*DECK SNLS1
SUBROUTINE SNLS1 (FCN, IOPT, M, N, X, FVEC, FJAC, LDFJAC, FTOL,
+ XTOL, GTOL, MAXFEV, EPSFCN, DIAG, MODE, FACTOR, NPRINT, INFO,
+ NFEV, NJEV, IPVT, QTF, WA1, WA2, WA3, WA4)
C***BEGIN PROLOGUE SNLS1
C***PURPOSE Minimize the sum of the squares of M nonlinear functions
C in N variables by a modification of the Levenberg-Marquardt
C algorithm.
C***LIBRARY SLATEC
C***CATEGORY K1B1A1, K1B1A2
C***TYPE SINGLE PRECISION (SNLS1-S, DNLS1-D)
C***KEYWORDS LEVENBERG-MARQUARDT, NONLINEAR DATA FITTING,
C NONLINEAR LEAST SQUARES
C***AUTHOR Hiebert, K. L., (SNLA)
C***DESCRIPTION
C
C 1. Purpose.
C
C The purpose of SNLS1 is to minimize the sum of the squares of M
C nonlinear functions in N variables by a modification of the
C Levenberg-Marquardt algorithm. The user must provide a subrou-
C tine which calculates the functions. The user has the option
C of how the Jacobian will be supplied. The user can supply the
C full Jacobian, or the rows of the Jacobian (to avoid storing
C the full Jacobian), or let the code approximate the Jacobian by
C forward-differencing. This code is the combination of the
C MINPACK codes (Argonne) LMDER, LMDIF, and LMSTR.
C
C
C 2. Subroutine and Type Statements.
C
C SUBROUTINE SNLS1(FCN,IOPT,M,N,X,FVEC,FJAC,LDFJAC,FTOL,XTOL,
C * GTOL,MAXFEV,EPSFCN,DIAG,MODE,FACTOR,NPRINT,INFO
C * ,NFEV,NJEV,IPVT,QTF,WA1,WA2,WA3,WA4)
C INTEGER IOPT,M,N,LDFJAC,MAXFEV,MODE,NPRINT,INFO,NFEV,NJEV
C INTEGER IPVT(N)
C REAL FTOL,XTOL,GTOL,EPSFCN,FACTOR
C REAL X(N),FVEC(M),FJAC(LDFJAC,N),DIAG(N),QTF(N),
C * WA1(N),WA2(N),WA3(N),WA4(M)
C
C
C 3. Parameters.
C
C Parameters designated as input parameters must be specified on
C entry to SNLS1 and are not changed on exit, while parameters
C designated as output parameters need not be specified on entry
C and are set to appropriate values on exit from SNLS1.
C
C FCN is the name of the user-supplied subroutine which calculates
C the functions. If the user wants to supply the Jacobian
C (IOPT=2 or 3), then FCN must be written to calculate the
C Jacobian, as well as the functions. See the explanation
C of the IOPT argument below.
C If the user wants the iterates printed (NPRINT positive), then
C FCN must do the printing. See the explanation of NPRINT
C below. FCN must be declared in an EXTERNAL statement in the
C calling program and should be written as follows.
C
C
C SUBROUTINE FCN(IFLAG,M,N,X,FVEC,FJAC,LDFJAC)
C INTEGER IFLAG,LDFJAC,M,N
C REAL X(N),FVEC(M)
C ----------
C FJAC and LDFJAC may be ignored , if IOPT=1.
C REAL FJAC(LDFJAC,N) , if IOPT=2.
C REAL FJAC(N) , if IOPT=3.
C ----------
C If IFLAG=0, the values in X and FVEC are available
C for printing. See the explanation of NPRINT below.
C IFLAG will never be zero unless NPRINT is positive.
C The values of X and FVEC must not be changed.
C RETURN
C ----------
C If IFLAG=1, calculate the functions at X and return
C this vector in FVEC.
C RETURN
C ----------
C If IFLAG=2, calculate the full Jacobian at X and return
C this matrix in FJAC. Note that IFLAG will never be 2 unless
C IOPT=2. FVEC contains the function values at X and must
C not be altered. FJAC(I,J) must be set to the derivative
C of FVEC(I) with respect to X(J).
C RETURN
C ----------
C If IFLAG=3, calculate the LDFJAC-th row of the Jacobian
C and return this vector in FJAC. Note that IFLAG will
C never be 3 unless IOPT=3. FVEC contains the function
C values at X and must not be altered. FJAC(J) must be
C set to the derivative of FVEC(LDFJAC) with respect to X(J).
C RETURN
C ----------
C END
C
C
C The value of IFLAG should not be changed by FCN unless the
C user wants to terminate execution of SNLS1. In this case, set
C IFLAG to a negative integer.
C
C
C IOPT is an input variable which specifies how the Jacobian will
C be calculated. If IOPT=2 or 3, then the user must supply the
C Jacobian, as well as the function values, through the
C subroutine FCN. If IOPT=2, the user supplies the full
C Jacobian with one call to FCN. If IOPT=3, the user supplies
C one row of the Jacobian with each call. (In this manner,
C storage can be saved because the full Jacobian is not stored.)
C If IOPT=1, the code will approximate the Jacobian by forward
C differencing.
C
C M is a positive integer input variable set to the number of
C functions.
C
C N is a positive integer input variable set to the number of
C variables. N must not exceed M.
C
C X is an array of length N. On input, X must contain an initial
C estimate of the solution vector. On output, X contains the
C final estimate of the solution vector.
C
C FVEC is an output array of length M which contains the functions
C evaluated at the output X.
