Category H. Differentiation, integration

H1. Numerical differentiation H2. Quadrature (numerical evaluation of definite integrals) H1. Numerical differentiation CHFDV-S Evaluate a cubic polynomial given in Hermite form and its DCHFDV-D first derivative at an array of points. While designed for use by PCHFD, it may be useful directly as an evaluator for a piecewise cubic Hermite function in applications, such as graphing, where the interval is known in advance. If only function values are required, use CHFEV instead. PCHFD-S Evaluate a piecewise cubic Hermite function and its first DPCHFD-D derivative at an array of points. May be used by itself for Hermite interpolation, or as an evaluator for PCHIM or PCHIC. If only function values are required, use PCHFE instead. H2. Quadrature (numerical evaluation of definite integrals) QPDOC-A Documentation for QUADPACK, a package of subprograms for automatic evaluation of one-dimensional definite integrals. H2A. One-dimensional integrals H2A1. Finite interval (general integrand) H2A1A. Integrand available via user-defined procedure H2A1A1. Automatic (user need only specify required accuracy) GAUS8-S Integrate a real function of one variable over a finite DGAUS8-D interval using an adaptive 8-point Legendre-Gauss algorithm. Intended primarily for high accuracy integration or integration of smooth functions. QAG-S The routine calculates an approximation result to a given DQAG-D definite integral I = integral of F over (A,B), hopefully satisfying following claim for accuracy ABS(I-RESULT)LE.MAX(EPSABS,EPSREL*ABS(I)). QAGE-S The routine calculates an approximation result to a given DQAGE-D definite integral I = Integral of F over (A,B), hopefully satisfying following claim for accuracy ABS(I-RESLT).LE.MAX(EPSABS,EPSREL*ABS(I)). QAGS-S The routine calculates an approximation result to a given DQAGS-D Definite integral I = Integral of F over (A,B), Hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). QAGSE-S The routine calculates an approximation result to a given DQAGSE-D definite integral I = Integral of F over (A,B), hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). QNC79-S Integrate a function using a 7-point adaptive Newton-Cotes DQNC79-D quadrature rule. QNG-S The routine calculates an approximation result to a DQNG-D given definite integral I = integral of F over (A,B), hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). H2A1A2. Nonautomatic QK15-S To compute I = Integral of F over (A,B), with error DQK15-D estimate J = integral of ABS(F) over (A,B) QK21-S To compute I = Integral of F over (A,B), with error DQK21-D estimate J = Integral of ABS(F) over (A,B) QK31-S To compute I = Integral of F over (A,B) with error DQK31-D estimate J = Integral of ABS(F) over (A,B) QK41-S To compute I = Integral of F over (A,B), with error DQK41-D estimate J = Integral of ABS(F) over (A,B) QK51-S To compute I = Integral of F over (A,B) with error DQK51-D estimate J = Integral of ABS(F) over (A,B) QK61-S To compute I = Integral of F over (A,B) with error DQK61-D estimate J = Integral of ABS(F) over (A,B) H2A1B. Integrand available only on grid H2A1B2. Nonautomatic AVINT-S Integrate a function tabulated at arbitrarily spaced DAVINT-D abscissas using overlapping parabolas. PCHIA-S Evaluate the definite integral of a piecewise cubic DPCHIA-D Hermite function over an arbitrary interval. PCHID-S Evaluate the definite integral of a piecewise cubic DPCHID-D Hermite function over an interval whose endpoints are data points. H2A2. Finite interval (specific or special type integrand including weight functions, oscillating and singular integrands, principal value integrals, splines, etc.) H2A2A. Integrand available via user-defined procedure H2A2A1. Automatic (user need only specify required accuracy) BFQAD-S Compute the integral of a product of a function and a DBFQAD-D derivative of a B-spline. BSQAD-S Compute the integral of a K-th order B-spline using the DBSQAD-D B-representation. PFQAD-S Compute the integral on (X1,X2) of a product of a function DPFQAD-D F and the ID-th derivative of a B-spline, (PP-representation). PPQAD-S Compute the integral on (X1,X2) of a K-th order B-spline DPPQAD-D using the piecewise polynomial (PP) representation. QAGP-S The routine calculates an approximation result to a given DQAGP-D definite integral I = Integral of F over (A,B), hopefully satisfying following claim for accuracy break points of the integration interval, where local difficulties of the integrand may occur(e.g. SINGULARITIES, DISCONTINUITIES), are provided by the user. QAGPE-S Approximate a given definite integral I = Integral of F DQAGPE-D over (A,B), hopefully satisfying the accuracy claim: ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). Break points of the integration interval, where local difficulties of the integrand may occur (e.g. singularities or discontinuities) are provided by the user. QAWC-S The routine