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

lmder() void lmder ( void(*)(int, int, double *, double *, int, double *, int *)  fcn, int  m, int  n, double  x[], double  fvec[], int  ldfjac, double  fjac[], double  ftol, double  xtol, double  gtol, int  maxfev, double  diag[], int  mode, double  factor, int  nprint, int *  nfev, int *  njev, int  ipvt[], double  work[], int  lwork, int *  info  ) Nonlinear least squares approximation by Levenberg-Marquardt method Purpose lmder 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 and the Jacobian. Parameters [in] fcn User supplied subroutine which calculates the functions fi(x) and the Jacobian defined as follows. void fcn(int m, int n, double x[], double fvec[], int ldfjac, double fjac[][ldfjac], int *iflag) { If iflag = 1: 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 = 2: Calculate the Jacobian matrix (dfi/dxj) and return in fjac[] (i = 1 to m, j = 1 to n). 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 variables 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,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). [out] njev Number of Jacobian evaluations (number of calls to fcn with iflag = 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 <= 0) = -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) = -12: The argument diag had an illegal value (diag[i] <= 0 when mode = 2) = -14: 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 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