Alan M. Cohen

Cautious romberg extrapolation

An automatic quadrature routine was presented by de Boor which was based on the use of the trapezoidal rule with “cautious Romberg extrapolation”. It is the object of this paper to modify the approach of de Boor to cautious Romberg extrapolation by making a comparison between the results obtained from the trapezoidal and Simpson rules. This has the advantage that it gives some guidance regarding the estimation of the dominant terms involved, requires no additional function evaluations, and only minor changes in de Boors program CADRE are necessary to effect the proposed modifications. Special account is taken of integrands with end-point singularities and the possibilities of logarithmic terms being involved in the extrapolation. Further, indications are given of where automatic Romberg extrapolation can fail.

The construction of quadrature rules by parameter optimisation

Given a symmetric rule for the approximate numerical integration of we choose criteria for optimising the parameters in that rule. The method is extended to find integration rules in higher dimensions.

Some integration formulae for symmetric functions of two variables

We shall establish a five point formula for evaluating ∫1 − 1 ∫1 − 1 f(x y)dx dy when f(x v) is a symmetric function composed of monomials x m y n , which is exact for m+ n≦5. When f(x y) is not symmetric then the rule is exact for monomials x m y n where m + n≦3. This result will be used 10 construct a family of nine point formulae which arc exact for m +n≦7 when f(x y) is symmetric.

On improving convergence of alternating series

The root α of the equation can be determined by using the iterative formula provided that x 0is a good starting approximation to α and It is noted here that there are advantages in using the formula and, where λ is suitably chosen, this formula is equivalent to Aitkens; δ2extrapolation formula. The iterative technique is found to be successful in speeding up the convergence of alternating series and has also been applied to finding zeros of the Riemann Zeta Function.

A method for double series summation

The method of summation by diagonals is shown to be a practical method for the summation of double series which can give good results. Where problems arise an elementary analysis can often be used which enables the sum to be calculated to a high degree of accuracy. The method can be extended in an obvious manner to higher dimensional series.

Acceleration of convergence of series for certain multiple integrals

Levin [1] has given a method for speeding up convergence of series. In this paper it is shown that Levins method—and similar methods—do not work for certain types of series and an analysis is given to explain this phenomenon. It is also shown that Levins method can be used to efficiently compute certain types of multiple integrals. A variation of a method of Van Wijngaarden is shown to be efficient in evaluating series of positive terms for which Levins method does not work.

A special property of the Lobatto weights

It is shown that if the abscissae of the Lobatto quadrature formula are known to accuracy ϵ the weights can be determined to accuracy O(ϵ2).

The Construction Of Some Optimal 2D Cubature Formulae

In this paper, we showcase some formulae that have been derived using the least squares optimisation method, which produces cubatures that are superior for use with singular integrals. We consider some formulae of standard and well-known structure for two differing weight functions, (i) w(x, y)=1 and (ii) w(x, y)= \sqrt{x^2 + y^2} .

Quadrature rules by parameter optimisation II

The basis for the construction of optimum quadrature rules has been given in Cohen and Gismalla [1]. In this paper the ideas are extended to the construction of optimal Lobatto type formulae and the construction of optimal rules for where w(x) is a singular symmetric weight function.