aaaa Subroutine

public subroutine aaaa()

AAAA is a dummy subroutine with QUADPACK documentation in its comments.

  1. introduction

quadpack is a fortran subroutine package for the numerical computation of definite one-dimensional integrals. it originated from a joint project of r. piessens and e. de doncker (appl. math. and progr. div.- k.u.leuven, belgium), c. ueberhuber (inst. fuer math.-, austria), and d. kahaner (nation. bur. of standards- washington d.c., u.s.a.).

  1. survey

  2. qags : is an integrator based on globally adaptive interval subdivision in connection with extrapolation (de doncker, 1978) by the epsilon algorithm (wynn, 1956).

  3. qagp : serves the same purposes as qags, but also allows for eventual user-supplied information, i.e. the abscissae of internal singularities, discontinuities and other difficulties of the integrand function. the algorithm is a modification of that in qags.

  4. qagi : handles integration over infinite intervals. the infinite range is mapped onto a finite interval and then the same strategy as in qags is applied.

  5. qawo : is a routine for the integration of cos(omegax)f(x) or sin(omegax)f(x) over a finite interval (a,b). omega is is specified by the user the rule evaluation component is based on the modified clenshaw-curtis technique. an adaptive subdivision scheme is used connected with an extrapolation procedure, which is a modification of that in qags and provides the possibility to deal even with singularities in f.

  6. qawf : calculates the fourier cosine or fourier sine transform of f(x), for user-supplied interval (a, infinity), omega, and f. the procedure of qawo is used on successive finite intervals, and convergence acceleration by means of the epsilon algorithm (wynn, 1956) is applied to the series of the integral contributions.

  7. qaws : integrates w(x)f(x) over (a,b) with a < b finite, and w(x) = ((x-a)alfa)((b-x)beta)v(x) where v(x) = 1 or log(x-a) or log(b-x) or log(x-a)log(b-x) and -1 < alfa, -1 < beta. the user specifies a, b, alfa, beta and the type of the function v. a globally adaptive subdivision strategy is applied, with modified clenshaw-curtis integration on the subintervals which contain a or b.

  8. qawc : computes the cauchy principal value of f(x)/(x-c) over a finite interval (a,b) and for user-determined c. the strategy is globally adaptive, and modified clenshaw-curtis integration is used on the subranges which contain the point x = c.

each of the routines above also has a "more detailed" version with a name ending in e, as qage. these provide more information and control than the easier versions.

the preceeding routines are all automatic. that is, the user inputs his problem and an error tolerance. the routine attempts to perform the integration to within the requested absolute or relative error. there are, in addition, a number of non-automatic integrators. these are most useful when the problem is such that the user knows that a fixed rule will provide the accuracy required. typically they return an error estimate but make no attempt to satisfy any particular input error request.

      estimate the integral on [a,b] using 15, 21,..., 61
      point rule and return an error estimate.
 qk15i 15 point rule for (semi)infinite interval.
 qk15w 15 point rule for special singular weight functions.
 qc25c 25 point rule for cauchy principal values
 qc25o 25 point rule for sin/cos integrand.
 qmomo integrates k-th degree chebychev polynomial times
       function with various explicit singularities.
  1. guidelines for the use of quadpack

here it is not our purpose to investigate the question when automatic quadrature should be used. we shall rather attempt to help the user who already made the decision to use quadpack, with selecting an appropriate routine or a combination of several routines for handling his problem.

for both quadrature over finite and over infinite intervals, one of the first questions to be answered by the user is related to the amount of computer time he wants to spend, versus his -own- time which would be needed, for example, for manual subdivision of the interval or other analytic manipulations.

(1) the user may not care about computer time, or not be willing to do any analysis of the problem. especially when only one or a few integrals must be calculated, this attitude can be perfectly reasonable. in this case it is clear that either the most sophisticated of the routines for finite intervals, qags, must be used, or its analogue for infinite intervals, qagi. these routines are able to cope with rather difficult, even with improper integrals. this way of proceeding may be expensive. but the integrator is supposed to give you an answer in return, with additional information in the case of a failure, through its error estimate and flag. yet it must be stressed that the programs cannot be totally reliable.

(2) the user may want to examine the integrand function. if bad local difficulties occur, such as a discontinuity, a singularity, derivative singularity or high peak at one or more points within the interval, the first advice is to split up the interval at these points. the integrand must then be examinated over each of the subintervals separately, so that a suitable integrator can be selected for each of them. if this yields problems involving relative accuracies to be imposed on -finite- subintervals, one can make use of qagp, which must be provided with the positions of the local difficulties. however, if strong singularities are present and a high accuracy is requested, application of qags on the subintervals may yield a better result.

   for quadrature over finite intervals we thus dispose of qags
   - qng for well-behaved integrands,
   - qag for functions with an oscillating behavior of a non
     specific type,
   - qawo for functions, eventually singular, containing a
     factor cos(omega*x) or sin(omega*x) where omega is known,
   - qaws for integrands with algebraico-logarithmic end point
     singularities of known type,
   - qawc for cauchy principal values.


   on return, the work arrays in the argument lists of the
   adaptive integrators contain information about the interval
   subdivision process and hence about the integrand behavior:
   the end points of the subintervals, the local integral
   contributions and error estimates, and eventually other
   characteristics. for this reason, and because of its simple
   globally adaptive nature, the routine qag in particular is
   well-suited for integrand examination. difficult spots can
   be located by investigating the error estimates on the

   for infinite intervals we provide only one general-purpose
   routine, qagi. it is based on the qags algorithm applied
   after a transformation of the original interval into (0,1).
   yet it may eventuate that another type of transformation is
   more appropriate, or one might prefer to break up the
   original interval and use qagi only on the infinite part
   and so on. these kinds of actions suggest a combined use of
   different quadpack integrators. note that, when the only
   difficulty is an integrand singularity at the finite
   integration limit, it will in general not be necessary to
   break up the interval, as qagi deals with several types of
   singularity at the boundary point of the integration range.
   it also handles slowly convergent improper integrals, on
   the condition that the integrand does not oscillate over
   the entire infinite interval. if it does we would advise
   to sum succeeding positive and negative contributions to
   the integral -e.g. integrate between the zeros- with one
   or more of the finite-range integrators, and apply
   convergence acceleration eventually by means of quadpack
   subroutine qelg which implements the epsilon algorithm.
   such quadrature problems include the fourier transform as
   a special case. yet for the latter we have an automatic
   integrator available, qawf.