C
C FJAC is an output array. For IOPT=1 and 2, FJAC is an M by N
C array. For IOPT=3, FJAC is an N by N array. The upper N by N
C submatrix of FJAC contains an upper triangular matrix R with
C diagonal elements of nonincreasing magnitude such that
C
C T T T
C P *(JAC *JAC)*P = R *R,
C
C where P is a permutation matrix and JAC is the final calcu-
C lated Jacobian. Column J of P is column IPVT(J) (see below)
C of the identity matrix. The lower part of FJAC contains
C information generated during the computation of R.
C
C LDFJAC is a positive integer input variable which specifies
C the leading dimension of the array FJAC. For IOPT=1 and 2,
C LDFJAC must not be less than M. For IOPT=3, LDFJAC must not
C be less than N.
C
C FTOL is a non-negative input variable. Termination occurs when
C both the actual and predicted relative reductions in the sum
C of squares are at most FTOL. Therefore, FTOL measures the
C relative error desired in the sum of squares. Section 4 con-
C tains more details about FTOL.
C
C XTOL is a non-negative input variable. Termination occurs when
C the relative error between two consecutive iterates is at most
C XTOL. Therefore, XTOL measures the relative error desired in
C the approximate solution. Section 4 contains more details
C about XTOL.
C
C GTOL is a non-negative input variable. Termination occurs when
C the cosine of the angle between FVEC and any column of the
C Jacobian is at most GTOL in absolute value. Therefore, GTOL
C measures the orthogonality desired between the function vector
C and the columns of the Jacobian. Section 4 contains more
C details about GTOL.
C
C MAXFEV is a positive integer input variable. Termination occurs
C when the number of calls to FCN to evaluate the functions
C has reached MAXFEV.
C
C EPSFCN is an input variable used in determining a suitable step
C for the forward-difference approximation. This approximation
C assumes that the relative errors in the functions are of the
C order of EPSFCN. If EPSFCN is less than the machine preci-
C sion, it is assumed that the relative errors in the functions
C are of the order of the machine precision. If IOPT=2 or 3,
C then EPSFCN can be ignored (treat it as a dummy argument).
C
C DIAG is an array of length N. If MODE = 1 (see below), DIAG is
C internally set. If MODE = 2, DIAG must contain positive
C entries that serve as implicit (multiplicative) scale factors
C for the variables.
C
C MODE is an integer input variable. If MODE = 1, the variables
C will be scaled internally. If MODE = 2, the scaling is speci-
C fied by the input DIAG. Other values of MODE are equivalent
C to MODE = 1.
C
C FACTOR is a positive input variable used in determining the ini-
C tial step bound. This bound is set to the product of FACTOR
C and the Euclidean norm of DIAG*X if nonzero, or else to FACTOR
C itself. In most cases FACTOR should lie in the interval
C (.1,100.). 100. is a generally recommended value.
C
C NPRINT is an integer input variable that enables controlled
C printing of iterates if it is positive. In this case, FCN is
C called with IFLAG = 0 at the beginning of the first iteration
C and every NPRINT iterations thereafter and immediately prior
C to return, with X and FVEC available for printing. Appropriate
C print statements must be added to FCN (see example) and
C FVEC should not be altered. If NPRINT is not positive, no
C special calls to FCN with IFLAG = 0 are made.
C
C INFO is an integer output variable. If the user has terminated
C execution, INFO is set to the (negative) value of IFLAG. See
C description of FCN and JAC. Otherwise, INFO is set as follows.
C
C INFO = 0 improper input parameters.
C
C INFO = 1 both actual and predicted relative reductions in the
C sum of squares are at most FTOL.
C
C INFO = 2 relative error between two consecutive iterates is
C at most XTOL.
C
C INFO = 3 conditions for INFO = 1 and INFO = 2 both hold.
C
C INFO = 4 the cosine of the angle between FVEC and any column
C of the Jacobian is at most GTOL in absolute value.
C
C INFO = 5 number of calls to FCN for function evaluation
C has reached MAXFEV.
C
C INFO = 6 FTOL is too small. No further reduction in the sum
C of squares is possible.
C
C INFO = 7 XTOL is too small. No further improvement in the
C approximate solution X is possible.
C
C INFO = 8 GTOL is too small. FVEC is orthogonal to the
C columns of the Jacobian to machine precision.
C
C Sections 4 and 5 contain more details about INFO.
C
C NFEV is an integer output variable set to the number of calls to
C FCN for function evaluation.
C
C NJEV is an integer output variable set to the number of
C evaluations of the full Jacobian. If IOPT=2, only one call to
C FCN is required for each evaluation of the full Jacobian.
C If IOPT=3, the M calls to FCN are required.
C If IOPT=1, then NJEV is set to zero.
C
C IPVT is an integer output array of length N. IPVT defines a
C permutation matrix P such that JAC*P = Q*R, where JAC is the
C final calculated Jacobian, Q is orthogonal (not stored), and R
C is upper triangular with diagonal elements of nonincreasing
C magnitude. Column J of P is column IPVT(J) of the identity
C matrix.
C
C QTF is an output array of length N which contains the first N
C elements of the vector (Q transpose)*FVEC.
C
C WA1, WA2, and WA3 are work arrays of length N.
C
C WA4 is a work array of length M.
C
C
C 4. Successful Completion.
C
C The accuracy of SNLS1 is controlled by the convergence parame-
C ters FTOL, XTOL, and GTOL. These parameters are used in tests
C which make three types of comparisons between the approximation
C X and a solution XSOL. SNLS1 terminates when any of the tests
C is satisfied. If any of the convergence parameters is less than
C the machine precision (as defined by the function R1MACH(4)),
C then SNLS1 only attempts to satisfy the test defined by the
C machine precision. Further progress is not usually possible.
