K)*PROD
430 CONTINUE
440 CONTINUE
FJAC(N,J) = PROD/TEMP
450 CONTINUE
GO TO 500
C
C OSBORNE 1 FUNCTION.
C
460 CONTINUE
DO 470 I = 1, 33
TEMP = TEN*FLOAT(I-1)
TMP1 = EXP(-X(4)*TEMP)
TMP2 = EXP(-X(5)*TEMP)
FJAC(I,1) = -ONE
FJAC(I,2) = -TMP1
FJAC(I,3) = -TMP2
FJAC(I,4) = TEMP*X(2)*TMP1
FJAC(I,5) = TEMP*X(3)*TMP2
470 CONTINUE
GO TO 500
C
C OSBORNE 2 FUNCTION.
C
480 CONTINUE
DO 490 I = 1, 65
TEMP = FLOAT(I-1)/TEN
TMP1 = EXP(-X(5)*TEMP)
TMP2 = EXP(-X(6)*(TEMP-X(9))**2)
TMP3 = EXP(-X(7)*(TEMP-X(10))**2)
TMP4 = EXP(-X(8)*(TEMP-X(11))**2)
FJAC(I,1) = -TMP1
FJAC(I,2) = -TMP2
FJAC(I,3) = -TMP3
FJAC(I,4) = -TMP4
FJAC(I,5) = TEMP*X(1)*TMP1
FJAC(I,6) = X(2)*(TEMP - X(9))**2*TMP2
FJAC(I,7) = X(3)*(TEMP - X(10))**2*TMP3
FJAC(I,8) = X(4)*(TEMP - X(11))**2*TMP4
FJAC(I,9) = -TWO*X(2)*X(6)*(TEMP - X(9))*TMP2
FJAC(I,10) = -TWO*X(3)*X(7)*(TEMP - X(10))*TMP3
FJAC(I,11) = -TWO*X(4)*X(8)*(TEMP - X(11))*TMP4
490 CONTINUE
500 CONTINUE
RETURN
C
C LAST CARD OF SUBROUTINE SSQJAC.
C
END
SUBROUTINE INITPT(N,X,NPROB,FACTOR)
INTEGER N,NPROB
REAL FACTOR
REAL X(N)
C **********
C
C SUBROUTINE INITPT
C
C THIS SUBROUTINE SPECIFIES THE STANDARD STARTING POINTS FOR THE
C FUNCTIONS DEFINED BY SUBROUTINE SSQFCN. THE SUBROUTINE RETURNS
C IN X A MULTIPLE (FACTOR) OF THE STANDARD STARTING POINT. FOR
C THE 11TH FUNCTION THE STANDARD STARTING POINT IS ZERO, SO IN
C THIS CASE, IF FACTOR IS NOT UNITY, THEN THE SUBROUTINE RETURNS
C 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 18.
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
REAL C1,C2,C3,C4,C5,C6,C7,C8,C9,C10,C11,C12,C13,C14,C15,C16,C17,
* FIVE,H,HALF,ONE,SEVEN,TEN,THREE,TWENTY,TWNTF,TWO,ZERO
REAL FLOAT
DATA ZERO,HALF,ONE,TWO,THREE,FIVE,SEVEN,TEN,TWENTY,TWNTF
* /0.0E0,5.0E-1,1.0E0,2.0E0,3.0E0,5.0E0,7.0E0,1.0E1,2.0E1,
* 2.5E1/
DATA C1,C2,C3,C4,C5,C6,C7,C8,C9,C10,C11,C12,C13,C14,C15,C16,C17
* /1.2E0,2.5E-1,3.9E-1,4.15E-1,2.0E-2,4.0E3,2.5E2,3.0E-1,
* 4.0E-1,1.5E0,1.0E-2,1.3E0,6.5E-1,7.0E-1,6.0E-1,4.5E0,
* 5.5E0/
FLOAT(IVAR) = IVAR
C
C SELECTION OF INITIAL POINT.
C
GO TO (10,10,10,30,40,50,60,70,80,90,100,120,130,140,150,170,
* 190,200), NPROB
C
C LINEAR FUNCTION - FULL RANK OR RANK 1.
