Qag Qag_r Qagi Qagi_r Qk15 Qk15_r

Qagi() Sub Qagi ( F As  LongPtr, Bound As  Double, Inf As  Long, Result As  Double, Info As  Long, Optional AbsErr As  Double, Optional Neval As  Long, Optional EpsAbs As  Double = -1, Optional EpsRel As  Double = -1, Optional Limit As  Long = -1, Optional Last As  Long  ) Infinite interval automatic quadrature (adaptive automatic quadrature) Purpose This routine computes I = integral of f(x) over [Bound, +inf], [-inf, Bound] or [-inf, +inf], satisfying the requested accuracy, where f(x) is a given function defined by a user supplied subroutine. 15-point Gauss-Kronrod rule is used, and the integration interval will be adaptively subdivided to satisfy the requested accuracy. The semi-infinite integration range is mapped onto the interval [0, 1], and then the integration rule is applied to compute the required integral. ∫ f(x)dx [Bound, +∞] = ∫ f(Bound + (1 - t)/t) / t^2 dt [0, 1] The infinite integral will be computed as the sum of two semi-infinite integrals. ∫ f(x)dx [-∞, +∞] = ∫ (f(x) + f(-x)) dx [0, +∞] 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] Bound The finite bound of original integration range. (Not referenced if interval is doubly infinite (Inf = 2)) [in] Inf The kind of integration range. = 1: Semi-infinite integral [Bound, +∞] = -1: Semi-infinite integral [-∞, Bound] = 2: Infinite integral [-∞, +∞] (If other value is specified, Inf = 2 is assumed) [out] Result Approximation to the integral. [out] Info = 0: Successful exit. = 1: Maximum number of subdivisions allowed has been achieved. = 2: The occurrence of roundoff error is detected, which prevents the requested tolerance from being achieved. = 3: Extremely bad integrand behavior occurs at some points of the integration interval. = 4: The algorithm does not converge. It is presumed that the requested tolerance cannot be achieved, and that the returned result is the best which can be obtained. = 5: The integral is probably divergent, or slowly convergent. [out] AbsErr (Optional) Estimate of the modulus of the absolute error, which should equal or exceed the true error. [out] Neval (Optional) Number of integrand evaluations. [in] EpsAbs (Optional) Absolute accuracy requested. (default = 0) The requested accuracy is assumed to be satisfied if AbsErr <= max(EpsAbs, EpsRel*|Result|)) (If EpsAbs < 0, the default value will be used) [in] EpsRel (Optional) Relative accuracy requested. (default = 1.0e-12) The requested accuracy is assumed to be satisfied if AbsErr <= max(EpsAbs, EpsRel*|Result|)) If EpsAbs <= 0 and EpsRel < 50*eps, EpsRel is assumed to be 50*eps, where eps is the machine precision. (If EpsRel < 0, the default value will be used) [in] Limit (Optional) Maximum number of subintervals in the partition of the given integration interval (limit >= 1) (default = 100) (If Limit < 1, the default value will be used) [out] Last (Optional) Number of subintervals produced in the subdivision process. Reference SLATEC (QUADPACK) Example Program Compute the following integral. ∫ 1/(1 + x^2) dx [0, +∞] (= π/2) Function F1(X As Double) As Double F1 = 1 / (1 + X ^ 2) End Function Sub Ex_Qagi() Dim Bound As Double, Inf As Long, Result As Double, Info As Long Bound = 0: Inf = 1 Call Qagi(AddressOf F1, Bound, Inf, Result, Info) Debug.Print "S =", Result, "S(true) =", Dconst(13) / 2 Debug.Print "Info =", Info End Sub Example Results S = 1.5707963267949 S(true) = 1.5707963267949 Info = 0