Dfmin Dfmin_r

Dfmin_r() Sub Dfmin_r ( A As  Double, B As  Double, Tol As  Double, XX As  Double, YY As  Double, IRev As  Long  ) Minimum of a single variable general nonlinear function (reverse communication version) 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] Tol Desired length of the interval of uncertainty of the final result. (Tol >= 0) [out] XX IRev = 0: The abscissa approximating the point where f(x) attains a minimum on the interval [A, B]. IRev = 1: The abscissa where the function value shoule be evaluated and given in the next call. [in] YY IRev = 1: The function value f(XX) should be given in YY 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 as follows and call the routine again. = 0: Computation finished. = 1: User should set the function value at XX (f(XX)) in YY. Do not alter any variables other than YY. 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_r() Dim A As Double, B As Double, Tol As Double Dim XX As Double, YY As Double, IRev As Long A = 0: B = 2 Tol = 0.0000000001 '1.0e-10 IRev = 0 Do Call Dfmin_r(A, B, Tol, XX, YY, IRev) If IRev <> 0 Then YY = FDfmin(XX) Loop While IRev <> 0 Debug.Print "X =", XX End Sub Example Results X = 0.816496587630398