Defin Defin_r Qag Qag_r Qags Qags_r Qng Qng_r

Qags() Sub Qags ( 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 Limit As  Long = -1, Optional Last As  Long  ) Finite interval adaptive quadrature with sigularities (21 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 with singularities defined by a user supplied subroutine. 21 point Gauss-Kronrod rule is used, and the integration interval will be adaptively subdivided to satisfy the requested accuracy in connection with extrapolation, which will eliminate the effects of integrand singularities of several types. 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 reached. = 2: Requested tolerance cannot be achieved due to roundoff error. = 3: Bad integrand behavior found in the integration interval. = 4: Algorithm does not converge due to the roundoff error in the extrapolation table. = 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 [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_Qags() Dim A As Double, B As Double, Result As Double, Info As Long A = 0: B = 4 Call Qags(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