NAG C Library Chapter Introduction

d01 — Quadrature

1  Scope of the Chapter

2  Background to the Problems

1. the value of a one-dimensional definite integral of the form

   ∫abfxdx

   (1)

   where fx is defined by you, either at a set of points xi,fxi, for i=1,2,…,n, where a=x1<x2<⋯<xn=b, or in the form of a function; and the limits of integration a,b may be finite or infinite.
   Some methods are specially designed for integrands of the form

   fx=wxgx

   (2)

   which contain a factor wx, called the weight-function, of a specific form. These methods take full account of any peculiar behaviour attributable to the wx factor.
2. the values of the one-dimensional indefinite integrals arising from (1) where the ranges of integration are interior to the interval a,b.
3. the value of a multi-dimensional definite integral of the form

   ∫Rnfx1,x2,…,xndxn⋯dx2dx1

   (3)

   where fx1,x2,…,xn is a function defined by you and Rn is some region of n-dimensional space.
   The simplest form of Rn is the n-rectangle defined by

   ai≤xi≤bi,  i=1,2,…,n

   (4)

   where ai and bi are constants. When ai and bi are functions of xj (j<i), the region can easily be transformed to the rectangular form (see page 266 of Davis and Rabinowitz (1975)). Some of the methods described incorporate the transformation procedure.

2.1  One-dimensional Integrals

2.2  Multi-dimensional Integrals

1. Products of one-dimensional rules
   Using a two-dimensional integral as an example, we have

   ∫a1b1∫a2b2fx,ydy dx≃∑i=1Nwi ∫a2b2fxi,ydy

   (7)

   ∫a1b1∫a2b2fx,ydy dx≃∑i=1N∑j=1Nwivjfxi,yj

   (8)

   where wi,xi and vi,yi are the weights and abscissae of the rules used in the respective dimensions.
   A different one-dimensional rule may be used for each dimension, as appropriate to the range and any weight function present, and a different strategy may be used, as appropriate to the integrand behaviour as a function of each independent variable.
   For a rule-evaluation strategy in all dimensions, the formula (8) is applied in a straightforward manner. For automatic strategies (i.e., attempting to attain a requested accuracy), there is a problem in deciding what accuracy must be requested in the inner integral(s). Reference to formula (7) shows that the presence of a limited but random error in the y-integration for different values of xi can produce a ‘jagged’ function of x, which may be difficult to integrate to the desired accuracy and for this reason products of automatic one-dimensional functions should be used with caution (see Lyness (1983)).
2. Monte Carlo methods
   These are based on estimating the mean value of the integrand sampled at points chosen from an appropriate statistical distribution function. Usually a variance reducing procedure is incorporated to combat the fundamentally slow rate of convergence of the rudimentary form of the technique. These methods can be effective by comparison with alternative methods when the integrand contains singularities or is erratic in some way, but they are of quite limited accuracy.
3. Automatic adaptive procedures
   An automatic adaptive strategy in several dimensions normally involves division of the region into subregions, concentrating the divisions in those parts of the region where the integrand is worst behaved. It is difficult to arrange with any generality for variable limits in the inner integral(s). For this reason, some methods use a region where all the limits are constants; this is called a hyper-rectangle. Integrals over regions defined by variable or infinite limits may be handled by transformation to a hyper-rectangle. Integrals over regions so irregular that such a transformation is not feasible may be handled by surrounding the region by an appropriate hyper-rectangle and defining the integrand to be zero outside the desired region. Such a technique should always be followed by a Monte Carlo method for integration.
   The method used locally in each subregion produced by the adaptive subdivision process is usually one of three types: Monte Carlo, number theoretic or deterministic. Deterministic methods are usually the most rapidly convergent but are often expensive to use for high dimensionality and not as robust as the other techniques.

3  Recommendations on Choice and Use of Available Functions

3.1  One-dimensional Integrals over a Finite Interval

1. Integrand defined at a set of points
   If fx is defined numerically at four or more points, then the Gill–Miller finite difference method (nag_1d_quad_vals (d01gac)) should be used. The interval of integration is taken to coincide with the range of x values of the points supplied. It is in the nature of this problem that any function may be unreliable. In order to check results independently and so as to provide an alternative technique you may fit the integrand by Chebyshev-series using nag_1d_cheb_fit (e02adc) and then use functions nag_1d_cheb_intg (e02ajc) and nag_1d_cheb_eval2 (e02akc) to evaluate its integral (which need not be restricted to the range of the integration points, as is the case for nag_1d_quad_vals (d01gac)). A further alternative is to fit a cubic spline to the data using nag_1d_spline_fit_knots (e02bac) and then to evaluate its integral using nag_1d_spline_intg (e02bdc).
2. Integrand defined as a function
   If the functional form of fx is known, then one of the following approaches should be taken. They are arranged in the order from most specific to most general, hence the first applicable procedure in the list will be the most efficient. However, if you do not wish to make any assumptions about the integrand, the most reliable functions to use will be nag_1d_quad_gen_1 (d01sjc).
   1. Rule-evaluation functions
      If fx is known to be sufficiently well behaved (more precisely, can be closely approximated by a polynomial of moderate degree), a Gaussian function with a suitable number of abscissae may be used.
      nag_1d_quad_gauss (d01bac) may be used if it is not required to examine the weights and abscissae.
   2. Automatic adaptive functions
      Firstly, several functions are available for integrands of the form wxgx where gx is a ‘smooth’ function (i.e., has no singularities, sharp peaks or violent oscillations in the interval of integration) and wx is a weight function of one of the following forms.
      1. if wx= b-x α x-a β log⁡b-xklog⁡x-al, where k,l=0​ or ​1, α,β>-1: use nag_1d_quad_wt_alglog (d01apc);
      2. if wx=1x-c : use nag_1d_quad_wt_cauchy (d01aqc) (this integral is called the Hilbert transform of g);
      3. if wx=cos⁡ωx or sin⁡ωx: use nag_1d_quad_wt_trig (d01anc) (this function can also handle certain types of singularities in gx).
      Secondly, there are some functions for general fx. If fx is known to be free of singularities, though it may be oscillatory, nag_1d_quad_osc_1 (d01skc) may be used.
      The most powerful of the finite interval integration functions is nag_1d_quad_gen_1 (d01sjc), which can cope with singularities of several types. It may be used if none of the more specific situations described above applies. nag_1d_quad_gen_1 (d01sjc) is somewhat more reliable, particularly where the integrand has singularities other than at an end point, or has discontinuities or cusps, and is therefore recommended where the integrand is known to be badly behaved, or where its nature is completely unknown.
      Most of the functions in this chapter require you to supply a function or function to evaluate the integrand at a single point.
      If fx has singularities of certain types, discontinuities or sharp peaks occurring at known points, the integral should be evaluated separately over each of the subranges or nag_1d_quad_inf_wt_trig (d01asc) may be used.