C
C The tests assume that the functions are reasonably well behaved,
C and, if the Jacobian is supplied by the user, that the functions
C and the Jacobian are coded consistently. If these conditions
C are not satisfied, then SNLS1 may incorrectly indicate conver-
C gence. If the Jacobian is coded correctly or IOPT=1,
C then the validity of the answer can be checked, for example, by
C rerunning SNLS1 with tighter tolerances.
C
C First Convergence Test. If ENORM(Z) denotes the Euclidean norm
C of a vector Z, then this test attempts to guarantee that
C
C ENORM(FVEC) .LE. (1+FTOL)*ENORM(FVECS),
C
C where FVECS denotes the functions evaluated at XSOL. If this
C condition is satisfied with FTOL = 10**(-K), then the final
C residual norm ENORM(FVEC) has K significant decimal digits and
C INFO is set to 1 (or to 3 if the second test is also satis-
C fied). Unless high precision solutions are required, the
C recommended value for FTOL is the square root of the machine
C precision.
C
C Second Convergence Test. If D is the diagonal matrix whose
C entries are defined by the array DIAG, then this test attempts
C to guarantee that
C
C ENORM(D*(X-XSOL)) .LE. XTOL*ENORM(D*XSOL).
C
C If this condition is satisfied with XTOL = 10**(-K), then the
C larger components of D*X have K significant decimal digits and
C INFO is set to 2 (or to 3 if the first test is also satis-
C fied). There is a danger that the smaller components of D*X
C may have large relative errors, but if MODE = 1, then the
C accuracy of the components of X is usually related to their
C sensitivity. Unless high precision solutions are required,
C the recommended value for XTOL is the square root of the
C machine precision.
C
C Third Convergence Test. This test is satisfied when the cosine
C of the angle between FVEC and any column of the Jacobian at X
C is at most GTOL in absolute value. There is no clear rela-
C tionship between this test and the accuracy of SNLS1, and
C furthermore, the test is equally well satisfied at other crit-
C ical points, namely maximizers and saddle points. Therefore,
C termination caused by this test (INFO = 4) should be examined
C carefully. The recommended value for GTOL is zero.
C
C
C 5. Unsuccessful Completion.
C
C Unsuccessful termination of SNLS1 can be due to improper input
C parameters, arithmetic interrupts, or an excessive number of
C function evaluations.
C
C Improper Input Parameters. INFO is set to 0 if IOPT .LT. 1
C or IOPT .GT. 3, or N .LE. 0, or M .LT. N, or for IOPT=1 or 2
C LDFJAC .LT. M, or for IOPT=3 LDFJAC .LT. N, or FTOL .LT. 0.E0,
C or XTOL .LT. 0.E0, or GTOL .LT. 0.E0, or MAXFEV .LE. 0, or
C FACTOR .LE. 0.E0.
C
C Arithmetic Interrupts. If these interrupts occur in the FCN
C subroutine during an early stage of the computation, they may
C be caused by an unacceptable choice of X by SNLS1. In this
C case, it may be possible to remedy the situation by rerunning
C SNLS1 with a smaller value of FACTOR.
C
C Excessive Number of Function Evaluations. A reasonable value
C for MAXFEV is 100*(N+1) for IOPT=2 or 3 and 200*(N+1) for
C IOPT=1. If the number of calls to FCN reaches MAXFEV, then
C this indicates that the routine is converging very slowly
C as measured by the progress of FVEC, and INFO is set to 5.
C In this case, it may be helpful to restart SNLS1 with MODE
C set to 1.
C
C
C 6. Characteristics of the Algorithm.
C
C SNLS1 is a modification of the Levenberg-Marquardt algorithm.
C Two of its main characteristics involve the proper use of
C implicitly scaled variables (if MODE = 1) and an optimal choice
C for the correction. The use of implicitly scaled variables
C achieves scale invariance of SNLS1 and limits the size of the
C correction in any direction where the functions are changing
C rapidly. The optimal choice of the correction guarantees (under
C reasonable conditions) global convergence from starting points
C far from the solution and a fast rate of convergence for
C problems with small residuals.
C
C Timing. The time required by SNLS1 to solve a given problem
C depends on M and N, the behavior of the functions, the accu-
C racy requested, and the starting point. The number of arith-
C metic operations needed by SNLS1 is about N**3 to process each
C evaluation of the functions (call to FCN) and to process each
C evaluation of the Jacobian it takes M*N**2 for IOPT=2 (one
C call to FCN), M*N**2 for IOPT=1 (N calls to FCN) and
C 1.5*M*N**2 for IOPT=3 (M calls to FCN). Unless FCN
C can be evaluated quickly, the timing of SNLS1 will be
C strongly influenced by the time spent in FCN.
C
C Storage. SNLS1 requires (M*N + 2*M + 6*N) for IOPT=1 or 2 and
C (N**2 + 2*M + 6*N) for IOPT=3 single precision storage
C locations and N integer storage locations, in addition to
C the storage required by the program. There are no internally
C declared storage arrays.
C
C *Long Description:
C
C 7. Example.
C
C The problem is to determine the values of X(1), X(2), and X(3)
C which provide the best fit (in the least squares sense) of
C
C X(1) + U(I)/(V(I)*X(2) + W(I)*X(3)), I = 1, 15
C
C to the data
C
C Y = (0.14,0.18,0.22,0.25,0.29,0.32,0.35,0.39,
C 0.37,0.58,0.73,0.96,1.34,2.10,4.39),
C
C where U(I) = I, V(I) = 16 - I, and W(I) = MIN(U(I),V(I)). The
C I-th component of FVEC is thus defined by
C
C Y(I) - (X(1) + U(I)/(V(I)*X(2) + W(I)*X(3))).
