--- /dev/null
+*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
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