◆ N2g1()

Sub N2g1	(	M As	Long,
		N As	Long,
		X() As	Double,
		F As	LongPtr,
		Fj As	LongPtr,
		Info As	Long,
		Optional NFcall As	Long,
		Optional NFjcall As	Long,
		Optional Niter As	Long
	)

Nonlinear least squares approximation (adaptive algorithm) (simple driver)

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 and Jacobian values.

N2g1 is equivalent to using N2g with setting default parameters. Covariance matrix and regression diagnostic vector are not computed.

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. For same X() value, same Nf will be passed to F and Fj. End Sub
[in]	Fj	The user-supplied subroutine which calculates the Jacobian of the function fi(x1, x2, ..., xn) defined as follows. Sub Fj(M As Long, N As Long, X() As Double, Nf As Long, Fjac() As Double) Calculate the Jacobian (dFi/dXj) from given X() and return in Fjac(i-1,j-1) (i = 1 to M, j = 1 to N). Nf is invocation counter. If given X() is out of bounds, Nf should be set to 0. The other variables should not be altered. For same X() value, same Nf will be passed to F and Fj. End Sub
[out]	Info	= 0: Successful exit. = -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. = 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.
[out]	NFcall	(Optional) Number of function evaluations of F.
[out]	NFjcall	(Optional) Number of function evaluations of Fj.
[out]	Niter	(Optional) Number of iterations.

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.
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 FN2g(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 JN2g(M As Long, N As Long, X() As Double, Nf As Long, Fjac() 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

Fjac(I, 0) = Exp(-Xdata(I) * X(1)) - 1

Fjac(I, 1) = -Xdata(I) * X(0) * Exp(-X(1) * Xdata(I))

Next

End Sub

Sub Ex_N2g1()

Const M = 4, N = 2

Dim X(N - 1) As Double, Info As Long

X(0) = 500: X(1) = 0.0001

Call N2g1(M, N, X(), AddressOf FN2g, AddressOf JN2g, Info)

Debug.Print "C1, C2 =", X(0), X(1)

Debug.Print "Info =", Info

End Sub

Example Results: C1, C2 = 241.084896112856 5.44942234058364E-04

Info = 0