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

N2f() Sub N2f ( M As  Long, N As  Long, X() As  Double, F As  LongPtr, Cov() As  Double, Rd() As  Double, Info As  Long, Optional Itsum As  LongPtr = NullPtr, Optional Info2 As  Long, Optional NFcall As  Long, Optional NFjcall As  Long, Optional Niter As  Long, Optional S As  Double, Optional NFcov As  Long, Optional NFjcov As  Long, Optional Rtol As  Double = -1, Optional Atol As  Double = -1, Optional Rdreq As  Long = -1, Optional Covreq As  Long = 0, Optional MaxFcall As  Long = -1, Optional MaxIter As  Long = -1, Optional Dtype As  Long = -1, Optional Dfac As  Double = -1, Optional Dtol As  Double = -1, Optional D0 As  Double = -1, Optional Tuner1 As  Double = -1, Optional Xctol As  Double = -1, Optional Xftol As  Double = -1, Optional Lmax0 As  Double = -1, Optional Lmaxs As  Double = -1, Optional Sctol As  Double = -1, Optional Dltfdc As  Double = -1, Optional Dltfdj As  Double = -1  ) Nonlinear least squares approximation (adaptive algorithm) (Jacobian not required) Purpose This routine minimizes the sum of the squares of M nonlinear functions in N variables by the adaptive algorithm which combines and augments a Gauss-Newton, Levenberg- Marquardt and other techniques for better convergence. minimize the sum of fi(x1, x2, ..., xn)^2 (sum for i = 1 to M) The user must provide a subroutine which calculates the function values. Since the Jacobian is calculated by finite difference approximation within the routine, the user is not required to calculate the Jacobian. Parameters [in] M Number of data. (M > 0) [in] N Number of parameters. (0 < N <= M) [in,out] X() Array X(LX - 1) (LX >= N) [in] X must contain an initial estimate of the solution vector. [out] X contains the final estimate of the solution vector. [in] F The user-supplied subroutine which calculates the function fi(x1, x2, ..., xn) defined as follows. Sub F(M As Long, N As Long, X() As Double, Nf As Long, Fvec() As Double) Calculate the Fi value from given X() and return in Fvec(i-1) (i = 1 to M). Nf is invocation counter. If given X() is out of bounds, Nf should be set to 0. The other variables should not be altered. End Sub [out] Cov() Array Cov(LCov - 1) (LCov >= N(N + 1)/2) Covariance matrix. (Lower triangular part stored in column major order) (Computed only when Rdreq = 1 or 3 and normally terminated) [out] Rd() Array Rd(LRd - 1) (LRd >= M) Regression diagnostic vector. (Rd(i) is an index of the change that would occur in the residual sum of squares if the i-th data is deleted) (Computed only when Rdreq = 2 or 3 and normally terminated) [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 < 1) = -3: The argument X() is invalid. = -5: The argument Cov() is invalid. = -6: The argument Rd() is invalid. = 7: Singular convergence. (Hessian near the current iterate appears to be singular) = 8: False convergence. (Iterate appears to be converging to a noncritical point. Tolerances may be too small) = 9: Function evaluation limit reached. = 10: Iteration limit reached. = 63: F(X) cannot be computed at the initial X. = 65: The gradient could not be computed at X. [in] Itsum (Optional) The user supplied subroutine to print the intermediate results defined as follows. (default = NUllPtr) If the address is supplied (if Itsum <> NullPtr), the subroutine is called after every iteration. Sub Itsum(N As Long, X() As Double, NIter As Long, Nf As Long, Nfj As Long, S As Double) Output the following information in desired format. N: Number of variables. X(): Current approximation of the solution vector. NIter: Iteration counter. Nf: Number of function calls of F excluding those for computing Jacobian. Nfj: Number of function calls of F for computing Jacobian. S: Residual sum of squares at X(). End Sub Argument values should not be altered. [out] Info2 (Optional) Sub-code for Info = 0. = 1: X convergence. = 2: Relative function convergence. = 3: Both X and relative function convergence. = 4: Absolute function convergence. [out] NFcall (Optional) Number of function evaluations of F excluding those used in computing the Jacobian matrix and including those used in computing the covariance matrix. [out] NFjcall (Optional) Number of function evaluations of F used in computing the Jacobian matrix including those used in computing the covariance matrix. [out] Niter (Optional) Number of iterations. [out] S (Optional) Residual sum of squares at the obtained solution vector X(). [out] NFcov (Optional) Number of function evaluations of F to compute the covariance matrix (excluding those for computing Jacobian). [out] NFjcov (Optional) Number of function evaluations of F for computing Jacobian to compute the covariance matrix. [in] Rtol (Optional) Relative function convergence tolerance. (Eps <= Rtol <= 0.1) (default = 1e-10) (Eps shows the machine epsilon