◆ lmder_r()

void lmder_r	(	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,
		double	xx[],
		double	yy[],
		int *	irev
	)

Nonlinear least squares approximation by Levenberg-Marquardt method (reverse communication version)

Purpose: lmder_r 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 the calculated function values and the Jacobian according to irev.

Parameters

[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] irev = 0: The obtained solution vector. irev = 30: The abscissa where the Jacobian shoule be evaluated. irev = 50, 51: Recent approximation of the solution vector.
[out]	fvec[]	Array fvec[lfvec] (lfvec >= m) irev = 0: The function values evaluated at the solution vector x[]. irev = 50, 51: The function values evaluated at the recent approximation of the solution vector.
[in]	ldfjac	Leading dimension of two dimensional array fjac[]. (ldfjac >= m)
[in,out]	fjac[][]	Array fjac[lfjac][ldfjac] (lfjac >= n) [in] irev = 30: Jacobian(dfi/dxj) at x[] should be given in fjac[][] in the next call. [out] irev = 0: 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 function evaluations with irev = 1 to 3 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: Returns with irev = 50 or 51 at the beginning of the first iteration and every nprint iterations thereafter and at the end of last iteration for printing intermediate result. <= 0: Does not return for printing intermediate result.
[out]	nfev	Number of function evaluations with irev = 1 or 2.
[out]	njev	Number of Jacobian evaluations with irev = 30.
[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 >= 4*n + m)
[out]	info	= 0: Successful exit (sub-code is set to work[0]) = -1: The argument m had an illegal value (m < n) = -2: The argument n had an illegal value (n <= 0) = -5: The argument ldfjac had an illegal value (ldfjac < m) = -7: The argument ftol had an illegal value (ftol < 0) = -8: The argument xtol had an illegal value (xtol < 0) = -9: The argument gtol had an illegal value (gtol < 0) = -10: The argument maxfev had an illegal value (maxfev <= 0) = -11: The argument diag had an illegal value (diag[i] <= 0 when mode = 2) = -13: The argument factor had an illegal value (factor <= 0) = -19: The argument lwork had an illegal value (lwork too small) = 1: Number of function evaluations with irev=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
[out]	xx[]	Array xx[lxx] (lxx >= n) When returned with irev = 1 or 2, xx[] contains the abscissa where the function value should be evaluated and given in the next call.
[in]	yy[]	Array yy[lyy] (lyy >= m) When returned with irev = 1 or 2, the function value fi(xx[]) (i = 1 to m) should be given in yy[] in the next call.
[in,out]	irev	Control variable for reverse communication. [in] Before first call, irev should be initialized to zero. On succeeding calls, irev should not be altered. [out] If irev is not zero, complete the following process and call this routine again. = 0: Computation finished. See return code in info. = 1 or 2: User should set the function values at xx[] in yy[]. Do not alter any variables other than yy[]. = 30: User should set the Jacobian at x[] in fjac[][]. Do not alter any variables other than fjac[][]. = 50 or 51: Display the intermediate result (x[], fvec[], etc.) (in the case of nprint > 0). Do not alter any variables.

Reference: netlib/minpack