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

Lmstr1_r() Sub Lmstr1_r ( M As  Long, N As  Long, X() As  Double, Fvec() As  Double, Fjac() As  Double, Tol As  Double, Ipvt() As  Long, Info As  Long, XX() As  Double, YY() As  Double, YYpr() As  Double, IRev As  Long, Optional Info2 As  Long  ) Nonlinear least squares approximation by Levenberg-Marquardt method (limited storage) (simple driver) (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 (row-by-row) according to irev. Lmstr1_r is equivalent to using Lmstr_r with setting FTol = Tol, XTol = Tol, GTol = 0, Maxfev = 100*(n+1), Mode = 1, Factor = 100 and Nprint = 0. 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 >= 10: The abscissa where the Jacobian shoule be evaluated. [out] Fvec() Array Fvec(LFvec - 1) (LFvec >= M) IRev = 0: The function values evaluated at the solution vector X(). [out] Fjac() Array Fjac(LFjac1 - 1, LFjac2 - 1) (LFjac1 >= M, LFjac2 >= N) 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] Tol Relative error desired in the sum of squares and the approximate solution. (Tol >= 0) [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 Tol had an illegal value. (Tol < 0) = -7: The argument Ipiv() is invalid. (Array Ipiv() is not big enough) = -9: The argument XX() is invalid. (Array XX() is not big enough) = -10: The argument YY() is invalid. (Array YY() is not big enough) = -11: The argument YYpr() is invalid. (Array YYpr() 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] YYpr() Array YYpr(LYYpr - 1) (LYYpr >= N) When returned with IRev >= 10, the (IRev-10+1)-th row of Jacobian (dfi/dxj) (i = IRev-10+1, j = 1 to N) should be given in YYpr() 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(). >= 10: User should set the (IRev-10+1)-th row of Jacobian at X() in YYpr(). Do not alter any variables other than YYpr(). [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. 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 FLmstr(M As Long, N As Long, X() As Double, Fvec() As Double, Fjrow() 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 And IFlag <= M + 1 Then Fjrow(0) = Exp(-Xdata(IFlag - 2) * X(1)) - 1 Fjrow(1) = -Xdata(IFlag - 2) * X(0) * Exp(-X(1) * Xdata(IFlag - 2)) End If End Sub Sub Ex_Lmstr1_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 Tol As Double, Ipvt(N - 1) As Long, Info As Long Dim XX(N - 1) As Double, YY(M - 1) As Double, YYpr(N - 1) As Double Dim IRev As Long, IFlag As Long Tol = 0.00000001 '1.0e-8 X(0) = 500: X(1) = 0.0001 IRev = 0 Do Call Lmstr1_r(M, N, X(), Fvec(), Fjac(), Tol, Ipvt(), Info, XX(), YY(), YYpr(), IRev) If IRev = 1 Then IFlag = 1 Call FLmstr(M, N, XX(), YY(), YYpr(), IFlag) ElseIf IRev >= 10 Then IFlag = IRev - 10 + 2 Call FLmstr(M, N, XX(), YY(), YYpr(), 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