hereafter) (If Rtol < Eps or Rtol > 0.1, the default value will be used) [in] Atol (Optional) Absolute function convergence tolerance. (default = 1e-20) (If Atol < 0, the default value will be used) [in] Rdreq (Optional) Whether to compute a covariance matrix or regression diagnostic vector. (default = 3) = 0: Compute neither. = 1: Compute covariance matrix only. = 2: Compute regression diagnostic vector only. = 3: Compute both covariance matrix and regression diagnostic vector. (For other values, the default value will be used) [in] Covreq (Optional) Which covariance matrix of the form is to be computed. (default = -1) = -1: sigma * H^(-1) * (J^T * J) * H^(-1) = -2: sigma * H^(-1) = -3: sigma * (J^T * J) where sigma = S / max(1, M-N) (S is residual sum of squares), H is the Hessian matrix and J is the Jacobian matrix. (If Covreq = 1, 2 or 3, then Covreq = -1, -2 or -3 respectively are assumed. For other values, the default value will be used) [in] MaxFcall (Optional) Maximum number of function evaluations of F. (default = 200) (If MaxFcall <= 0, the default value will be used) [in] MaxIter (Optional) Maximum number of iterations. (default = 150) (If MaxIter <= 0, the default value will be used) [in] Dtype (Optional) Choice of adaptive scaling. (Dtype = 0, 1 or 2) (default = 1) = 0: Disable adaptive scaling. (scale factor = 1) = 1: Enable adaptive scaling during all iterations. = 2: Enable adaptive scaling during the first iteration and scale factor is left unchanged thereafter. (For other values, the default value will be used) [in] Dfac (Optional) Factor for adaptive scaling (0 <= Dfac <= 1) (default = 0.6) A scale factor D(i) is chosen by adaptive scaling so that D(i)*X(i) has about the same magnitude for all i. Let D1(i) = max(||Ji||, Dfac*D(i)) where ||Ji|| is the 2-norm of the i-th column of Jacobian matrix, then D(i) is chosen as follows.   if D1(i) >= Dtol: D(i) = D1(i)   if D1(i) < Dtol: D(i) = D0 (If Dfac < 0 or Dfac > 1, the default value will be used) [in] Dtol (Optional) Tolerance for adaptive scaling. (Dtol > 0) (default = 1.0e-6) (If Dtol <= 0, the default value will be used) [in] D0 (Optional) Initial value for adaptive scaling. (D0 > 0) (default = 1) (If D0 <= 0, the default value will be used) [in] Tuner1 (Optional) Parameter to check for false convergence. (0 <= Tuner1 <= 0.5) (default = 0.1) (If Tuner1 < 0 or Tuner1 > 0.5, the default value will be used) [in] Xctol (Optional) X convergence tolerance. (0 <= Xctol <= 1) (default = Eps^(1/2)) (If Xctol < 0 or Xctol > 1, the default value will be used) [in] Xftol (Optional) False convergence tolerance. (0 <= Xftol <= 1) (default = 100*Eps) (If Xftol < 0 or Xftol > 1, the default value will be used) [in] Lmax0 (Optional) Maximum 2-norm allowed for scaled very first step. (Lmax0 > 0) (default = 1) (If Lmax0 <= 0, the default value will be used) [in] Lmaxs (Optional) [in] Sctol (Optional) Lmaxs and Sctol are the singular convergence test parameters. (Lmaxs > 0, 0 <= Sctol <= 1) (default: Lmaxs = 1, Sctol = 1e-10) If the function reduction predicted for a step of length bounded by Lmaxs is less than Sctol*abs(f), returns with Info = 7 (f is the function value at the start of the current iteration). (If Lmaxs <= 0, the default value will be used) (If Sctol < 0 or Sctol > 1, the default value will be used) [in] Dltfdc (Optional) Step size used to compute a covariance matrix is chosen as follows (in the case that Covreq = -1 or -2). (Eps <= Dltfdc <= 1) (default = Eps^(1/3))   Dltfdc*max(|X(I)|, 1/D(I)) (If Dltfdc < Eps or Dltfdc > 1, the default value will be used) [in] Dltfdj (Optional) Step size used to compute finite difference Jacobian approximation is chosen as follows. (Eps <= Dltfdj <= 1) (default = Sqrt(Eps))   Dltfdj*max(|X(I)|, 1/D(I)) (If Dltfdj < Eps or Dltfdj > 1, the default value will be used) Reference netlib/port Example Program Approximate the following data with model function f(x) = c1*(1 - exp(-c2*x)). Determine two parameters c1 and c2 by the nonlinear least squares method, and compute the covariance matrix. 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 FN2f(M As Long, N As Long, X() As Double, Nf As Long, Fvec() As Double) 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 For I = 0 To M - 1 Fvec(I) = Ydata(I) - X(0) * (1 - Exp(-Xdata(I) * X(1))) Next End Sub Sub Ex_N2f() Const M = 4, N = 2 Dim X(N - 1) As Double, Cov(N * (N + 1) / 2 - 1) As Double, Rd(M - 1) As Double Dim Info As Long X(0) = 500: X(1) = 0.0001 Call N2f(M, N, X(), AddressOf FN2f, Cov(), Rd(), Info) Debug.Print "C1, C2 =", X(0), X(1) Debug.Print "Cov =" Debug.Print Cov(0), Cov(1) Debug.Print Cov(1), Cov(2) Debug.Print "Info =", Info End Sub Example Results C1, C2 = 241.084897030993 5.44942231587108E-04 Cov = 20.9247050299849 -5.61462410710359E-05 -5.61462410710359E-05 1.51120280693799E-10 Info = 0