3.2  One-dimensional Integrals over a Semi-infinite or Infinite Interval

1. Integrand defined at a set of points
   If fx is defined numerically at four or more points, and the portion of the integral lying outside the range of the points supplied may be neglected, then the Gill–Miller finite difference method, nag_1d_quad_vals (d01gac), should be used.
2. Integrand defined as a function
   1. Rule evaluation functions
      If fx behaves approximately like a polynomial in x, apart from a weight function of the form:
      1. e-βx,β>0 (semi-infinite interval, lower limit finite); or
      2. e-βx,β<0 (semi-infinite interval, upper limit finite); or
      3. e-β x-α 2,β>0 (infinite interval),
      or if fx behaves approximately like a polynomial in x+b-1 (semi-infinite range), then the Gaussian functions may be used.
      nag_1d_quad_gauss_1 (d01tac) may be used if it is not required to examine the weights and abscissae.
   2. Automatic adaptive functions
      nag_1d_quad_inf_1 (d01smc) may be used, except for integrands which decay slowly towards an infinite end point, and oscillate in sign over the entire range. For this class, it may be possible to calculate the integral by integrating between the zeros and invoking some extrapolation process.
      nag_1d_quad_inf_wt_trig_1 (d01ssc) may be used for integrals involving weight functions of the form cos⁡ωx and sin⁡ωx  over a semi-infinite interval (lower limit finite).
      The following alternative procedures are mentioned for completeness, though their use will rarely be necessary.
      1. If the integrand decays rapidly towards an infinite end point, a finite cut-off may be chosen, and the finite range methods applied.
      2. If the only irregularities occur in the finite part (apart from a singularity at the finite limit, with which nag_1d_quad_inf_1 (d01smc) can cope), the range may be divided, with nag_1d_quad_inf_1 (d01smc) used on the infinite part.
      3. A transformation to finite range may be employed,e.g.,

         x=1-tt  or  x=-loge⁡t

         will transform 0,∞ to 1,0 while for infinite ranges we have

         ∫-∞∞fxdx=∫0∞fx+f-xdx.

         If the integrand behaves badly on -∞,0 and well on 0,∞ or vice versa it is better to compute it as ∫-∞0fxdx+∫0∞fxdx. This saves computing unnecessary function values in the semi-infinite range where the function is well behaved.

3.3  Multi-dimensional Integrals

1. Automatic functions (nag_multid_quad_monte_carlo (d01gbc) and nag_multid_quad_adapt (d01fcc))
   Both functions are for integrals of the form

   ∫a1b1 ∫a2b2 ⋯ ∫anbn fx1,x2,…,xndxndxn-1⋯dx1.

   nag_multid_quad_monte_carlo (d01gbc)

   is an adaptive Monte Carlo function. This function is usually slow and not recommended for high-accuracy work. It is a robust function that can often be used for low-accuracy results with highly irregular integrands or when n is large.

   nag_multid_quad_adapt (d01fcc)

   is an adaptive deterministic function. Convergence is fast for well behaved integrands. Highly accurate results can often be obtained for n between 2 and 5, using significantly fewer integrand evaluations than would be required by nag_multid_quad_monte_carlo (d01gbc). The function will usually work when the integrand is mildly singular and for n≤10 should be used before nag_multid_quad_monte_carlo (d01gbc). If it is known in advance that the integrand is highly irregular, it is best to compare results from at least two different functions.

   There are many problems for which one or both of the functions will require large amounts of computing time to obtain even moderately accurate results. The amount of computing time is controlled by the number of integrand evaluations allowed by you, and you should set this argument carefully, with reference to the time available and the accuracy desired.

4  Decision Trees

Tree 1: One-dimensional integrals over a finite interval

Tree 2: One-dimensional integrals over a semi-infinite or infinite interval

Tree 3: Multi-dimensional integrals

5  Index

6  Functions Withdrawn or Scheduled for Withdrawal

7  References