QUADPACK

The main focus of QUADPACK is on automatic integration routines in which the user inputs the problem and an absolute or relative error tolerance and the routine attempts to perform the integration with an error no larger than that requested. There are nine such automatic routines in QUADPACK, in addition to a number of non-automatic routines. All but one of the automatic routines use adaptive quadrature.

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Each of the adaptive routines also have versions suffixed by E that have an extended parameter list that provides more information and allows more control. Double precision versions of all routines were released with prefix D.

The two general-purpose routines most suitable for use without further analysis of the integrand are QAGS for integration over a finite interval and QAGI for integration over an infinite interval. These two routines are used in GNU Octave (the quad command){{cite web|url=http://nines.cs.kuleuven.be/software/QUADPACK|title=QUADPACK|publisher=Numerical Integration, Nonlinear Equations & Software (NINES) Group, Katholieke Universiteit Leuven|access-date=November 13, 2010}} and R (the integrate function).{{cite web |title=integrate {stats}: Integration of One-Dimensional Functions |url=http://stat.ethz.ch/R-manual/R-devel/library/stats/html/integrate.html |work=Documentation for package ‘stats’ version 2.13.0 |access-date=16 November 2010 |date=October 2010}}

;QAGS : uses global adaptive quadrature based on 21-point Gauss–Kronrod quadrature within each subinterval, with acceleration by Peter Wynn's epsilon algorithm.{{cite web|title=17.4 QAGS adaptive integration with singularities|url=https://www.gnu.org/software/gsl/manual/html_node/QAGS-adaptive-integration-with-singularities.html|work=GNU Scientific Library -- Reference| publisher=Free Software Foundation|access-date=16 November 2010}}

;QAGI : is the only general-purpose routine for infinite intervals, and maps the infinite interval onto the semi-open interval (0,1] using a transformation then uses the same approach as QAGS, except with 15-point rather than 21-point Gauss–Kronrod quadrature. For an integral over the whole real line, the transformation used is $x\; =\; (1-t)/t$:{{cite book| last1=Piessens | first1=Robert |last2=de Doncker-Kapenga | first2=Elise | last3=Überhuber | first3=Christoph W. | last4=Kahaner | first4=David | title=QUADPACK: A subroutine package for automatic integration | year=1983 | publisher=Springer-Verlag | isbn=978-3-540-12553-2 }} $$$$

\int_{-\infty}^{+\infty} f(x) dx = \int_0^1 {dt\over t^2} \left(f\left(\frac{1-t}{t}\right)

+ f\left(-\frac{1-t}{t}\right)\right) \;.

This is not the best approach for all integrands: another transformation may be appropriate, or one might prefer to break up the original interval and use QAGI only on the infinite part.{{cite web |last=Piessens |first=Robert |title=Subroutine QPDOC |url=http://www.netlib.org/quadpack/doc |work=QUADPACK |publisher=netlib |access-date=16 November 2010 |display-authors=et al |date=1984-04-17}}

;QNG : simple non-adaptive integrator

;QAG : simple adaptive integrator

;QAGP : similar to QAGS but allows user to specify locations of internal singularities, discontinuities etc.

;QAWO : integral of {{math|cos(ωx) f(x)}} or {{math|sin(ωx) f(x)}} over a finite interval

;QAWF : Fourier transform

;QAWS : integral of {{math|w(x) f(x)}} from {{math|a}} to {{math|b}}, where {{math|f}} is smooth and {{math|w(x) {{=}} (x–a)^α (b–x)^β log^k(x–a) log^l(b–x)}}, with {{math|k, l {{=}} 0 or 1}} and {{math|α, β > –1}}

;QAWC : Cauchy principal value of the integral of {{math|f(x)/(x–c)}} for user-specified {{math|c}} and {{math|f}}

• List of numerical libraries

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• {{Cite journal | last1 = Favati | first1 = P. | last2 = Lotti | first2 = G. | last3 = Romani | first3 = F. | title = Algorithm 691; Improving QUADPACK automatic integration routines | journal = ACM Transactions on Mathematical Software | volume = 17 | issue = 2 | pages = 218–232 | year = 1991 | doi = 10.1145/108556.108580| s2cid = 19675880 | doi-access = free }}
• {{Cite journal | last1 = Cools | first1 = R. | last2 = Haegemans | first2 = A. | doi = 10.1145/838250.838253 | title = Algorithm 824: CUBPACK: a package for automatic cubature; framework description| journal = ACM Transactions on Mathematical Software | volume = 29 | issue = 3 | pages = 287–296| year = 2003 | s2cid = 6855610 }}

Category:Fortran libraries

Category:Numerical integration

Category:Numerical software

Category:Public-domain software with source code