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

Lmdif1() Sub Lmdif1 ( F As  LongPtr, M As  Long, N As  Long, X() As  Double, Fvec() As  Double, Tol As  Double, Info As  Long, Optional Info2 As  Long  ) Nonlinear least squares approximation by Levenberg-Marquardt method (Jacobian not required) (simple driver) 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 a subroutine which calculates the functions. Since the Jacobian is calculated by a forward-difference approximation within the routine, the user is not required to provide the Jacobian. Lmdif1 is equivalent to using Lmdif with setting FTol = Tol, XTol = Tol, GTol = 0 and Mode = 1. Parameters [in] F User supplied subroutine which calculates the functions fi(x) defined as follows. Sub F(M As Long, N As Long, X() As Double, Fvec() As Double, IFlag As Long) If IFlag = 1 or 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. End Sub 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 - 1) (LX >= N) [in] An initial estimate of the solution vector. [out] The obtained solution vector. [out] Fvec() Array Fvec(LFvec - 1) (LFvec >= M) The function values evaluated at the solution vector X(). [in] Tol Relative error desired in the sum of squares and the approximate solution. (tol >= 0) [out] Info = 0: Successful exit. (Sub-code is set to Info2) = -2: The argument M had an illegal value. (M < N) = -3: The argument N had an illegal value. (N <= 0) = -4: The argument X() is invalid. (Array X() is not big enough) = -5: The argument Fvec() is invalid. (Array Fvec() is not big enough) = -6: The argument Tol had an illegal value. (Tol < 0) = 1: Number of calls to F with IFlag = 1 or 2 has reached Maxfev. = 2: Tol is too small. No further reduction in the sum of squares is possible. = 3: Tol is too small. No further improvement in the approximate solution X is possible. = 5: User imposed termination (returned from F with IFlag < 0). [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 Tol. = 2: Relative error between two consecutive iterates is at most Tol. = 3: Both of above are satisfied. = 4: Fvec is orthogonal to the columns of the Jacobian to machine precision. 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 FLmdif(M As Long, N As Long, X() As Double, Fvec() 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 Or IFlag = 2 Then For I = 0 To M - 1 Fvec(I) = Ydata(I) - X(0) * (1 - Exp(-Xdata(I) * X(1))) Next End If End Sub Sub Ex_Lmdif1() Const M As Long = 4, N As Long = 2 Dim X(N - 1) As Double, Fvec(M - 1) As Double, Tol As Double, Info As Long Tol = 0.00000001 '1.0e-8 X(0) = 500: X(1) = 0.0001 Call Lmdif1(AddressOf FLmdif, M, N, X(), Fvec(), Tol, Info) Debug.Print "C1, C2 =", X(0), X(1) Debug.Print "Info =", Info End Sub Example Results C1, C2 = 241.084897132837 5.44942231312974E-04 Info = 0