An extended split-radix fast Fourier transform (FFT) algorithm is proposed. The extended split-radix FFT algorithm hits the same asymptotic arithmetic complexity as the conventional split-radix FFT algorithm. Moreover...
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An extended split-radix fast Fourier transform (FFT) algorithm is proposed. The extended split-radix FFT algorithm hits the same asymptotic arithmetic complexity as the conventional split-radix FFT algorithm. Moreover, this algorithm has the advantage of fewer loads and stores than either the conventional split-radix FFT algorithm or the radix-4 FFT algorithm.
We study asymptotically fast multiplication algorithms for matrix pairs of arbitrary dimensions, and optimize the exponents of their arithmeticcomplexity bounds. For a large class of input matrix pairs, we improve th...
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We study asymptotically fast multiplication algorithms for matrix pairs of arbitrary dimensions, and optimize the exponents of their arithmeticcomplexity bounds. For a large class of input matrix pairs, we improve the known exponents. We also show some applications of our results: (i) we decrease from O(n 2 + n 1+o(1)logq) to O(n 1.9998 + n 1+o(1)logq) the known arithmeticcomplexity bound for the univariate polynomial factorization of degree n over a finite field with q elements; (ii) we decrease from 2.837 to 2.7945 the known exponent of the work and arithmetic processor bounds for fast deterministic (NC) parallel evaluation of the determinant, the characteristic polynomial, and the inverse of an n × n matrix, as well as for the solution to a nonsingular linear system of n equations; (iii) we decrease from O(m 1.575 n) to O(m 1.5356 n) the known bound for computing basic solutions to a linear programming problem with m constraints and n variables.
First we study asymptotically fast algorithms for rectangular matrix multiplication. We begin with new algorithms for multiplication of an n x n matrix by an n x n(2) matrix in arithmetic time O(n(omega)), omega = 3.3...
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First we study asymptotically fast algorithms for rectangular matrix multiplication. We begin with new algorithms for multiplication of an n x n matrix by an n x n(2) matrix in arithmetic time O(n(omega)), omega = 3.333953..., which is less by 0.041 than the previous record 3.375477.... Then we present fast multiplication algorithms for matrix pairs of arbitrary dimensions, estimate the asymptotic running time as a function of the dimensions, and optimize the exponents of the complexity estimates. For a large class of input matrix pairs, we improve the known exponents. Finally we show three applications of our results: (a) we decrease from 2.851 to 2.837 the known exponent of *** bounds for fast deterministic (NC) parallel evaluation of the determinant, the characteristic polynomial, and the inverse of an n x n matrix, as well as for the solution to a nonsingular linear system of n equations, (b) we asymptotically accelerate the known sequential algorithms for the univariate polynomial composition mod x(n), yielding the complexity bound O(n(1.667)) versus the old record of O(n(1.688)), and for the univariate polynomial factorization over a finite field, and (c) we improve slightly the known complexity estimates for computing basic solutions to the linear programming problem with n constraints and n variables. (C) 1998 Academic Press.
The partial fraction expansion problem and its inverse are studied and it is shown that these two problems can be solved in O(Nlog2<span style="display: i
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