Dfmin Dfmin_r

Dfmin() Sub Dfmin ( A As  Double, B As  Double, F As  LongPtr, Tol As  Double, X As  Double  ) Minimum of a single variable general nonlinear function Purpose This routine finds a minimum of a function f(x) between the given values A and B. The method used is a combination of golden section search and successive parabolic interpolation. Convergence is never much slower than that for a Fibonacci search. If the function f has a continuous second derivative which is positive at the minimum (which is not at A or B), then convergence is superlinear, and usually of the order of about 1.324.... The function f is never evaluated at two points closer together than eps*abs(Dfmin) + (Tol/3), where eps is approximately the square root of the relative machine precision. If f is a unimodal function and the computed values of f are always unimodal when separated by at least eps*abs(xstar) + (Tol/3), then Dfmin approximates the abscissa of the global minimum of f on the interval [A, B] with an error less than 3*eps*abs(Dfmin) + Tol. If f is not unimodal, then Dfmin may approximate a local, but perhaps non-global, minimum to the same accuracy. Parameters [in] A Left endpoint of initial interval. [in] B Right endpoint of initial interval. [in] F User supplied function, which evaluates f(x) defined as follows: Function F(X As Double) As Double F = (computed function value f(x)) End Function X should not be changed. [in] Tol Desired length of the interval of uncertainty of the final result. (Tol >= 0) [out] X Abscissa approximating the point where f(x) attains a minimum on the interval [A, B]. Reference D. Kahaner, C. Moler, S. Nash, "Numerical Methods and Software", Prentice-Hall (1989) Example Program Find the minimum point of the following function in the interval [0, 2]. f(x) = x^3 - 2x - 5 Function FDfmin(X As Double) As Double FDfmin = X ^ 3 - 2 * X - 5 End Function Sub Ex_Dfmin() Dim A As Double, B As Double, Tol As Double, X As Double A = 0: B = 2 Tol = 0.0000000001 '1.0e-10 Call Dfmin(A, B, AddressOf FDfmin, Tol, X) Debug.Print "X =", X End Sub Example Results X = 0.816496587630398