C
10 CONTINUE
DO 20 J = 1, N
X(J) = ONE
20 CONTINUE
GO TO 210
C
C ROSENBROCK FUNCTION.
C
30 CONTINUE
X(1) = -C1
X(2) = ONE
GO TO 210
C
C HELICAL VALLEY FUNCTION.
C
40 CONTINUE
X(1) = -ONE
X(2) = ZERO
X(3) = ZERO
GO TO 210
C
C POWELL SINGULAR FUNCTION.
C
50 CONTINUE
X(1) = THREE
X(2) = -ONE
X(3) = ZERO
X(4) = ONE
GO TO 210
C
C FREUDENSTEIN AND ROTH FUNCTION.
C
60 CONTINUE
X(1) = HALF
X(2) = -TWO
GO TO 210
C
C BARD FUNCTION.
C
70 CONTINUE
X(1) = ONE
X(2) = ONE
X(3) = ONE
GO TO 210
C
C KOWALIK AND OSBORNE FUNCTION.
C
80 CONTINUE
X(1) = C2
X(2) = C3
X(3) = C4
X(4) = C3
GO TO 210
C
C MEYER FUNCTION.
C
90 CONTINUE
X(1) = C5
X(2) = C6
X(3) = C7
GO TO 210
C
C WATSON FUNCTION.
C
100 CONTINUE
DO 110 J = 1, N
X(J) = ZERO
110 CONTINUE
GO TO 210
C
C BOX 3-DIMENSIONAL FUNCTION.
C
120 CONTINUE
X(1) = ZERO
X(2) = TEN
X(3) = TWENTY
GO TO 210
C
C JENNRICH AND SAMPSON FUNCTION.
C
130 CONTINUE
X(1) = C8
X(2) = C9
GO TO 210
C
C BROWN AND DENNIS FUNCTION.
C
140 CONTINUE
X(1) = TWNTF
X(2) = FIVE
X(3) = -FIVE
X(4) = -ONE
GO TO 210
C
C CHEBYQUAD FUNCTION.
C
150 CONTINUE
H = ONE/FLOAT(N+1)
DO 160 J = 1, N
X(J) = FLOAT(J)*H
160 CONTINUE
GO TO 210
C
C BROWN ALMOST-LINEAR FUNCTION.
C
170 CONTINUE
DO 180 J = 1, N
X(J) = HALF
180 CONTINUE
GO TO 210
C
C OSBORNE 1 FUNCTION.
C
190 CONTINUE
X(1) = HALF
X(2) = C10
X(3) = -ONE
X(4) = C11
X(5) = C5
GO TO 210
C
C OSBORNE 2 FUNCTION.
C
200 CONTINUE
X(1) = C12
X(2) = C13
X(3) = C13
X(4) = C14
X(5) = C15
X(6) = THREE
X(7) = FIVE
X(8) = SEVEN
X(9) = TWO
X(10) = C16
X(11) = C17
210 CONTINUE
C
C COMPUTE MULTIPLE OF INITIAL POINT.
C
IF (FACTOR .EQ. ONE) GO TO 260
IF (NPROB .EQ. 11) GO TO 230
DO 220 J = 1, N
X(J) = FACTOR*X(J)
220 CONTINUE
GO TO 250
230 CONTINUE
DO 240 J = 1, N
X(J) = FACTOR
240 CONTINUE
250 CONTINUE
260 CONTINUE
RETURN
C
C LAST CARD OF SUBROUTINE INITPT.
C
END
SUBROUTINE SSQFCN(M,N,X,FVEC,NPROB)
INTEGER M,N,NPROB
REAL X(N),FVEC(M)
C **********
C
C SUBROUTINE SSQFCN
C
C THIS SUBROUTINE DEFINES THE FUNCTIONS OF EIGHTEEN NONLINEAR
C LEAST SQUARES PROBLEMS. THE ALLOWABLE VALUES OF (M,N) FOR
C FUNCTIONS 1,2 AND 3 ARE VARIABLE BUT WITH M .GE. N.
