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

Covar() Sub Covar ( N As  Long, R() As  Double, Ipvt() As  Long, Tol As  Double, Info As  Long  ) Covariance matrix of nonlinear least squares approximation Purpose This routine computes the covariance matrix of nonlinear least squares approximation computed by Lmder, Lmder1, Lmdif, Lmstr or Lmstr1. Given an M x N matrix A, the N x N symmetric matrix corresponding to A, called as the covariance matrix C, is defined as follows. C = (A^T*A)^(-1) This routine completes this equation using the necessary information from the QR factorization of A which is provided by Lmder, Lmder1, Lmdif, Lmstr or Lmstr1. Parameters [in] N Order of covariance matrix. (N > 0) [in,out] R() Array R(LR1 - 1, LR2 - 1) (LR1 >= N, LR2 >= N) [in] The upper triangle matrix R obtained from QR factorization AP = QR (P is the permutation matrix given by Ipvt(), and Q is M x N orthogonal matrix) [out] The square symmetric covariance matrix. [in] Ipvt() Array Ipvt(LIpvt - 1) (LIpvt >= N) The pivot indices that define the permutation matrix P. [in] Tol The tolerance to determine the numerical rank of the matrix A. (Tol > 0) If some diagonal element of R is less than (the maximum absolute value of diagonal elements of R)*Tol in magnitude, it will be regarded as rank deficient. [out] Info = 0: Successful exit. = -1: The argument N had an illegal value. (N <= 0) = -2: The argument R() is invalid. (Array R() is not big enough) = -3: The argument Ipvt() is invalid. (Array Ipvt() is not big enough) = -4: The argument Tol had an illegal value. (Tol <= 0) Reference netlib/minpack Example Program Approximate the following data with model function f(x) = c1*(1 - exp(-c2*x)). Determine two parameters c1 and c2 by Lmder1, and compute the covariance matrix by Covar. f(x) x 10.07 77.6 29.61 239.9 50.76 434.8 81.78 760.0 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_Covar() Const M = 4, N = 2 Dim X(N - 1) As Double, Fvec(M - 1) As Double, R(M - 1, N - 1) As Double Dim Tol As Double, Ipvt(N - 1) As Long, Info As Long Dim S As Double, I As Long Tol = 0.00000001 '1.0e-8 X(0) = 500: X(1) = 0.0001 Call Lmder1(AddressOf FLmder, M, N, X(), Fvec(), R(), Tol, Ipvt(), Info) Debug.Print "Info =", Info Call Covar(N, R(), Ipvt(), Tol, Info) For I = 0 To M - 1 S = S + Fvec(I) ^ 2 Next S = S / (M - N) Debug.Print "Cov =" Debug.Print R(0, 0) * S, R(0, 1) * S Debug.Print R(1, 0) * S, R(1, 1) * S Debug.Print "Info =", Info End Sub Example Results Info = 0 Cov = 20.5868644246757 -5.52380489748924E-05 -5.52380489748924E-05 1.48678858325168E-10 Info = 0