Qk15 Qk15_r Qk21 Qk21_r Qk31 Qk31_r Qk41 Qk41_r Qk51 Qk51_r Qk61 Qk61_r

Qk21_r() Sub Qk21_r ( A As  Double, B As  Double, Result As  Double, XX As  Double, YY As  Double, IRev As  Long, Optional AbsErr As  Double, Optional ResAbs As  Double, Optional ResAsc As  Double  ) Finite interval quadrature (21 point Gauss-Kronrod rule) (reverse communication version) Purpose This routine routine computes   I = integral of f over [a, b] with error estimate, and   J = integral of abs(f) over [a, b], where f is a given function. User should provide the necessary computed values of f according to the argument IRev. The integral will be evaluated with the 21 point Gauss-Kronrod rule. Parameters [in] A Lower limit of integration a. [in] B Upper limit of integration b. [out] Result Approximation to I = integral of f over [a, b]. [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 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. [out] ResAbs (Optional) Approximation to J = integral of abs(f) over [a, b]. [out] ResAsc (Optional) Approximation to the integral of abs(f - I/(b - a)) over [a, b]. Reference SLATEC (QUADPACK) Example Program Compute the following integral. ∫ 1/(1 + x^2) dx [0, 4] (= atan(4)) Sub Ex_Qk21_r() Dim A As Double, B As Double, Result As Double Dim XX As Double, YY As Double, IRev As Long A = 0: B = 4 IRev = 0 Do Call Qk21_r(A, B, Result, XX, YY, IRev) If IRev = 1 Then YY = 1 / (1 + XX ^ 2) Loop While IRev <> 0 Debug.Print "S =", Result, "S(true) =", Atn(4) End Sub Example Results S = 1.3258176636671 S(true) = 1.32581766366803