C
C **********
C
C PROGRAM TEST
C C
C C Driver for SNLS1 example.
C C
C INTEGER J,IOPT,M,N,LDFJAC,MAXFEV,MODE,NPRINT,INFO,NFEV,NJEV,
C * NWRITE
C INTEGER IPVT(3)
C REAL FTOL,XTOL,GTOL,FACTOR,FNORM,EPSFCN
C REAL X(3),FVEC(15),FJAC(15,3),DIAG(3),QTF(3),
C * WA1(3),WA2(3),WA3(3),WA4(15)
C REAL ENORM,R1MACH
C EXTERNAL FCN
C DATA NWRITE /6/
C C
C IOPT = 1
C M = 15
C N = 3
C C
C C The following starting values provide a rough fit.
C C
C X(1) = 1.E0
C X(2) = 1.E0
C X(3) = 1.E0
C C
C LDFJAC = 15
C C
C C Set FTOL and XTOL to the square root of the machine precision
C C and GTOL to zero. Unless high precision solutions are
C C required, these are the recommended settings.
C C
C FTOL = SQRT(R1MACH(4))
C XTOL = SQRT(R1MACH(4))
C GTOL = 0.E0
C C
C MAXFEV = 400
C EPSFCN = 0.0
C MODE = 1
C FACTOR = 1.E2
C NPRINT = 0
C C
C CALL SNLS1(FCN,IOPT,M,N,X,FVEC,FJAC,LDFJAC,FTOL,XTOL,
C * GTOL,MAXFEV,EPSFCN,DIAG,MODE,FACTOR,NPRINT,
C * INFO,NFEV,NJEV,IPVT,QTF,WA1,WA2,WA3,WA4)
C FNORM = ENORM(M,FVEC)
C WRITE (NWRITE,1000) FNORM,NFEV,NJEV,INFO,(X(J),J=1,N)
C STOP
C 1000 FORMAT (5X,' FINAL L2 NORM OF THE RESIDUALS',E15.7 //
C * 5X,' NUMBER OF FUNCTION EVALUATIONS',I10 //
C * 5X,' NUMBER OF JACOBIAN EVALUATIONS',I10 //
C * 5X,' EXIT PARAMETER',16X,I10 //
C * 5X,' FINAL APPROXIMATE SOLUTION' // 5X,3E15.7)
C END
C SUBROUTINE FCN(IFLAG,M,N,X,FVEC,DUM,IDUM)
C C This is the form of the FCN routine if IOPT=1,
C C that is, if the user does not calculate the Jacobian.
C INTEGER M,N,IFLAG
C REAL X(N),FVEC(M)
C INTEGER I
C REAL TMP1,TMP2,TMP3,TMP4
C REAL Y(15)
C DATA Y(1),Y(2),Y(3),Y(4),Y(5),Y(6),Y(7),Y(8),
C * Y(9),Y(10),Y(11),Y(12),Y(13),Y(14),Y(15)
C * /1.4E-1,1.8E-1,2.2E-1,2.5E-1,2.9E-1,3.2E-1,3.5E-1,3.9E-1,
C * 3.7E-1,5.8E-1,7.3E-1,9.6E-1,1.34E0,2.1E0,4.39E0/
C C
C IF (IFLAG .NE. 0) GO TO 5
C C
C C Insert print statements here when NPRINT is positive.
C C
C RETURN
C 5 CONTINUE
C DO 10 I = 1, M
C TMP1 = I
C TMP2 = 16 - I
C TMP3 = TMP1
C IF (I .GT. 8) TMP3 = TMP2
C FVEC(I) = Y(I) - (X(1) + TMP1/(X(2)*TMP2 + X(3)*TMP3))
C 10 CONTINUE
C RETURN
C END
C
C
C Results obtained with different compilers or machines
C may be slightly different.
C
C FINAL L2 NORM OF THE RESIDUALS 0.9063596E-01
C
C NUMBER OF FUNCTION EVALUATIONS 25
C
C NUMBER OF JACOBIAN EVALUATIONS 0
C
C EXIT PARAMETER 1
C
C FINAL APPROXIMATE SOLUTION
C
C 0.8241058E-01 0.1133037E+01 0.2343695E+01
C
C
C For IOPT=2, FCN would be modified as follows to also
C calculate the full Jacobian when IFLAG=2.
C
C SUBROUTINE FCN(IFLAG,M,N,X,FVEC,FJAC,LDFJAC)
C C
C C This is the form of the FCN routine if IOPT=2,
C C that is, if the user calculates the full Jacobian.
C C
C INTEGER LDFJAC,M,N,IFLAG
C REAL X(N),FVEC(M)
C REAL FJAC(LDFJAC,N)
C INTEGER I
C REAL TMP1,TMP2,TMP3,TMP4
C REAL Y(15)
C DATA Y(1),Y(2),Y(3),Y(4),Y(5),Y(6),Y(7),Y(8),
C * Y(9),Y(10),Y(11),Y(12),Y(13),Y(14),Y(15)
C * /1.4E-1,1.8E-1,2.2E-1,2.5E-1,2.9E-1,3.2E-1,3.5E-1,3.9E-1,
C * 3.7E-1,5.8E-1,7.3E-1,9.6E-1,1.34E0,2.1E0,4.39E0/
C C
C IF (IFLAG .NE. 0) GO TO 5
C C
C C Insert print statements here when NPRINT is positive.
