Symbolic–numeric sparse interpolation of multivariate polynomials

We consider the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all numbers are represented in standard, fixed-precision, floating point arithmetic....

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Bibliographic Details
Published in:	Journal of symbolic computation Vol. 44; no. 8; pp. 943 - 959
Main Authors:	Giesbrecht, Mark, Labahn, George, Lee, Wen-shin
Format:	Journal Article
Language:	English
Published:	Elsevier Ltd 01.08.2009
Subjects:	Multivariate interpolation Symbolic–numeric computing Symbolic–numeric computing Multivariate interpolation
ISSN:	0747-7171, 1095-855X
Online Access:	Get full text
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Summary:	We consider the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all numbers are represented in standard, fixed-precision, floating point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give efficient and numerically robust solutions. We note the similarity between the exact Ben-Or/Tiwari sparse interpolation algorithm and the classical Prony’s method for interpolating a sum of exponential functions, and exploit the generalized eigenvalue reformulation of Prony’s method. We analyse the numerical stability of our algorithms and the sensitivity of the solutions, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques in practice through numerical experiments and applications.
ISSN:	0747-7171 1095-855X
DOI:	10.1016/j.jsc.2008.11.003