INTLIB
1-dimensional quadrature

INTLIB is a FORTRAN90 library which estimates integrals over 1D regions.

The integrand may be available as a function F(X), or as data at equally spaced or unequally spaced points.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

INTLIB is available in a FORTRAN90 version.

Related Data and Programs:

CUBPACK, a FORTRAN90 library which estimates the integral of a function over a collection of N-dimensional hyperrectangles and simplices.

NINTLIB, a FORTRAN90 library which estimates integrals over multidimensional regions.

PRODUCT_RULE, a FORTRAN90 program which constructs a product quadrature rule from 1D factor rules.

QUADRATURE_RULES, a dataset directory which contains files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADPACK, a FORTRAN90 library which numerically estimates integrals.

QUADRULE, a FORTRAN90 library which defines quadrature rules for 1D domains.

SIMPACK, a FORTRAN77 library which approximates the integral of a function over a multidimensional simplex.

STROUD, a FORTRAN90 library which defines quadrature rules for a variety of multidimensional reqions.

TANH_QUAD, a FORTRAN90 library which sets up the tanh quadrature rule;

TEST_INT, a FORTRAN90 library which defines test integrands for 1D quadrature rules.

TEST_INT_2D, a FORTRAN90 library which defines test integrands for 2D quadrature rules.

TOMS351, a FORTRAN77 library which estimates an integral using Romberg integration.

TOMS379, a FORTRAN77 library which estimates an integral.

TOMS418, a FORTRAN77 library which estimates the integral of a function with a sine or cosine factor.

TOMS424, a FORTRAN77 library which estimates the integral of a function using Clenshaw-Curtis quadrature.

TOMS468, a FORTRAN77 library which applies "automatic" integration to a function.

Reference:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Roland Bulirsch, Josef Stoer,
   Fehlerabschaetzungen und Extrapolation mit rationaled Funktionen bei Verfahren vom Richardson-Typus,
   (Error estimates and extrapolation with rational functions in processes of Richardson type),
   Numerische Mathematik,
   Volume 6, Number 1, December 1964, pages 413-427.
3. Stephen Chase, Lloyd Fosdick,
   An Algorithm for Filon Quadrature,
   Communications of the Association for Computing Machinery,
   Volume 12, Number 8, August 1969, pages 453-457.
4. Stephen Chase, Lloyd Fosdick,
   Algorithm 353: Filon Quadrature,
   Communications of the Association for Computing Machinery,
   Volume 12, Number 8, August 1969, pages 457-458.
5. William Cody,
   An Overview of Software Development for Special Functions, in Numerical Analysis Dundee, 1975,
   edited by GA Watson,
   Lecture Notes in Mathematics, 506,
   Springer, 1976.
6. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
7. Carl deBoor, John Rice,
   CADRE: An algorithm for numerical quadrature,
   in Mathematical Software,
   edited by John Rice,
   Academic Press, 1971,
   ISBN: 012587250X,
   LC: QA1.M766.
8. Augustin Dubrulle,
   A short note on the implicit QL algorithm for symmetric tridiagonal matrices,
   Numerische Mathematik,
   Volume 15, Number 5, September 1970, page 450.
9. Philip Gill, GF Miller,
   An algorithm for the integration of unequally spaced data,
   The Computer Journal,
   Number 15, Number 1, 1972, pages 80-83.
10. Gene Golub,
    Some Modified Matrix Eigenvalue Problems,
    SIAM Review,
    Volume 15, Number 2, Part 1, April 1973, pages 318-334.
11. Gene Golub, John Welsch,
    Calculation of Gaussian Quadrature Rules,
    Mathematics of Computation,
    Volume 23, Number 106, April 1969, pages 221-230.
12. John Hart, Ward Cheney, Charles Lawson, Hans Maehly, Charles Mesztenyi, John Rice, Henry Thatcher, Christoph Witzgall,
    Computer Approximations,
    Wiley, 1968.
13. Tore Havie,
    On a Modification of the Clenshaw Curtis Quadrature Rule,
    BIT,
    Volume 9, Number 4, December 1969, pages 338-350.
14. Paul Hennion,
    Algorithm 77: Interpolation, Differentiation and Integration,
    Communications of the ACM,
    Volume 5, 1962, page 96.
15. Robert Kubik,
    Algorithm 257: Havie Integrator,
    Communications of the ACM,
    Volume 8, Number 6, June 1965, page 381.
16. James Lyness,
    Algorithm 379: SQUANK (Simpson Quadrature Used Adaptively - Noise Killed),
    Communications of the ACM,
    Volume 13, Number 4, April 1970, pages 260-263.
17. Roger Martin, James Wilkinson,
    The Implicit QL Algorithm,
    Numerische Mathematik,
    Volume 12, Number 5, December 1968, pages 377-383.
18. William McKeeman, Lawrence Tesler,
    Algorithm 182: Nonrecursive adaptive integration,
    Communications of the ACM,
    Volume 6, 1963, page 315.
19. Arthur Stroud, Don Secrest,
    Gaussian Quadrature Formulas,
    Prentice Hall, 1966,
    LC: QA299.4G3S7.
20. James Wilkinson, Christian Reinsch,
    Handbook for Automatic Computation,
    Volume II, Linear Algebra, Part 2,
    Springer, 1971,
    ISBN: 0387054146.

Source Code:

• intlib.f90, the source code.

Examples and Tests:

• intlib_test.f90, a sample problem.
• intlib_test_output.txt, the output file.

List of Routines:

• AVINT estimates the integral of unevenly spaced data.
• CADRE estimates the integral of F(X) from A to B.
• CHINSP estimates an integral using a modified Clenshaw-Curtis scheme.
• CLASS sets recurrence coeeficients for various orthogonal polynomials.
• CSPINT estimates the integral of a tabulated function.
• CUBINT approximates an integral using cubic interpolation of data.
• FILON_COS uses Filon's method on integrals with a cosine factor.
• FILON_SIN uses Filon's method on integrals with a sine factor.
• GAMMA calculates the Gamma function for a real argument X.
• GAUS8 estimates the integral of a function.
• GAUSQ2 finds the eigenvalues of a symmetric tridiagonal matrix.
• GAUSSQ computes a Gauss quadrature rule.
• HIORDQ approximates the integral of a function using equally spaced data.
• IRATEX estimates the integral of a function.
• MONTE_CARLO estimates the integral of a function by Monte Carlo.
• PLINT approximates the integral of unequally spaced data.
• QNC79 approximates the integral of F(X) using Newton-Cotes quadrature.
• QUAD approximates the integral of F(X) by Romberg integration.
• R8VEC_EVEN returns N values, evenly spaced between ALO and AHI.
• RMINSP approximates the integral of a function using Romberg integration.
• SIMP approximates the integral of a function by an adaptive Simpson's rule.
• SIMPNE approximates the integral of unevenly spaced data.
• SIMPSN approximates the integral of evenly spaced data.
• SOLVE solves a special linear system.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• WEDINT uses Weddle's rule to integrate data at equally spaced points.

You can go up one level to the FORTRAN90 source codes.