Covar Lmder Lmder1 Lmder1_r Lmder_r Lmdif Lmdif1 Lmdif1_r Lmdif_r Lmstr Lmstr1 Lmstr1_r Lmstr_r N2f N2f1 N2f1_r N2f_r N2g N2g1 N2g1_r N2g_r N2p N2p_r

Lmder_r() Sub Lmder_r ( M As  Long, N As  Long, X() As  Double, Fvec() As  Double, Fjac() As  Double, FTol As  Double, XTol As  Double, GTol As  Double, Diag() As  Double, Mode As  Long, Ipvt() As  Long, Info As  Long, XX() As  Double, YY() As  Double, IRev As  Long, Optional Info2 As  Long, Optional Maxfev As  Long = 0, Optional Factor As  Double = 0, Optional Nprint As  Long = 0, Optional Nfev As  Long, Optional Njev As  Long  ) Nonlinear least squares approximation by Levenberg-Marquardt method (reverse communication version) Purpose This routine 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 - 1) (LX >= N) [in] An initial estimate of the solution vector. [out] IRev = 0: The obtained solution vector.   IRev = 2: The abscissa where the Jacobian shoule be evaluated.   IRev = 3: Recent approximation of the solution vector. [out] Fvec() Array Fvec(LFvec - 1) (LFvec >= M) IRev = 0: The function values evaluated at the solution vector X(). IRev = 3: The function values evaluated at the recent approximation of the solution vector. [in,out] Fjac() Array Fjac(LFjac1 - 1, LFjac2 - 1) (LFjac1 >= M, LFjac2 >= N) [in] IRev = 2: 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,out] Diag() Array Diag(LD - 1) (LD >= 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.) [out] Ipvt() Array Ipvt(LIpvt - 1) (LIpvt >= N) The pivot indices that define the permutation matrix P. [out] Info = 0: Successful exit. (Sub-code is set to Info2) = -1: The argument M had an illegal value. (M < N) = -2: The argument N had an illegal value. (N <= 0) = -3: The argument X() is invalid. (Array X() is not big enough) = -4: The argument Fvec() is invalid. (Array Fvec() is not big enough) = -5: The argument Fjac() is invalid. (Array Fjac() is not big enough) = -6: The argument FTol had an illegal value. (FTol < 0) = -7: The argument XTol had an illegal value. (XTol < 0) = -8: The argument GTol had an illegal value. (GTol < 0) = -9: The argument Diag() is invalid. (Array Diag() is not big enough) or had an illegal value (Diag(i) < 0 when Mode = 2) = -11: The argument Ipiv() is invalid. (Array Ipiv() is not big enough) = -13: The argument XX() is invalid. (Array XX() is not big enough) = -14: The argument YY() is invalid. (Array YY() is not big enough) = 1: Number of function evaluations with IRev = 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. [out] XX() Array XX(LXX - 1) (LXX >= N) When returned with IRev = 1, XX() contains the abscissa where the function value should be evaluated and given in the next call. [in] YY() Array YY(LYY - 1) (LYY >= M) When returned with IRev = 1, 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, set the required values to the specified variables or display the intermediate result as follows and call the routine again. = 0: Computation finished. See return code in Info. = 1: User should set the function values at XX() in YY(). Do not alter any variables other than YY(). = 2: User should set the Jacobian at X() in Fjac(). Do not alter any variables other than Fjac(). = 3: Display the intermediate result (X(), Fvec(), etc.) (in the case of Nprint > 0). Do not alter any variables. [out] Info2 (Optional) Sub-code on return with Info = 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] Maxfev (Optional) Termination occurs when the number of function evaluations (IRev = 1) has reached this value. (default = 100*(N + 1)) (if Maxfev <= 0, the default value will be used.) [in] Factor (Optional) 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. (default = 100) (if Factor <= 0, the default value will be used) [in] Nprint (Optional) > 0: Returns with IRev = 3 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. (default = 0) [out] Nfev (Optional) Number of function evaluations with IRev = 1. [out] Njev (Optional) Number of Jacobian evaluations with IRev = 2. Reference netlib/minpack Example Program Approximate the following data with model function f(x) = c1*(1 - exp(-c2*x)). Two parameters c1 and c2 are determined by the nonlinear least squares method. f(x) x 10.07 77.6 29.61 239.9 50.76 434.8 81.78 760.0 The initial estimates of the solution are c1 = 500 and c2 = 0.0001. Sub FLmder(M As Long, N As Long, X() As Double, Fvec() As Double, Fjac() As Double, IFlag As Long) Dim Xdata(3) As Double, Ydata(3) As Double, I As Long Ydata(0) = 10.07: Xdata(0) = 77.6 Ydata(1) = 29.61: Xdata(1) = 239.9 Ydata(2) = 50.76: Xdata(2) = 434.8 Ydata(3) = 81.78: Xdata(3) = 760 If IFlag = 1 Then For I = 0 To M - 1 Fvec(I) = Ydata(I) - X(0) * (1 - Exp(-Xdata(I) * X(1))) Next ElseIf IFlag = 2 Then For I = 0 To M - 1 Fjac(I, 0) = Exp(-Xdata(I) * X(1)) - 1 Fjac(I, 1) = -Xdata(I) * X(0) * Exp(-X(1) * Xdata(I)) Next End If End Sub Sub Ex_Lmder_r() Const M = 4, N = 2 Dim X(N - 1) As Double, Fvec(M - 1) As Double, Fjac(M - 1, N - 1) As Double Dim FTol As Double, XTol As Double, GTol As Double, Diag(N - 1) As Double Dim Mode As Long, Ipvt(N - 1) As Long, Info As Long Dim XX(N - 1) As Double, YY(M - 1) As Double, IRev As Long, IFlag As Long FTol = 0.00000001: XTol = 0.00000001: GTol = 0.00000001 '1.0e-8 X(0) = 500: X(1) = 0.0001 Mode = 1 IRev = 0 Do Call Lmder_r(M, N, X(), Fvec(), Fjac(), FTol, XTol, GTol, Diag(), Mode, Ipvt(), Info, XX(), YY(), IRev) If IRev = 1 Or IRev = 2 Then IFlag = IRev Call FLmder(M, N, XX(), YY(), Fjac(), IFlag) End If Loop While IRev <> 0 Debug.Print "C1, C2 =", X(0), X(1) Debug.Print "Info =", Info End Sub Example Results C1, C2 = 241.084896089973 5.44942234119956E-04 Info = 0