Dehint Dehint_r Deoint Deoint_r Qagi Qagi_r Qawf Qawf_r Qk15i Qk15i_r

Qk15i_r() Sub Qk15i_r ( A As  Double, Inf As  Long, Result As  Double, XX As  Double, YY As  Double, IRev As  Long, Optional AbsErr As  Double  ) Semi-infinite/infinite interval quadrature (15-point Gauss-Kronrod rule) (reverse communication version) Purpose The routine computes an approximation to the integral of function f(x) over a semi-infinite or infinite interval with the 15 point Gauss-Kronrod rule. The integrand f(x) is computed and provided by the user in accordance with IRev. The function f(x) is transformed to the function f01(t) so that the semi-infinite integration is obtained by computing the finite integration over [0, 1]. ∫ f(x)dx [a, +∞] = ∫ f01(t)dt [0, 1] where f01(t) = f(a + (1 - t)/t)/t^2 The infinite integral is computed as the sum of two semi-infinite integrals. ∫ f(x)dx [-∞, +∞] = ∫ (f(x) + f(-x)) dx [0, +∞] Parameters [in] A Lower or upper bound of semi-infinite integration interval a. (Not referenced if infinite interval (Inf = 2)) [in] Inf The kind of integration range. = 1: Semi-infinite integral [a, +∞] = -1: Semi-infinite integral [-∞, a] = 2: Infinite integral [-∞, +∞] (If other value is specified, Inf = 2 is assumed) [out] Result Approximation to the integral. [out] XX When returned with IRev = 1, XX contains the abscissa where the function value should be evaluated and given in the next call. [in] YY When returned with 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, complete the following tasks and call this routine again without changing IRev. = 0: Computation finished. See return code in Info. = 1: User should set the function values at XX in YY. Do not alter any variables other than YY. [out] AbsErr (Optional) Estimate of the modulus of the absolute error, which should equal or exceed the true error. Reference SLATEC (QUADPACK) Example Program Compute the following integral. ∫ 1/(1 + x^2) dx [0, +∞] (= π/2) Sub Ex_Qk15i_r() Dim A As Double, Inf As Long, Result As Double Dim XX As Double, YY As Double, IRev As Long A = 0 Inf = 1 IRev = 0 Do Call Qk15i_r(A, Inf, Result, XX, YY, IRev) If IRev = 1 Then YY = 1 / (1 + XX ^ 2) Loop While IRev <> 0 Debug.Print "S =", Result, "S(true) =", Dconst(13) / 2 End Sub Example Results S = 1.57079632684678 S(true) = 1.5707963267949