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Page 1 Documentation for MINPACK subroutine LMDIF Double precision version Argonne National Laboratory Burton S. Garbow, Kenneth E. Hillstrom, Jorge J. More March 1980 1. Purpose. The purpose of LMDIF is to minimize the sum of the squares of M nonlinear functions in N variables by a modification of the Levenberg-Marquardt algorithm. The user must provide a subrou- tine which calculates the functions. The Jacobian is then cal- culated by a forward-difference approximation. 2. Subroutine and type statements. SUBROUTINE LMDIF(FCN,M,N,X,FVEC,FTOL,XTOL,GTOL,MAXFEV,EPSFCN, * DIAG,MODE,FACTOR,NPRINT,INFO,NFEV,FJAC,LDFJAC, * IPVT,QTF,WA1,WA2,WA3,WA4) INTEGER M,N,MAXFEV,MODE,NPRINT,INFO,NFEV,LDFJAC INTEGER IPVT(N) DOUBLE PRECISION FTOL,XTOL,GTOL,EPSFCN,FACTOR DOUBLE PRECISION X(N),FVEC(M),DIAG(N),FJAC(LDFJAC,N),QTF(N), * WA1(N),WA2(N),WA3(N),WA4(M) EXTERNAL FCN 3. Parameters. Parameters designated as input parameters must be specified on entry to LMDIF and are not changed on exit, while parameters designated as output parameters need not be specified on entry and are set to appropriate values on exit from LMDIF. FCN is the name of the user-supplied subroutine which calculates the functions. FCN must be declared in an EXTERNAL statement in the user calling program, and should be written as follows. SUBROUTINE FCN(M,N,X,FVEC,IFLAG) INTEGER M,N,IFLAG DOUBLE PRECISION X(N),FVEC(M) ---------- CALCULATE THE FUNCTIONS AT X AND RETURN THIS VECTOR IN FVEC. ---------- RETURN END Page 2 The value of IFLAG should not be changed by FCN unless the user wants to terminate execution of LMDIF. In this case set IFLAG to a negative integer. M is a positive integer input variable set to the number of functions. N is a positive integer input variable set to the number of variables. N must not exceed M. X is an array of length N. On input X must contain an initial estimate of the solution vector. On output X contains the final estimate of the solution vector. FVEC is an output array of length M which contains the functions evaluated at the output X. FTOL is a nonnegative input variable. Termination occurs when both the actual and predicted relative reductions in the sum of squares are at most FTOL. Therefore, FTOL measures the relative error desired in the sum of squares. Section 4 con- tains more details about FTOL. XTOL is a nonnegative input variable. Termination occurs when the relative error between two consecutive iterates is at most XTOL. Therefore, XTOL measures the relative error desired in the approximate solution. Section 4 contains more details about XTOL. GTOL is a nonnegative input variable. Termination occurs when the cosine of the angle between FVEC and any column of the Jacobian is at most GTOL in absolute value. Therefore, GTOL measures the orthogonality desired between the function vector and the columns of the Jacobian. Section 4 contains more details about GTOL. MAXFEV is a positive integer input variable. Termination occurs when the number of calls to FCN is at least MAXFEV by the end of an iteration. EPSFCN is an input variable used in determining a suitable step for the forward-difference approximation. This approximation assumes that the relative errors in the functions are of the order of EPSFCN. If EPSFCN is less than the machine preci- sion, it is assumed that the relative errors in the functions are of the order of the machine precision. DIAG is an array of length N. If MODE = 1 (see below), DIAG is internally set. If MODE = 2, DIAG must contain positive entries that serve as multiplicative scale factors for the variables. MODE is an integer input variable. If MODE = 1, the variables will be scaled internally. If MODE = 2, the scaling is Page 3 specified by the input DIAG. Other values of MODE are equiva- lent to MODE = 1. FACTOR is a positive input variable used in determining the ini- tial step bound. This bound is set to the product of FACTOR and the Euclidean norm of DIAG*X if nonzero, or else to FACTOR itself. In most cases FACTOR should lie in the interval (.1,100.). 100. is a generally recommended value. NPRINT is an integer input variable that enables controlled printing of iterates if it is positive. In this case, FCN is called with IFLAG = 0 at the beginning of the first iteration and every NPRINT iterations thereafter and immediately prior to return, with X and FVEC available for printing. If NPRINT is not positive, no special calls of FCN with IFLAG = 0 are made. INFO is an integer output variable. If the user has terminated execution, INFO is set to the (negative) value of IFLAG. See description of FCN. Otherwise, INFO is set as follows. INFO = 0 Improper input parameters. INFO = 1 Both actual and predicted relative reductions in the sum of squares are at most FTOL. INFO = 2 Relative error between two consecutive iterates is at most XTOL. INFO = 3 Conditions for INFO = 1 and INFO = 2 both hold. INFO = 4 The cosine of the angle between FVEC and any column of the Jacobian is at most GTOL in absolute value. INFO = 5 Number of calls to FCN has reached or exceeded MAXFEV. INFO = 6 FTOL is too small. No further reduction in the sum of squares is possible. INFO = 7 XTOL is too small. No further improvement in the approximate solution X is possible. INFO = 8 GTOL is too small. FVEC is orthogonal to the columns of the Jacobian to machine precision. Sections 4 and 5 contain more details about INFO. NFEV is an integer output variable set to the number of calls to FCN. FJAC is an output M by N array. The upper N by N submatrix of FJAC contains an upper triangular matrix R with diagonal ele- ments of nonincreasing magnitude such that Page 4 T T T P *(JAC *JAC)*P = R *R, where P is a permutation matrix and JAC is the final calcu- lated J