On the automatic numerical evaluation of definite integrals

Stratified Sequences of Nested Quadrature Formulas

“…In a general purpose library it is desirable to have a routine that does not require its user to be aware of what singularities a function may have, or where they are located. For this reason, Cranley and Patterson [6] suggested the following model, which has become the standard for adaptive quadrature packages such as QUADPACK [22]:…”

“…It is possible to express the quadrature formulas leading to the diagonal elements of the Romberg table in the form (6). As we have seen, however, the degree of the Romberg formulas is low when compared to the number of nodes, which makes them less attractive.…”

“…In this section we show how to compute a particular family of stratified sequences in which a l l the formulas are interpolatory. We avoid the question of finding stratified sequences of the full generality allowed by the condition (6), and restrict our attention to the special case satisfying the following two conditions: 0 Each formula &k has the same number of points as the corresponding 0 The weights akj multiplying the old sums are preassigned.…”

On the use of Gauss quadrature in adaptive automatic integration schemes

“…In the numerical experiment we have used an adaptive strategy due to Cranley and Patterson [5]. The subinterval with the greatest estimated error is divided into two halves until the sum of the estimated errors is less than a given error-bound.…”

Increasing robustness in global adaptive quadrature through interval selection heuristics

“…2 Library implementations of local adaptive quadrature are not "pure" and do not conform to the definition given earlier. All versions allow intervals to be terminated in (1.3) due to relative error and make provision for wholesale termination in the event of excessive function evaluations.…”