calculates an approximation result to a DQAWC-D Cauchy principal value I = INTEGRAL of F*W over (A,B) (W(X) = 1/((X-C), C.NE.A, C.NE.B), hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABE,EPSREL*ABS(I)). QAWCE-S The routine calculates an approximation result to a DQAWCE-D CAUCHY PRINCIPAL VALUE I = Integral of F*W over (A,B) (W(X) = 1/(X-C), (C.NE.A, C.NE.B), hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)) QAWO-S Calculate an approximation to a given definite integral DQAWO-D I = Integral of F(X)*W(X) over (A,B), where W(X) = COS(OMEGA*X) or W(X) = SIN(OMEGA*X), hopefully satisfying the following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). QAWOE-S Calculate an approximation to a given definite integral DQAWOE-D I = Integral of F(X)*W(X) over (A,B), where W(X) = COS(OMEGA*X) or W(X) = SIN(OMEGA*X), hopefully satisfying the following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). QAWS-S The routine calculates an approximation result to a given DQAWS-D definite integral I = Integral of F*W over (A,B), (where W shows a singular behaviour at the end points see parameter INTEGR). Hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). QAWSE-S The routine calculates an approximation result to a given DQAWSE-D definite integral I = Integral of F*W over (A,B), (where W shows a singular behaviour at the end points, see parameter INTEGR). Hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). QMOMO-S This routine computes modified Chebyshev moments. The K-th DQMOMO-D modified Chebyshev moment is defined as the integral over (-1,1) of W(X)*T(K,X), where T(K,X) is the Chebyshev polynomial of degree K. H2A2A2. Nonautomatic QC25C-S To compute I = Integral of F*W over (A,B) with DQC25C-D error estimate, where W(X) = 1/(X-C) QC25F-S To compute the integral I=Integral of F(X) over (A,B) DQC25F-D Where W(X) = COS(OMEGA*X) Or (WX)=SIN(OMEGA*X) and to compute J=Integral of ABS(F) over (A,B). For small value of OMEGA or small intervals (A,B) 15-point GAUSS- KRONROD Rule used. Otherwise generalized CLENSHAW-CURTIS us QC25S-S To compute I = Integral of F*W over (BL,BR), with error DQC25S-D estimate, where the weight function W has a singular behaviour of ALGEBRAICO-LOGARITHMIC type at the points A and/or B. (BL,BR) is a part of (A,B). QK15W-S To compute I = Integral of F*W over (A,B), with error DQK15W-D estimate J = Integral of ABS(F*W) over (A,B) H2A3. Semi-infinite interval (including e**(-x) weight function) H2A3A. Integrand available via user-defined procedure H2A3A1. Automatic (user need only specify required accuracy) QAGI-S The routine calculates an approximation result to a given DQAGI-D INTEGRAL I = Integral of F over (BOUND,+INFINITY) OR I = Integral of F over (-INFINITY,BOUND) OR I = Integral of F over (-INFINITY,+INFINITY) Hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). QAGIE-S The routine calculates an approximation result to a given DQAGIE-D integral I = Integral of F over (BOUND,+INFINITY) or I = Integral of F over (-INFINITY,BOUND) or I = Integral of F over (-INFINITY,+INFINITY), hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)) QAWF-S The routine calculates an approximation result to a given DQAWF-D Fourier integral I = Integral of F(X)*W(X) over (A,INFINITY) where W(X) = COS(OMEGA*X) or W(X) = SIN(OMEGA*X). Hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.EPSABS. QAWFE-S The routine calculates an approximation result to a DQAWFE-D given Fourier integral I = Integral of F(X)*W(X) over (A,INFINITY) where W(X) = COS(OMEGA*X) or W(X) = SIN(OMEGA*X), hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.EPSABS. H2A3A2. Nonautomatic QK15I-S The original (infinite integration range is mapped DQK15I-D onto the interval (0,1) and (A,B) is a part of (0,1). it is the purpose to compute I = Integral of transformed integrand over (A,B), J = Integral of ABS(Transformed Integrand) over (A,B). H2A4. Infinite interval (including e**(-x**2)) weight function) H2A4A. Integrand available via user-defined procedure H2A4A1. Automatic (user need only specify required accuracy) QAGI-S The routine calculates an approximation result to a given DQAGI-D INTEGRAL I = Integral of F over (BOUND,+INFINITY) OR I = Integral of F over (-INFINITY,BOUND) OR I = Integral of F over (-INFINITY,+INFINITY) Hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). QAGIE-S The routine calculates an approximation result to a given DQAGIE-D integral I = Integral of F over (BOUND,+INFINITY) or I = Integral of F over (-INFINITY,BOUND) or I = Integral of F over (-INFINITY,+INFINITY), hopefully satisfying following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)) H2A4A2. Nonautomatic QK15I-S The original (infinite integration range is mapped DQK15I-D onto the interval (0,1) and (A,B) is a part of (0,1). it is the purpose to compute I = Integral of transformed integrand over (A,B), J = Integral of ABS(Transformed Integrand) over (A,B).