Integration Problems

Integration Problems


The goal of this toolbox is to provide integration test problems in order to test integration algorithms. This allows to test integration algorithms on a larger base of functions. Developping these functions is error-prone, so that using this module allows to save much time. Moreover, testing an algorithm leads to the difficulty of chosing a set of test functions. But all functions are not equally difficult to integrate; while some are simple, other might be particularily difficult, leading to poor results. Comparing two algorithms, each on a different set of functions will not lead to an accurate view of the algorithm. This module allows to compare the algorithms more precisely.

In spirit, this module has the same motivations as the Unconstrained Optimization Scilab toolbox based on the More, Garbow and Hillstrom collection of functions.

A manual, available in pdf, is provided in the doc directory of this module. It provides a detailed analysis of each function and references where each function is used or analysed.


The module currently provides 23 functions :

  • SUM (10) Sum of X
  • SQSUM (10) Sum of Squares
  • SUMSQROOT (10) Sum of Square Roots
  • PRODONES (10) Product of Signed Ones
  • PRODEXP (10) Product of Exponentials
  • PRODCUB (10) Product of Cubes
  • PRODX (10) Product of X
  • SUMFIFJ (10) Sum of fi fj
  • SUMF1FJ (10) Sum of f1 fj
  • HELLEKALEK (10) Hellekalek
  • ROOSARNOLD1 (10) Roos and Arnold 1
  • ROOSARNOLD2 (10) Roos and Arnold 2
  • ROOSARNOLD3 (10) Roos and Arnold 3
  • RST1 (10) Radovic, Sobol, Tichy (aj=1)
  • RST2 (10) Radovic, Sobol, Tichy (aj=j)
  • RST3 (10) Radovic, Sobol, Tichy (aj=j^2)
  • SOBOLPROD (10) Sobol Product
  • OSCILL (6) Genz - Oscillatory
  • PRPEAK (6) Genz - Product Peak
  • CORPEAK (6) Genz - Corner Peak
  • GAUSSIAN (6) Genz - Gaussian
  • C0 (6) Genz - C0
  • DISCONT (6) Genz - Discontinuous


The following is a list of functions of the module.

  • intprb_getexpect : Returns the expectation of the problem.
  • intprb_getfcn : Returns the function value.
  • intprb_getname : Returns the name.
  • intprb_getproblems : Lists the problems.
  • intprb_getsetup : Returns the parameters of the problem.
  • intprb_getvariance : Compute the variance.
  • intprb_getvarprbs : Returns the array of problems with variable n.
  • Integration Algorithms
    • intprb_crudeld : Estimates a multidimensional integral using a Low Discrepancy sequence.
    • intprb_crudemc : Estimates a multidimensional integral using Monte Carlo.
    • intprb_getcrudeld : Uses a crude Quasi Monte-Carlo to estimate the integral.
    • intprb_getcrudemc : Uses a crude Monte-Carlo to estimate the integral.
    • intprb_montecarlo : Estimates a multidimensional integral using iterated Monte Carlo.
  • Support Functions
    • intprb_contour : Draws a 2D contour of the function.
    • intprb_muvblockupdate : Update the mean and the variance of x, given the mean and the value of disjoint parts of x.
    • intprb_muvstepstart : Returns the starting values for the step-by-step (mean,variance) computation.
    • intprb_muvstepstop : Finish the computation of the step-by-step (mean,variance) computation.
    • intprb_muvstepupdate : Update values for the step-by-step (mean,variance) computation.
    • intprb_shragerandom : A portable random number generator by Schrage.
    • intprb_tuplenext : Computes the next element of a tuple space.


  • genzphi : Estimates the normal cumulative density function
  • bench_mc : Test intprb_getcrudemc on all problems
  • bench_lowdisc : Test intprb_getcrudeld on all problems


  • This module depends on the apifun module (>= v0.2).
  • This module depends on the helptbx module (to build the help pages).
  • This module depends on the assert module.
  • This module depends on the lowdisc module (>= v0.2).



  • "Integration Test Problems User's Manual", Michael Baudin, 2010.
  • "Computational investigations of low-discrepancy sequences", Kocis, L. and Whiten, W. J. 1997. ACM Trans. Math. Softw. 23, 2 (Jun. 1997), 266-294.
  • "Quadrature Rule Applied to Test Integrals",, John Burkardt, 2007
  • "Testing Multidimensional Integration Routines",, John Burkardt, 2007
  • "Multi-dimensional Integration Test Functions",, John Burkardt, 2007
  • "Testpack",, Alan Genz, 1987


  • 2010, Michael Baudin
  • 2007, John Burkardt
  • 1987, Alan Genz


This toolbox is released under the GNU LGPL license.

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