Then 3. Theoretically the N - d smallest singular values of HN a r e equal to zero. If the numerical rank d of HN is difficult to determine, it is safe to consider HN a s a matrix of full rank and to solve an N x N generalized eigenvalue problem and associated Vandermonde system. This follows from Theorem 3.

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The quotient of two such corresponding diagonal entries is an eigenvalue that is not the nth component of a zero of f. Fortunately, the corresponding Vandermonde system enables us to detect such spurious "nth components". Indeed, the Vandermonde matrix whose nodes are given by the eigenvalues of 3. Therefore we will use shifted versions of the integrals sp and tk,p, denoted by 8p and tk,p.

The results of this section then lead to the following algorithm. Calculate the eigenvalues q ,. Solve the Vandermonde system The components of the solution of the latter will be rounded to the nearest integers, and eigenvalues that have "multiplicity" zero are thrown away 9 REMARK 3. W h a t happens if our algorithm is applied in case the nth components of the zeros of f that lie in D are not mutually distinct? In this case only a subset of the zeros will be determined correctly, namely the zeros whose nth component occurs only once.

It follows that Algorithm 3. This algorithm takes full account of the structure of a Vandermonde matrix and is not only faster but also more accurate than general purpose algorithms such as Gaussian elimina- tion with partial pivoting. This Vandermonde solver has arithmetic complexity O d2. It has also been used to solve the Vandermonde system for the first n - 1 components Step 9 in Algorithm 3. The results are shown in the following table.

The efficient numerical evaluation of these integrals is a problem that remains to be tackled. Nevertheless we would like to present a small example in which these integrals were calculated numerically. This package implements an adaptive algorithm for numerical integration over hyperrectangular regions.

First we calculated Re I 1. We requested an absolute accuracy of 0. Next we calculated the arithmetic mean of the second components. A crude approximation to this mean is sufficient to reduce ill-conditioning, and therefore we requested a relative accuracy of only 0. Finally we calculated all the other integrals needed by Algorithm 3.

One of the very interesting features of DCUHRE is that it is able to integrate a vector of similar integrals over a common integration region. Since a significant part of the computation required for each integrand is the same for all of the integrands, these common calculations need be done only once for each integrand evaluation point. We requested a relative accuracy of 10 We refined the approximations for these zeros iteratively via Newton's method.

The zeros of f that lie in D are given by 0, 0 and Note that the number of functional evaluations needed to approximate the integrals to the requested accuracy is independent of the number of zeros of f that lie in D. We had interesting conversations with Franki Dillen about differential forms, and stimulating discussions with Alan Genz on the logarithmic residue integrals that appear in this paper. We thank Johan Quaegebeur for having read the proof of Theorem 2. Aizenberg and A.

Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Croz, A. Green- baum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Berntsen, T. Espelid, and A. Software, 17 , pp.

**dcovanunan.tk**

## Asymptotic and Computational Analysis: Conference in Honor of Frank W.j. Olver's 65th Birthday

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## Floyd B. Hanson -- Publications

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Limited preview. Contributor Ehrenpreis, Leon. Grinberg, Eric, Bibliography Includes bibliographical references. Contents On a problem of injectivity for the Radon transform on a paraboloid by M. Agranovsky Anti-self-dual symplectic forms and integral geometry by J. Alvarez Holomorphic extendibility of functions via nonlinear Fourier transforms by T.

Banh Division-interpolation methods and Nullstellensatze by C. Berenstein and A. Yger Uniqueness and non-uniqueness for microanalytic continuation of ultradistributions by J. Boman A rational Landen transformation. The case of degree six by G. Boros and V. Braun, R.