Yagati N. Lakshman

Improved Sparse Multivariate Polynomial Interpolation Algorithms

We consider the problem of interpolating sparse multivariate polynomials from their values. We discuss two algorithms for sparse interpolation, one due to Ben-Or and Tiwari (1988) and the other due to Zippel (1988). We present efficient algorithms for finding the rank of certain special Toeplitz systems arising in the Ben-Or and Tiwari algorithm and for solving transposed Vandermonde systems of equations, the use of which greatly improves the time complexities of the two interpolation algorithms.

Approximate polynomial greatest common divisors and nearest singular polynomials

The problem of computing the greatest common divisor (gcd) of two polynomials ~, g ~ A[z], A being a unique factorization domain, is well understood and there area number of efficient algorithms for computing polynomial gcds beginning with the the work of Collins and Brown [3, 4, 9]. In this paper, we investigate the problem of finding approximate gcds. Given a pair of polynomials ~, g with real/complex coefficients, we wish to determine a small perturbation of the coefficients of ~, g such that the perturbed polynomials have a non-trivial gtd. We treat several variations of this problem and apply our techniques to the problem of finding a polynomial having multiple roots clo~seto a given polynomial. In this paper, F[a, b] denotes the polynomial ring in a, b over the field ~ and F(a, b) denotes the field of rational functions in a, b over f. C denotes the field of complex numbers and ‘R denotes the field of real numbers. For a polynomial j = f.z” + j.–lz”–l + . . . + fo ~ C[x], 11~11denotes

On Approximate GCDs of Univariate Polynomials

In this paper, we consider computations involving polynomials with inexact coefficients, i.e. with bounded coefficient errors. The presence of input errors changes the nature of questions traditionally asked in computer algebra. For instance, given two polynomials, instead of trying to compute their greatest common divisor, one might now try to compute a pair of polynomials with a non-trivial common divisor close to the input polynomials. We consider the problem of finding approximate common divisors in the context of inexactly specified polynomials. We develop efficient algorithms for the so-called nearest common divisor problem and several of its variants.

Efficient algorithms for computing the nearest polynomial with a real root and related problems

We present three new algorithms in the general area of input-sensitivity analysis: a problem formulation, possibly with floating point coefficients, lacks an expected property because the inputs are slightly perturbed. A task is to efficiently compute the nearest problem that has the desired property. Nearness to the desired property can lead to problems for numerical algorithms: for example, an almost singular linear system cannot be solved by classical numerical techniques. In such case one can approach the problem of locating the nearest problem with the desired property by symbolic computation techniques, for instance, by exact arithmetic. Our three properties are:

Modular rational sparse multivariate polynomial interpolation

The problem of interpolating multivariate polynomials whose coefficient domain is the rational numbers is considered. The effect of intermediate number growth on a speeded Ben-Or and Tiwari algorithm is studied. Then the newly developed modular algorithm is presented. The computing times for the speeded Ben-Or and Tiwari and the modular algorithm are compared, and it is shown that the modular algorithm is markedly superior.

Sparse Polynomial Interpolation in Nonstandard Bases

In this paper, we consider the problem of interpolating univariate polynomials over a field of characteristic zero that are sparse in (a) the Pochhammer basis or, (b) the Chebyshev basis. The polynomials are assumed to be given by black boxes, i.e., one can obtain the value of a polynomial at any point by querying its black box. We describe efficient new algorithms for these problems. Our algorithms may be regarded as generalizations of Ben-Or and Tiwaris (1988) algorithm (based on the BCH decoding algorithm) for interpolating polynomials that are sparse in the standard basis. The arithmetic complexity of the algorithms is

On the complexity of computing a Gröbner basis for the radical of a zero dimensional ideal

O(t^2 + t\log d)

Protium, an infrastructure for partitioned applications

which is also the complexity of the univariate version of the Ben-Or and Tiwari algorithm. That algorithm and those presented here also share the requirement of

Network processors applied to IPv4/IPv6 transition

2t

Algorithms for computing sparse shifts for multivariate polynomials

evaluation points.