C FOR FUNCTIONS 4,5,6,7,8,9 AND 10 THE VALUES OF (M,N) ARE
C (2,2),(3,3),(4,4),(2,2),(15,3),(11,4) AND (16,3), RESPECTIVELY.
C FUNCTION 11 (WATSON) HAS M = 31 WITH N USUALLY 6 OR 9.
C HOWEVER, ANY N, N = 2,...,31, IS PERMITTED.
C FUNCTIONS 12,13 AND 14 HAVE N = 3,2 AND 4, RESPECTIVELY, BUT
C ALLOW ANY M .GE. N, WITH THE USUAL CHOICES BEING 10,10 AND 20.
C FUNCTION 15 (CHEBYQUAD) ALLOWS M AND N VARIABLE WITH M .GE. N.
C FUNCTION 16 (BROWN) ALLOWS N VARIABLE WITH M = N.
C FOR FUNCTIONS 17 AND 18, THE VALUES OF (M,N) ARE
C (33,5) AND (65,11), RESPECTIVELY.
C
C THE SUBROUTINE STATEMENT IS
C
C SUBROUTINE SSQFCN(M,N,X,FVEC,NPROB)
C
C WHERE
C
C M AND N ARE POSITIVE INTEGER INPUT VARIABLES. N MUST NOT
C EXCEED M.
C
C X IS AN INPUT ARRAY OF LENGTH N.
C
C FVEC IS AN OUTPUT ARRAY OF LENGTH M WHICH CONTAINS THE NPROB
C FUNCTION EVALUATED AT X.
C
C NPROB IS A POSITIVE INTEGER INPUT VARIABLE WHICH DEFINES THE
C NUMBER OF THE PROBLEM. NPROB MUST NOT EXCEED 18.
C
C SUBPROGRAMS CALLED
C
C FORTRAN-SUPPLIED ... ATAN,COS,EXP,SIN,SQRT,SIGN
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,NM1
REAL C13,C14,C29,C45,DIV,DX,EIGHT,FIVE,ONE,PROD,SUM,S1,S2,TEMP,
* TEN,TI,TMP1,TMP2,TMP3,TMP4,TPI,TWO,ZERO,ZP25,ZP5
REAL V(11),Y1(15),Y2(11),Y3(16),Y4(33),Y5(65)
REAL FLOAT
DATA ZERO,ZP25,ZP5,ONE,TWO,FIVE,EIGHT,TEN,C13,C14,C29,C45
* /0.0E0,2.5E-1,5.0E-1,1.0E0,2.0E0,5.0E0,8.0E0,1.0E1,1.3E1,
* 1.4E1,2.9E1,4.5E1/
DATA V(1),V(2),V(3),V(4),V(5),V(6),V(7),V(8),V(9),V(10),V(11)
* /4.0E0,2.0E0,1.0E0,5.0E-1,2.5E-1,1.67E-1,1.25E-1,1.0E-1,
* 8.33E-2,7.14E-2,6.25E-2/
DATA Y1(1),Y1(2),Y1(3),Y1(4),Y1(5),Y1(6),Y1(7),Y1(8),Y1(9),
* Y1(10),Y1(11),Y1(12),Y1(13),Y1(14),Y1(15)
* /1.4E-1,1.8E-1,2.2E-1,2.5E-1,2.9E-1,3.2E-1,3.5E-1,3.9E-1,
* 3.7E-1,5.8E-1,7.3E-1,9.6E-1,1.34E0,2.1E0,4.39E0/
DATA Y2(1),Y2(2),Y2(3),Y2(4),Y2(5),Y2(6),Y2(7),Y2(8),Y2(9),
* Y2(10),Y2(11)
* /1.957E-1,1.947E-1,1.735E-1,1.6E-1,8.44E-2,6.27E-2,4.56E-2,
* 3.42E-2,3.23E-2,2.35E-2,2.46E-2/
