Defin Defin_r Qag Qag_r Qags Qags_r Qng Qng_r

Qag_r() Sub Qag_r ( A As  Double, B As  Double, 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 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) (reverse communication version) Purpose This routine computes I = integral of f over [a, b], satisfying the requested accuracy, where f is a given function. User should provide the necessary computed values of f according to the argument IRev. 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] A Lower limit of integration a. [in] B Upper limit of integration b. [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. [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 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] 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 assumed. If Key > 6, Key = 6 is assumed. [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)) Sub Ex_Qag_r() Dim A As Double, B As Double, Result As Double, Info As Long Dim XX As Double, YY As Double, IRev As Long A = 0: B = 4 IRev = 0 Do Call Qag_r(A, B, Result, Info, XX, YY, IRev) If IRev = 1 Then YY = 1 / (1 + XX ^ 2) Loop While IRev <> 0 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