C C
C RETURN
C 5 CONTINUE
C IF(IFLAG.NE.1) GO TO 20
C DO 10 I = 1, M
C TMP1 = I
C TMP2 = 16 - I
C TMP3 = TMP1
C IF (I .GT. 8) TMP3 = TMP2
C FVEC(I) = Y(I) - (X(1) + TMP1/(X(2)*TMP2 + X(3)*TMP3))
C 10 CONTINUE
C RETURN
C C
C C Below, calculate the full Jacobian.
C C
C 20 CONTINUE
C C
C DO 30 I = 1, M
C TMP1 = I
C TMP2 = 16 - I
C TMP3 = TMP1
C IF (I .GT. 8) TMP3 = TMP2
C TMP4 = (X(2)*TMP2 + X(3)*TMP3)**2
C FJAC(I,1) = -1.E0
C FJAC(I,2) = TMP1*TMP2/TMP4
C FJAC(I,3) = TMP1*TMP3/TMP4
C 30 CONTINUE
C RETURN
C END
C
C
C For IOPT = 3, FJAC would be dimensioned as FJAC(3,3),
C LDFJAC would be set to 3, and FCN would be written as
C follows to calculate a row of the Jacobian when IFLAG=3.
C
C SUBROUTINE FCN(IFLAG,M,N,X,FVEC,FJAC,LDFJAC)
C C This is the form of the FCN routine if IOPT=3,
C C that is, if the user calculates the Jacobian row by row.
C INTEGER M,N,IFLAG
C REAL X(N),FVEC(M)
C REAL FJAC(N)
C INTEGER I
C REAL TMP1,TMP2,TMP3,TMP4
C REAL Y(15)
C DATA Y(1),Y(2),Y(3),Y(4),Y(5),Y(6),Y(7),Y(8),
C * Y(9),Y(10),Y(11),Y(12),Y(13),Y(14),Y(15)
C * /1.4E-1,1.8E-1,2.2E-1,2.5E-1,2.9E-1,3.2E-1,3.5E-1,3.9E-1,
C * 3.7E-1,5.8E-1,7.3E-1,9.6E-1,1.34E0,2.1E0,4.39E0/
C C
C IF (IFLAG .NE. 0) GO TO 5
C C
C C Insert print statements here when NPRINT is positive.
C C
C RETURN
C 5 CONTINUE
C IF( IFLAG.NE.1) GO TO 20
C DO 10 I = 1, M
C TMP1 = I
C TMP2 = 16 - I
C TMP3 = TMP1
C IF (I .GT. 8) TMP3 = TMP2
C FVEC(I) = Y(I) - (X(1) + TMP1/(X(2)*TMP2 + X(3)*TMP3))
C 10 CONTINUE
C RETURN
C C
C C Below, calculate the LDFJAC-th row of the Jacobian.
C C
C 20 CONTINUE
C
C I = LDFJAC
C TMP1 = I
C TMP2 = 16 - I
C TMP3 = TMP1
C IF (I .GT. 8) TMP3 = TMP2
C TMP4 = (X(2)*TMP2 + X(3)*TMP3)**2
C FJAC(1) = -1.E0
C FJAC(2) = TMP1*TMP2/TMP4
C FJAC(3) = TMP1*TMP3/TMP4
C RETURN
C END
C
C***REFERENCES Jorge J. More, The Levenberg-Marquardt algorithm:
C implementation and theory. In Numerical Analysis
C Proceedings (Dundee, June 28 - July 1, 1977, G. A.
C Watson, Editor), Lecture Notes in Mathematics 630,
C Springer-Verlag, 1978.
C***ROUTINES CALLED CHKDER, ENORM, FDJAC3, LMPAR, QRFAC, R1MACH,
C RWUPDT, XERMSG
C***REVISION HISTORY (YYMMDD)
C 800301 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890531 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ)
C 900510 Convert XERRWV calls to XERMSG calls. (RWC)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE SNLS1
INTEGER IOPT,M,N,LDFJAC,MAXFEV,MODE,NPRINT,INFO,NFEV,NJEV
INTEGER IJUNK,NROW,IPVT(*)
REAL FTOL,XTOL,GTOL,FACTOR,EPSFCN
REAL X(*),FVEC(*),FJAC(LDFJAC,*),DIAG(*),QTF(*),WA1(*),WA2(*),
1 WA3(*),WA4(*)
LOGICAL SING
EXTERNAL FCN
INTEGER I,IFLAG,ITER,J,L,MODECH
REAL ACTRED,DELTA,DIRDER,EPSMCH,FNORM,FNORM1,GNORM,ONE,PAR,
1 PNORM,PRERED,P1,P5,P25,P75,P0001,RATIO,SUM,TEMP,TEMP1,
2 TEMP2,XNORM,ZERO
REAL R1MACH,ENORM,ERR,CHKLIM
CHARACTER*8 XERN1
CHARACTER*16 XERN3
C
SAVE CHKLIM, ONE, P1, P5, P25, P75, P0001, ZERO
DATA CHKLIM/.1E0/
DATA ONE,P1,P5,P25,P75,P0001,ZERO
1 /1.0E0,1.0E-1,5.0E-1,2.5E-1,7.5E-1,1.0E-4,0.0E0/
C
C***FIRST EXECUTABLE STATEMENT SNLS1
EPSMCH = R1MACH(4)
C
INFO = 0
IFLAG = 0
NFEV = 0
NJEV = 0
C
C CHECK THE INPUT PARAMETERS FOR ERRORS.
C
IF (IOPT .LT. 1 .OR. IOPT .GT. 3 .OR. N .LE. 0 .OR.