DATA Y3(1),Y3(2),Y3(3),Y3(4),Y3(5),Y3(6),Y3(7),Y3(8),Y3(9),
* Y3(10),Y3(11),Y3(12),Y3(13),Y3(14),Y3(15),Y3(16)
* /3.478E4,2.861E4,2.365E4,1.963E4,1.637E4,1.372E4,1.154E4,
* 9.744E3,8.261E3,7.03E3,6.005E3,5.147E3,4.427E3,3.82E3,
* 3.307E3,2.872E3/
DATA Y4(1),Y4(2),Y4(3),Y4(4),Y4(5),Y4(6),Y4(7),Y4(8),Y4(9),
* Y4(10),Y4(11),Y4(12),Y4(13),Y4(14),Y4(15),Y4(16),Y4(17),
* Y4(18),Y4(19),Y4(20),Y4(21),Y4(22),Y4(23),Y4(24),Y4(25),
* Y4(26),Y4(27),Y4(28),Y4(29),Y4(30),Y4(31),Y4(32),Y4(33)
* /8.44E-1,9.08E-1,9.32E-1,9.36E-1,9.25E-1,9.08E-1,8.81E-1,
* 8.5E-1,8.18E-1,7.84E-1,7.51E-1,7.18E-1,6.85E-1,6.58E-1,
* 6.28E-1,6.03E-1,5.8E-1,5.58E-1,5.38E-1,5.22E-1,5.06E-1,
* 4.9E-1,4.78E-1,4.67E-1,4.57E-1,4.48E-1,4.38E-1,4.31E-1,
* 4.24E-1,4.2E-1,4.14E-1,4.11E-1,4.06E-1/
DATA Y5(1),Y5(2),Y5(3),Y5(4),Y5(5),Y5(6),Y5(7),Y5(8),Y5(9),
* Y5(10),Y5(11),Y5(12),Y5(13),Y5(14),Y5(15),Y5(16),Y5(17),
* Y5(18),Y5(19),Y5(20),Y5(21),Y5(22),Y5(23),Y5(24),Y5(25),
* Y5(26),Y5(27),Y5(28),Y5(29),Y5(30),Y5(31),Y5(32),Y5(33),
* Y5(34),Y5(35),Y5(36),Y5(37),Y5(38),Y5(39),Y5(40),Y5(41),
* Y5(42),Y5(43),Y5(44),Y5(45),Y5(46),Y5(47),Y5(48),Y5(49),
* Y5(50),Y5(51),Y5(52),Y5(53),Y5(54),Y5(55),Y5(56),Y5(57),
* Y5(58),Y5(59),Y5(60),Y5(61),Y5(62),Y5(63),Y5(64),Y5(65)
* /1.366E0,1.191E0,1.112E0,1.013E0,9.91E-1,8.85E-1,8.31E-1,
* 8.47E-1,7.86E-1,7.25E-1,7.46E-1,6.79E-1,6.08E-1,6.55E-1,
* 6.16E-1,6.06E-1,6.02E-1,6.26E-1,6.51E-1,7.24E-1,6.49E-1,
* 6.49E-1,6.94E-1,6.44E-1,6.24E-1,6.61E-1,6.12E-1,5.58E-1,
* 5.33E-1,4.95E-1,5.0E-1,4.23E-1,3.95E-1,3.75E-1,3.72E-1,
* 3.91E-1,3.96E-1,4.05E-1,4.28E-1,4.29E-1,5.23E-1,5.62E-1,
* 6.07E-1,6.53E-1,6.72E-1,7.08E-1,6.33E-1,6.68E-1,6.45E-1,
* 6.32E-1,5.91E-1,5.59E-1,5.97E-1,6.25E-1,7.39E-1,7.1E-1,
* 7.29E-1,7.2E-1,6.36E-1,5.81E-1,4.28E-1,2.92E-1,1.62E-1,
* 9.8E-2,5.4E-2/
FLOAT(IVAR) = IVAR
C
C FUNCTION ROUTINE SELECTOR.
C
GO TO (10,40,70,110,120,130,140,150,170,190,210,250,270,290,310,
* 360,390,410), NPROB
C
C LINEAR FUNCTION - FULL RANK.
