Defin Defin_r Qag Qag_r Qags Qags_r Qng Qng_r

Qag() Sub Qag ( F As  LongPtr, A As  Double, B As  Double, 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 Key As  Long = -1, Optional Limit As  Long = -1, Optional Last As  Long  ) Finite interval adaptive quadrature (15/21/31/41/51/61 point Gauss-Kronrod rule) Purpose This routine computes I = integral of f over [a, b], satisfying the requested accuracy, where f is a given function defined by a user supplied subroutine. 15, 21, 31, 41, 51 or 61-point Gauss-Kronrod rule is used, and the integration interval will be adaptively subdivided to satisfy the requested accuracy. 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 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. [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] Key (Optional) Choice of local integration rule. (default = 1) = 1: Qk15 = 2: Qk21 = 3: Qk31 = 4: Qk41 = 5: Qk51 = 6: Qk61 (If Key < 1, Key = 1 is used. If Key > 6, Key = 6 is used) [in] Limit (Optional) Maximum number of subintervals in the partition of the given integration interval [a, b] (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, 4] (= atan(4)) Function F1(X As Double) As Double F1 = 1 / (1 + X ^ 2) End Function Sub Ex_Qag() Dim A As Double, B As Double, Result As Double, Info As Long A = 0: B = 4 Call Qag(AddressOf F1, A, B, Result, Info) Debug.Print "S =", Result, "S(true) =", Atn(4) Debug.Print "Info =", Info End Sub Example Results S = 1.32581766366803 S(true) = 1.32581766366803 Info = 0