1 M .LT. N .OR. LDFJAC .LT. N .OR. FTOL .LT. ZERO
2 .OR. XTOL .LT. ZERO .OR. GTOL .LT. ZERO
3 .OR. MAXFEV .LE. 0 .OR. FACTOR .LE. ZERO) GO TO 300
IF (IOPT .LT. 3 .AND. LDFJAC .LT. M) GO TO 300
IF (MODE .NE. 2) GO TO 20
DO 10 J = 1, N
IF (DIAG(J) .LE. ZERO) GO TO 300
10 CONTINUE
20 CONTINUE
C
C EVALUATE THE FUNCTION AT THE STARTING POINT
C AND CALCULATE ITS NORM.
C
IFLAG = 1
IJUNK = 1
CALL FCN(IFLAG,M,N,X,FVEC,FJAC,IJUNK)
NFEV = 1
IF (IFLAG .LT. 0) GO TO 300
FNORM = ENORM(M,FVEC)
C
C INITIALIZE LEVENBERG-MARQUARDT PARAMETER AND ITERATION COUNTER.
C
PAR = ZERO
ITER = 1
C
C BEGINNING OF THE OUTER LOOP.
C
30 CONTINUE
C
C IF REQUESTED, CALL FCN TO ENABLE PRINTING OF ITERATES.
C
IF (NPRINT .LE. 0) GO TO 40
IFLAG = 0
IF (MOD(ITER-1,NPRINT) .EQ. 0)
1 CALL FCN(IFLAG,M,N,X,FVEC,FJAC,IJUNK)
IF (IFLAG .LT. 0) GO TO 300
40 CONTINUE
C
C CALCULATE THE JACOBIAN MATRIX.
C
IF (IOPT .EQ. 3) GO TO 475
C
C STORE THE FULL JACOBIAN USING M*N STORAGE
C
IF (IOPT .EQ. 1) GO TO 410
C
C THE USER SUPPLIES THE JACOBIAN
C
IFLAG = 2
CALL FCN(IFLAG,M,N,X,FVEC,FJAC,LDFJAC)
NJEV = NJEV + 1
C
C ON THE FIRST ITERATION, CHECK THE USER SUPPLIED JACOBIAN
C
IF (ITER .LE. 1) THEN
IF (IFLAG .LT. 0) GO TO 300
C
C GET THE INCREMENTED X-VALUES INTO WA1(*).
C
MODECH = 1
CALL CHKDER(M,N,X,FVEC,FJAC,LDFJAC,WA1,WA4,MODECH,ERR)
C
C EVALUATE FUNCTION AT INCREMENTED VALUE AND PUT IN WA4(*).
C
IFLAG = 1
CALL FCN(IFLAG,M,N,WA1,WA4,FJAC,LDFJAC)
NFEV = NFEV + 1
IF(IFLAG .LT. 0) GO TO 300
DO 350 I = 1, M
MODECH = 2
CALL CHKDER(1,N,X,FVEC(I),FJAC(I,1),LDFJAC,WA1,
1 WA4(I),MODECH,ERR)
IF (ERR .LT. CHKLIM) THEN
WRITE (XERN1, '(I8)') I
WRITE (XERN3, '(1PE15.6)') ERR
CALL XERMSG ('SLATEC', 'SNLS1', 'DERIVATIVE OF ' //
* 'FUNCTION ' // XERN1 // ' MAY BE WRONG, ERR = ' //
* XERN3 // ' TOO CLOSE TO 0.', 7, 0)
ENDIF
350 CONTINUE
ENDIF
C
GO TO 420
C
C THE CODE APPROXIMATES THE JACOBIAN
C
410 IFLAG = 1
CALL FDJAC3(FCN,M,N,X,FVEC,FJAC,LDFJAC,IFLAG,EPSFCN,WA4)
NFEV = NFEV + N
420 IF (IFLAG .LT. 0) GO TO 300
C
C COMPUTE THE QR FACTORIZATION OF THE JACOBIAN.
C
CALL QRFAC(M,N,FJAC,LDFJAC,.TRUE.,IPVT,N,WA1,WA2,WA3)
C
C FORM (Q TRANSPOSE)*FVEC AND STORE THE FIRST N COMPONENTS IN
C QTF.
C
DO 430 I = 1, M
WA4(I) = FVEC(I)
430 CONTINUE
DO 470 J = 1, N
IF (FJAC(J,J) .EQ. ZERO) GO TO 460
SUM = ZERO
DO 440 I = J, M
SUM = SUM + FJAC(I,J)*WA4(I)
440 CONTINUE
TEMP = -SUM/FJAC(J,J)
DO 450 I = J, M
WA4(I) = WA4(I) + FJAC(I,J)*TEMP
450 CONTINUE
460 CONTINUE
FJAC(J,J) = WA1(J)
QTF(J) = WA4(J)
470 CONTINUE
GO TO 560
C
C ACCUMULATE THE JACOBIAN BY ROWS IN ORDER TO SAVE STORAGE.
C COMPUTE THE QR FACTORIZATION OF THE JACOBIAN MATRIX
C CALCULATED ONE ROW AT A TIME, WHILE SIMULTANEOUSLY
C FORMING (Q TRANSPOSE)*FVEC AND STORING THE FIRST
C N COMPONENTS IN QTF.
C
475 DO 490 J = 1, N
QTF(J) = ZERO
DO 480 I = 1, N
FJAC(I,J) = ZERO
480 CONTINUE
490 CONTINUE
DO 500 I = 1, M
NROW = I
IFLAG = 3
CALL FCN(IFLAG,M,N,X,FVEC,WA3,NROW)
IF (IFLAG .LT. 0) GO TO 300
C
C ON THE FIRST ITERATION, CHECK THE USER SUPPLIED JACOBIAN.