C
10 CONTINUE
SUM = ZERO
DO 20 J = 1, N
SUM = SUM + X(J)
20 CONTINUE
TEMP = TWO*SUM/FLOAT(M) + ONE
DO 30 I = 1, M
FVEC(I) = -TEMP
IF (I .LE. N) FVEC(I) = FVEC(I) + X(I)
30 CONTINUE
GO TO 430
C
C LINEAR FUNCTION - RANK 1.
C
40 CONTINUE
SUM = ZERO
DO 50 J = 1, N
SUM = SUM + FLOAT(J)*X(J)
50 CONTINUE
DO 60 I = 1, M
FVEC(I) = FLOAT(I)*SUM - ONE
60 CONTINUE
GO TO 430
C
C LINEAR FUNCTION - RANK 1 WITH ZERO COLUMNS AND ROWS.
C
70 CONTINUE
SUM = ZERO
NM1 = N - 1
IF (NM1 .LT. 2) GO TO 90
DO 80 J = 2, NM1
SUM = SUM + FLOAT(J)*X(J)
80 CONTINUE
90 CONTINUE
DO 100 I = 1, M
FVEC(I) = FLOAT(I-1)*SUM - ONE
100 CONTINUE
FVEC(M) = -ONE
GO TO 430
C
C ROSENBROCK FUNCTION.
C
110 CONTINUE
FVEC(1) = TEN*(X(2) - X(1)**2)
FVEC(2) = ONE - X(1)
GO TO 430
C
C HELICAL VALLEY FUNCTION.
C
120 CONTINUE
TPI = EIGHT*ATAN(ONE)
TMP1 = SIGN(ZP25,X(2))
IF (X(1) .GT. ZERO) TMP1 = ATAN(X(2)/X(1))/TPI
IF (X(1) .LT. ZERO) TMP1 = ATAN(X(2)/X(1))/TPI + ZP5
TMP2 = SQRT(X(1)**2+X(2)**2)
FVEC(1) = TEN*(X(3) - TEN*TMP1)
FVEC(2) = TEN*(TMP2 - ONE)
FVEC(3) = X(3)
GO TO 430
C
C POWELL SINGULAR FUNCTION.
C
130 CONTINUE
FVEC(1) = X(1) + TEN*X(2)
FVEC(2) = SQRT(FIVE)*(X(3) - X(4))
FVEC(3) = (X(2) - TWO*X(3))**2
FVEC(4) = SQRT(TEN)*(X(1) - X(4))**2
GO TO 430
C
C FREUDENSTEIN AND ROTH FUNCTION.
C
140 CONTINUE
FVEC(1) = -C13 + X(1) + ((FIVE - X(2))*X(2) - TWO)*X(2)
FVEC(2) = -C29 + X(1) + ((ONE + X(2))*X(2) - C14)*X(2)
GO TO 430
C
C BARD FUNCTION.
C
150 CONTINUE
DO 160 I = 1, 15
TMP1 = FLOAT(I)
TMP2 = FLOAT(16-I)
TMP3 = TMP1
IF (I .GT. 8) TMP3 = TMP2
FVEC(I) = Y1(I) - (X(1) + TMP1/(X(2)*TMP2 + X(3)*TMP3))
160 CONTINUE
GO TO 430
C
C KOWALIK AND OSBORNE FUNCTION.
C
170 CONTINUE
DO 180 I = 1, 11
TMP1 = V(I)*(V(I) + X(2))
TMP2 = V(I)*(V(I) + X(3)) + X(4)
FVEC(I) = Y2(I) - X(1)*TMP1/TMP2
180 CONTINUE
GO TO 430
C
C MEYER FUNCTION.
C
190 CONTINUE
DO 200 I = 1, 16
TEMP = FIVE*FLOAT(I) + C45 + X(3)
TMP1 = X(2)/TEMP
TMP2 = EXP(TMP1)
FVEC(I) = X(1)*TMP2 - Y3(I)
200 CONTINUE
GO TO 430
C
C WATSON FUNCTION.
