Sparse Polynomial Interpolation in Nonstandard Bases

In this paper, we consider the problem of interpolating univariate polynomials over a field of characteristic zeros 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 p...

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Veröffentlicht in:SIAM journal on computing Jg. 24; H. 2; S. 387 - 397
Hauptverfasser: Lakshman, Y. N., Saunders, B. David
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia, PA Society for Industrial and Applied Mathematics 01.04.1995
Schlagworte:
Algorithms
Codes
Exact sciences and technology
Mathematics
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis. Scientific computation
Polynomials
Sciences and techniques of general use
Sparse polynomial
BCH code
Tchebychev polynomial
Polynomial interpolation
Algorithm complexity
ISSN:0097-5397, 1095-7111
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Abstract In this paper, we consider the problem of interpolating univariate polynomials over a field of characteristic zeros 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 Tiwari's (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 $O(t^{2} + t \log d)$, 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 $2t$ evaluation points.
AbstractList In this paper, we consider the problem of interpolating univariate polynomials over a field of characteristic zeros 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 Tiwari's (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 $O(t^{2} + t \log d)$, 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 $2t$ evaluation points.
In this paper, we consider the problem of interpolating univariate polynomials over a field of characteristic zeros 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 Tiwari's (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 O(t super(2) + t log d), 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 2t evaluation points.
Author Saunders, B. David
Lakshman, Y. N.
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Cites_doi 10.1016/0196-8858(91)90005-4
10.1007/3-540-56686-4_41
10.1137/S0097539791194069
10.1016/S0747-7171(08)80015-6
10.1137/0212017
10.1016/S0747-7171(08)80018-1
10.1016/0196-8858(91)90004-3
10.1137/0219073
10.1007/3-540-51084-2_44
10.1109/TASSP.1982.1163845
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BCH code
Tchebychev polynomial
Polynomial interpolation
Algorithm complexity
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  doi: 10.1016/0196-8858(91)90005-4
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SubjectTerms Algorithms
Codes
Exact sciences and technology
Mathematics
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis. Scientific computation
Polynomials
Sciences and techniques of general use
Title Sparse Polynomial Interpolation in Nonstandard Bases
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