C
IF(ITER .GT. 1) GO TO 498
C
C GET THE INCREMENTED X-VALUES INTO WA1(*).
C
MODECH = 1
CALL CHKDER(M,N,X,FVEC,FJAC,LDFJAC,WA1,WA4,MODECH,ERR)
C
C EVALUATE AT INCREMENTED VALUES, IF NOT ALREADY EVALUATED.
C
IF(I .NE. 1) GO TO 495
C
C EVALUATE FUNCTION AT INCREMENTED VALUE AND PUT INTO WA4(*).
C
IFLAG = 1
CALL FCN(IFLAG,M,N,WA1,WA4,FJAC,NROW)
NFEV = NFEV + 1
IF(IFLAG .LT. 0) GO TO 300
495 CONTINUE
MODECH = 2
CALL CHKDER(1,N,X,FVEC(I),WA3,1,WA1,WA4(I),MODECH,ERR)
IF (ERR .LT. CHKLIM) THEN
WRITE (XERN1, '(I8)') I
WRITE (XERN3, '(1PE15.6)') ERR
CALL XERMSG ('SLATEC', 'SNLS1', 'DERIVATIVE OF FUNCTION '
* // XERN1 // ' MAY BE WRONG, ERR = ' // XERN3 //
* ' TOO CLOSE TO 0.', 7, 0)
ENDIF
498 CONTINUE
C
TEMP = FVEC(I)
CALL RWUPDT(N,FJAC,LDFJAC,WA3,QTF,TEMP,WA1,WA2)
500 CONTINUE
NJEV = NJEV + 1
C
C IF THE JACOBIAN IS RANK DEFICIENT, CALL QRFAC TO
C REORDER ITS COLUMNS AND UPDATE THE COMPONENTS OF QTF.
C
SING = .FALSE.
DO 510 J = 1, N
IF (FJAC(J,J) .EQ. ZERO) SING = .TRUE.
IPVT(J) = J
WA2(J) = ENORM(J,FJAC(1,J))
510 CONTINUE
IF (.NOT.SING) GO TO 560
CALL QRFAC(N,N,FJAC,LDFJAC,.TRUE.,IPVT,N,WA1,WA2,WA3)
DO 550 J = 1, N
IF (FJAC(J,J) .EQ. ZERO) GO TO 540
SUM = ZERO
DO 520 I = J, N
SUM = SUM + FJAC(I,J)*QTF(I)
520 CONTINUE
TEMP = -SUM/FJAC(J,J)
DO 530 I = J, N
QTF(I) = QTF(I) + FJAC(I,J)*TEMP
530 CONTINUE
540 CONTINUE
FJAC(J,J) = WA1(J)
550 CONTINUE
560 CONTINUE
C
C ON THE FIRST ITERATION AND IF MODE IS 1, SCALE ACCORDING
C TO THE NORMS OF THE COLUMNS OF THE INITIAL JACOBIAN.
C
IF (ITER .NE. 1) GO TO 80
IF (MODE .EQ. 2) GO TO 60
DO 50 J = 1, N
DIAG(J) = WA2(J)
IF (WA2(J) .EQ. ZERO) DIAG(J) = ONE
50 CONTINUE
60 CONTINUE
C
C ON THE FIRST ITERATION, CALCULATE THE NORM OF THE SCALED X
C AND INITIALIZE THE STEP BOUND DELTA.
C
DO 70 J = 1, N
WA3(J) = DIAG(J)*X(J)
70 CONTINUE
XNORM = ENORM(N,WA3)
DELTA = FACTOR*XNORM
IF (DELTA .EQ. ZERO) DELTA = FACTOR
80 CONTINUE
C
C COMPUTE THE NORM OF THE SCALED GRADIENT.
C
GNORM = ZERO
IF (FNORM .EQ. ZERO) GO TO 170
DO 160 J = 1, N
L = IPVT(J)
IF (WA2(L) .EQ. ZERO) GO TO 150
SUM = ZERO
DO 140 I = 1, J
SUM = SUM + FJAC(I,J)*(QTF(I)/FNORM)
140 CONTINUE
GNORM = MAX(GNORM,ABS(SUM/WA2(L)))
150 CONTINUE
160 CONTINUE
170 CONTINUE
C
C TEST FOR CONVERGENCE OF THE GRADIENT NORM.
C
IF (GNORM .LE. GTOL) INFO = 4
IF (INFO .NE. 0) GO TO 300
C
C RESCALE IF NECESSARY.
C
IF (MODE .EQ. 2) GO TO 190
DO 180 J = 1, N
DIAG(J) = MAX(DIAG(J),WA2(J))
180 CONTINUE
190 CONTINUE
C
C BEGINNING OF THE INNER LOOP.
C
200 CONTINUE
C
C DETERMINE THE LEVENBERG-MARQUARDT PARAMETER.
C
CALL LMPAR(N,FJAC,LDFJAC,IPVT,DIAG,QTF,DELTA,PAR,WA1,WA2,
1 WA3,WA4)
C
C STORE THE DIRECTION P AND X + P. CALCULATE THE NORM OF P.
C
DO 210 J = 1, N
WA1(J) = -WA1(J)
WA2(J) = X(J) + WA1(J)
WA3(J) = DIAG(J)*WA1(J)
210 CONTINUE
PNORM = ENORM(N,WA3)
C
C ON THE FIRST ITERATION, ADJUST THE INITIAL STEP BOUND.