C
210 CONTINUE
DO 240 I = 1, 29
DIV = FLOAT(I)/C29
S1 = ZERO
DX = ONE
DO 220 J = 2, N
S1 = S1 + FLOAT(J-1)*DX*X(J)
DX = DIV*DX
220 CONTINUE
S2 = ZERO
DX = ONE
DO 230 J = 1, N
S2 = S2 + DX*X(J)
DX = DIV*DX
230 CONTINUE
FVEC(I) = S1 - S2**2 - ONE
240 CONTINUE
FVEC(30) = X(1)
FVEC(31) = X(2) - X(1)**2 - ONE
GO TO 430
C
C BOX 3-DIMENSIONAL FUNCTION.
C
250 CONTINUE
DO 260 I = 1, M
TEMP = FLOAT(I)
TMP1 = TEMP/TEN
FVEC(I) = EXP(-TMP1*X(1)) - EXP(-TMP1*X(2))
* + (EXP(-TEMP) - EXP(-TMP1))*X(3)
260 CONTINUE
GO TO 430
C
C JENNRICH AND SAMPSON FUNCTION.
C
270 CONTINUE
DO 280 I = 1, M
TEMP = FLOAT(I)
FVEC(I) = TWO + TWO*TEMP - EXP(TEMP*X(1)) - EXP(TEMP*X(2))
280 CONTINUE
GO TO 430
C
C BROWN AND DENNIS FUNCTION.
C
290 CONTINUE
DO 300 I = 1, M
TEMP = FLOAT(I)/FIVE
TMP1 = X(1) + TEMP*X(2) - EXP(TEMP)
TMP2 = X(3) + SIN(TEMP)*X(4) - COS(TEMP)
FVEC(I) = TMP1**2 + TMP2**2
300 CONTINUE
GO TO 430
C
C CHEBYQUAD FUNCTION.
C
310 CONTINUE
DO 320 I = 1, M
FVEC(I) = ZERO
320 CONTINUE
DO 340 J = 1, N
TMP1 = ONE
TMP2 = TWO*X(J) - ONE
TEMP = TWO*TMP2
DO 330 I = 1, M
FVEC(I) = FVEC(I) + TMP2
TI = TEMP*TMP2 - TMP1
TMP1 = TMP2
TMP2 = TI
330 CONTINUE
340 CONTINUE
DX = ONE/FLOAT(N)
IEV = -1
DO 350 I = 1, M
FVEC(I) = DX*FVEC(I)
IF (IEV .GT. 0) FVEC(I) = FVEC(I) + ONE/(FLOAT(I)**2 - ONE)
IEV = -IEV
350 CONTINUE
GO TO 430
C
C BROWN ALMOST-LINEAR FUNCTION.
C
360 CONTINUE
SUM = -FLOAT(N+1)
PROD = ONE
DO 370 J = 1, N
SUM = SUM + X(J)
PROD = X(J)*PROD
370 CONTINUE
DO 380 I = 1, N
FVEC(I) = X(I) + SUM
380 CONTINUE
FVEC(N) = PROD - ONE
GO TO 430
C
C OSBORNE 1 FUNCTION.
C
390 CONTINUE
DO 400 I = 1, 33
TEMP = TEN*FLOAT(I-1)
TMP1 = EXP(-X(4)*TEMP)
TMP2 = EXP(-X(5)*TEMP)
FVEC(I) = Y4(I) - (X(1) + X(2)*TMP1 + X(3)*TMP2)
400 CONTINUE
GO TO 430
C
C OSBORNE 2 FUNCTION.
C
410 CONTINUE
DO 420 I = 1, 65
TEMP = FLOAT(I-1)/TEN
TMP1 = EXP(-X(5)*TEMP)
TMP2 = EXP(-X(6)*(TEMP-X(9))**2)
TMP3 = EXP(-X(7)*(TEMP-X(10))**2)
TMP4 = EXP(-X(8)*(TEMP-X(11))**2)
FVEC(I) = Y5(I)
* - (X(1)*TMP1 + X(2)*TMP2 + X(3)*TMP3 + X(4)*TMP4)
420 CONTINUE
430 CONTINUE
RETURN
C
C LAST CARD OF SUBROUTINE SSQFCN.
C
END
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