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

Qk31() Sub Qk31 ( F As  LongPtr, A As  Double, B As  Double, Result As  Double, Optional AbsErr As  Double, Optional ResAbs As  Double, Optional ResAsc As  Double  ) Finite interval quadrature (31 point Gauss Kronrod formula) 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 defined by a user supplied subroutine. The integral will be evaluated with a 31 point Gauss Kronrod formula. Parameters [in] F The user supplied subroutine which calculates the integrand function f(x) defined as follows. Function F(X As Double) As Double F = f(X) End Function X should not be changed. [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] 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)) Function F1(X As Double) As Double F1 = 1 / (1 + X ^ 2) End Function Sub Ex_Qk31() Dim A As Double, B As Double, Result As Double A = 0: B = 4 Call Qk31(AddressOf F1, A, B, Result) Debug.Print "S =", Result, "S(true) =", Atn(4) End Sub Example Results S = 1.32581766366803 S(true) = 1.32581766366803