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16.1 Introduction

Each algorithm computes an approximation to a definite integral of the form, 

     I = \int_a^b f(x) w(x) dx

where w(x) is a weight function (for general integrands w(x)=1). The user provides absolute and relative error bounds (epsabs, epsrel) which specify the following accuracy requirement, 

     |RESULT - I|  <= max(epsabs, epsrel |I|)

where RESULT is the numerical approximation obtained by the algorithm. The algorithms attempt to estimate the absolute error ABSERR = |RESULT - I| in such a way that the following inequality holds, 

     |RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)

The routines will fail to converge if the error bounds are too stringent, but always return the best approximation obtained up to that stage.

The algorithms in quadpack use a naming convention based on the following letters, 

     Q - quadrature routine
     
     N - non-adaptive integrator
     A - adaptive integrator
     
     G - general integrand (user-defined)
     W - weight function with integrand
     
     S - singularities can be more readily integrated
     P - points of special difficulty can be supplied
     I - infinite range of integration
     O - oscillatory weight function, cos or sin
     F - Fourier integral
     C - Cauchy principal value

The algorithms are built on pairs of quadrature rules, a higher order rule and a lower order rule. The higher order rule is used to compute the best approximation to an integral over a small range. The difference between the results of the higher order rule and the lower order rule gives an estimate of the error in the approximation.