covar lmder lmder1 lmder1_r lmder_r lmdif lmdif1 lmdif1_r lmdif_r lmstr lmstr1 lmstr1_r lmstr_r n2f n2f_r n2g n2g_r n2p n2p_r

lmdif() void lmdif ( void(*)(int, int, double *, double *, int *)  fcn, int  m, int  n, double  x[], double  fvec[], int  ldfjac, double  fjac[], double  ftol, double  xtol, double  gtol, int  maxfev, double  epsfcn, double  diag[], int  mode, double  factor, int  nprint, int *  nfev, int  ipvt[], double  work[], int  lwork, int *  info  ) Nonlinear least squares approximation by Levenberg-Marquardt method (Jacobian not required) Purpose lmdif minimizes the sum of the squares of m nonlinear functions in n variables by a modification of the Levenberg-Marquardt algorithm. minimize the sum of fi(x1, x2, ..., xn)^2 (sum for i = 1 to m) The user must provide a subroutine which calculates the function values. Since the Jacobian is calculated by finite difference approximation within the routine, the user is not required to calculate the Jacobian. Parameters [in] fcn User supplied subroutine which calculates the functions fi(x) defined as follows. void fcn(int m, int n, double x[], double fvec[], int *iflag) { If iflag = 1 or iflag = 2: Calculate the function values fi(x) at x[] and return in fvec[i-1] (i = 1 to m). Other variables should not be changed. If iflag = 0: If nprint is set to positive value, fcn is called every nprint iterations with iflag = 0 for printing intermediate result x[] and fvec[]. Any variable should not be changed. } The value of iflag should not be changed unless the user wants to terminate the execution. In this case, set iflag to a negative integer. [in] m Number of functions. (m > 0) [in] n Number of variables. (0 < n <= m) [in,out] x[] Array x[lx] (lx >= n) [in] An initial estimate of the solution vector. [out] The obtained solution vector. [out] fvec[] Array fvec[lfvec] (lfvec >= m) The function values evaluated at the solution vector x[]. [in] ldfjac Leading dimension of the two dimensional array fjac[][]. (ldfjac >= m) [out] fjac[][] Array fjac[lfjac][ldfjac] (lfjac >= n) The upper n x n submatrix contains an upper triangular matrix R with diagonal elements of nonincreasing magnitude such that   P^T * (J^T * J)*P = R^T * R where P is a permutation matrix and J is the final calculated Jacobian. [in] ftol Relative error desired in the sum of squares. Termination occurs when both the actual and predicted relative reductions in the sum of squares are at most ftol. (ftol >= 0) [in] xtol Relative error desired in the approximate solution. Termination occurs when the relative error between two consecutive iterates is at most xtol. (xtol >= 0) [in] gtol Orthogonality desired between the function vector and the columns of the Jacobian. Termination occurs when the cosine of the angle between fvec[] and any column of the jacobian is at most gtol in absolute value. (gtol >= 0) [in] maxfev Termination occurs when the number of calls to fcn with iflag = 1 or 2 has reached this value. (maxfev > 0) [in] epsfcn Used in determining a suitable step length 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 precision, it is assumed that the relative errors in the functions are of the order of the machine precision. [in,out] diag[] Array diag[ldiag] (ldiag >= n) [in] If mode = 2, diag must contain positive entries that serve as multiplicative scale factors for the variables. (diag[i] > 0) [out] If mode = 1, diag[] is set by the subroutine. [in] mode = 1: The variables will be automatically scaled by the subroutine. = 2: The scaling is specified by the input diag[]. (For other values, mode = 1 is assumed.) [in] factor Used in determining the initial 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 (0.1, 100). 100 is a generally recommended value. (factor > 0) [in] nprint > 0: fcn is called with iflag = 0 at the beginning of the first iteration and every nprint iterations thereafter and at the end of last iteration for printing intermediate result. <= 0: No special calls of fcn for printing intermediate results are made. [out] nfev Number of function evaluations (number of calls to fcn with iflag = 1 or 2). [out] ipvt[] Array ipvt[lipvt] (lipvt >= n) The pivot indices that define the permutation matrix P. [out] work[] Array work[lwork] Work array. On return with info = 0, sub-code is set to work[0]. = 1: Both actual and predicted relative reductions in the sum of squares are at most ftol. = 2: Relative error between two consecutive iterates is at most xtol. = 3: Both of above are satisfied. = 4: The cosine of the angle between fvec and any column of the Jacobian is at most gtol in absolute value. [in] lwork The length of work[]. (lwork >= 5*n + 2*m) [out] info = 0: Successful exit (sub-code is set to work[0]) = -2: The argument m had an illegal value (m < n) = -3: The argument n had an illegal value (n < 1) = -6: The argument ldfjac had an illegal value (ldfjac < m) = -8: The argument ftol had an illegal value (ftol < 0) = -9: The argument xtol had an illegal value (xtol < 0) = -10: The argument gtol had an illegal value (gtol < 0) = -11: The argument maxfev had an illegal value (maxfev <= 0) = -13: The argument diag had an illegal value (diag[i] <= 0 when mode = 2) = -15: The argument factor had an illegal value (factor <= 0) = -20: The argument lwork had an illegal value (lwork too small) = 1: Number of calls to fcn with iflag = 1 or 2 has reached maxfev = 2: ftol is too small. No further reduction in the sum of squares is possible = 3: xtol is too small. No further improvement in the approximate solution x is possible = 4: gtol is too small. fvec is orthogonal to the columns of the Jacobian to machine precision = 5: User imposed termination (returned from fcn with iflag < 0) Reference netlib/minpack