C
IF (ITER .EQ. 1) DELTA = MIN(DELTA,PNORM)
C
C EVALUATE THE FUNCTION AT X + P AND CALCULATE ITS NORM.
C
IFLAG = 1
CALL FCN(IFLAG,M,N,WA2,WA4,FJAC,IJUNK)
NFEV = NFEV + 1
IF (IFLAG .LT. 0) GO TO 300
FNORM1 = ENORM(M,WA4)
C
C COMPUTE THE SCALED ACTUAL REDUCTION.
C
ACTRED = -ONE
IF (P1*FNORM1 .LT. FNORM) ACTRED = ONE - (FNORM1/FNORM)**2
C
C COMPUTE THE SCALED PREDICTED REDUCTION AND
C THE SCALED DIRECTIONAL DERIVATIVE.
C
DO 230 J = 1, N
WA3(J) = ZERO
L = IPVT(J)
TEMP = WA1(L)
DO 220 I = 1, J
WA3(I) = WA3(I) + FJAC(I,J)*TEMP
220 CONTINUE
230 CONTINUE
TEMP1 = ENORM(N,WA3)/FNORM
TEMP2 = (SQRT(PAR)*PNORM)/FNORM
PRERED = TEMP1**2 + TEMP2**2/P5
DIRDER = -(TEMP1**2 + TEMP2**2)
C
C COMPUTE THE RATIO OF THE ACTUAL TO THE PREDICTED
C REDUCTION.
C
RATIO = ZERO
IF (PRERED .NE. ZERO) RATIO = ACTRED/PRERED
C
C UPDATE THE STEP BOUND.
C
IF (RATIO .GT. P25) GO TO 240
IF (ACTRED .GE. ZERO) TEMP = P5
IF (ACTRED .LT. ZERO)
1 TEMP = P5*DIRDER/(DIRDER + P5*ACTRED)
IF (P1*FNORM1 .GE. FNORM .OR. TEMP .LT. P1) TEMP = P1
DELTA = TEMP*MIN(DELTA,PNORM/P1)
PAR = PAR/TEMP
GO TO 260
240 CONTINUE
IF (PAR .NE. ZERO .AND. RATIO .LT. P75) GO TO 250
DELTA = PNORM/P5
PAR = P5*PAR
250 CONTINUE
260 CONTINUE
C
C TEST FOR SUCCESSFUL ITERATION.
C
IF (RATIO .LT. P0001) GO TO 290
C
C SUCCESSFUL ITERATION. UPDATE X, FVEC, AND THEIR NORMS.
C
DO 270 J = 1, N
X(J) = WA2(J)
WA2(J) = DIAG(J)*X(J)
270 CONTINUE
DO 280 I = 1, M
FVEC(I) = WA4(I)
280 CONTINUE
XNORM = ENORM(N,WA2)
FNORM = FNORM1
ITER = ITER + 1
290 CONTINUE
C
C TESTS FOR CONVERGENCE.
C
IF (ABS(ACTRED) .LE. FTOL .AND. PRERED .LE. FTOL
1 .AND. P5*RATIO .LE. ONE) INFO = 1
IF (DELTA .LE. XTOL*XNORM) INFO = 2
IF (ABS(ACTRED) .LE. FTOL .AND. PRERED .LE. FTOL
1 .AND. P5*RATIO .LE. ONE .AND. INFO .EQ. 2) INFO = 3
IF (INFO .NE. 0) GO TO 300
C
C TESTS FOR TERMINATION AND STRINGENT TOLERANCES.
C
IF (NFEV .GE. MAXFEV) INFO = 5
IF (ABS(ACTRED) .LE. EPSMCH .AND. PRERED .LE. EPSMCH
1 .AND. P5*RATIO .LE. ONE) INFO = 6
IF (DELTA .LE. EPSMCH*XNORM) INFO = 7
IF (GNORM .LE. EPSMCH) INFO = 8
IF (INFO .NE. 0) GO TO 300
C
C END OF THE INNER LOOP. REPEAT IF ITERATION UNSUCCESSFUL.
C
IF (RATIO .LT. P0001) GO TO 200
C
C END OF THE OUTER LOOP.
C
GO TO 30
300 CONTINUE
C
C TERMINATION, EITHER NORMAL OR USER IMPOSED.
C
IF (IFLAG .LT. 0) INFO = IFLAG
IFLAG = 0
IF (NPRINT .GT. 0) CALL FCN(IFLAG,M,N,X,FVEC,FJAC,IJUNK)
IF (INFO .LT. 0) CALL XERMSG ('SLATEC', 'SNLS1',
+ 'EXECUTION TERMINATED BECAUSE USER SET IFLAG NEGATIVE.', 1, 1)
IF (INFO .EQ. 0) CALL XERMSG ('SLATEC', 'SNLS1',
+ 'INVALID INPUT PARAMETER.', 2, 1)
IF (INFO .EQ. 4) CALL XERMSG ('SLATEC', 'SNLS1',
+ 'THIRD CONVERGENCE CONDITION, CHECK RESULTS BEFORE ACCEPTING.',
+ 1, 1)
IF (INFO .EQ. 5) CALL XERMSG ('SLATEC', 'SNLS1',
+ 'TOO MANY FUNCTION EVALUATIONS.', 9, 1)
IF (INFO .GE. 6) CALL XERMSG ('SLATEC', 'SNLS1',
+ 'TOLERANCES TOO SMALL, NO FURTHER IMPROVEMENT POSSIBLE.', 3, 1)
RETURN
C
C LAST CARD OF SUBROUTINE SNLS1.
C
END