A proposal for toeplitz matrix calculations
Studies in Applied Mathematics
Algebraic computations of scaled pade´ fractions
SIAM Journal on Computing
Superfast solution of real positive definite toeplitz systems
SIAM Journal on Matrix Analysis and Applications
The inverses of block Hankel and block Toeplitz matrices
SIAM Journal on Computing
Subresultants and Reduced Polynomial Remainder Sequences
Journal of the ACM (JACM)
On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors
Journal of the ACM (JACM)
On Euclid's Algorithm and the Theory of Subresultants
Journal of the ACM (JACM)
The Subresultant PRS Algorithm
ACM Transactions on Mathematical Software (TOMS)
Symbolic Computation of Padé Approximants
ACM Transactions on Mathematical Software (TOMS)
Fraction-free computation of matrix Padé systems
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
GCD of polynomials and Bezout matrices
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Algorithms for computing the sparsest shifts of polynomials via the Berlekamp/Massey algorithm
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Computing high precision Matrix Padé approximants
Numerical Algorithms
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For univariate power series with coefficients over an integral domain, the notions of power series pseudo-division and power series remainder sequences are introduced and developed. By appealing to the theory of resultants, an algorithm is developed for constructing power series remainder sequences in which the sizes of the coefficients of successive remainders grow linearly, only. The algorithm computes also a co-factor sequence, which is shown to be associated with a given power series remainder sequence. The co-factor sequence yields directly all the Pade fractions along some specific off-diagonal path of the Pade table for a pair of power series (A(z), B(z)). When the sizes of the coefficients of A(z) and B(z) are bounded by k, in the normal case, the algorithm computes Pade fractions of type (m,n) in time O(k^2(m+n)^4). In the abnormal case, and depending on the nature of abnormalities, the time complexity can reach O(k^2(m+n)^5). For computing Pade fractions, the algorithm compares favorably with fraction-free methods, which have a time complexity of O(k^2(m+n)^5), even in the normal case. When applied to polynomials, rather than to power series, the algorithm, for one specific off-diagonal path, corresponds to Euclid's extended algorithm for computing the greatest common divisor of two polynomials.