Dehint Dehint_r Deoint Deoint_r Qagi Qagi_r Qawf Qawf_r Qk15i Qk15i_r

Qagi_r() Sub Qagi_r ( Bound As  Double, Inf As  Long, Result As  Double, Info As  Long, XX As  Double, YY As  Double, IRev 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) (reverse communication version) Purpose This routine computes I = integral of f(x) over [Bound, +inf], [-inf, Bound] or [-inf, +inf], satisfying the requested accuracy. The integrand function f(x) is computed and provided by the user in accordance with IRev. 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] 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. = -7: The argument IRev had an illegal value. = 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] 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. [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) Sub Ex_Qagi_r() Dim Bound As Double, Inf As Long, Result As Double, Info As Long Dim XX As Double, YY As Double, IRev As Long Bound = 0: Inf = 1 IRev = 0 Do Call Qagi_r(Bound, Inf, Result, Info, 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 Debug.Print "Info =", Info End Sub Example Results S = 1.5707963267949 S(true) = 1.5707963267949 Info = 0