Constructive approach to develop vectorized and fast algorithms for special-type matrices

Abstract

The paper deals with the development of the approach to the transformation of sequential algorithms (with certain properties) into vectorized ones, which was suggested earlier by the author. Vectorized algorithms are constructed for linear systems with special-type matrices such as Toeplitz, Hankel, Toeplitz-plus-Hankel, Cauchy and Vandermonde matrices. The vectorized algorithms are used for obtaining new fast algorithms for the Hankel and Toeplitz systems, which require that 5η\ο%\η multiplications and lOnlogjn additions be performed for matrices of order n. In all the algorithms, no other assumption but that of the non-singularity of the leading submatrices is imposed on the coefficient matrix.

Fast algorithms for Toeplitz and quasi-Toeplitz Systems

“…Moreover, the results of numerical experiments reveal their satisfactory numerical stability. Such new fast algorithms for Toeplitz linear systems were suggested in [1,2,11,22]. It is worth mentioning that they originated just from the Levinson-type algorithms.…”

“…In [13], a Levmson-type algorithm for solving Toeplitz linear systems was vectorized. It was shown in [22] how a fast algorithm requiring Snlog^n complex multiplications and lOnlog^n complex additions-subtractions could be obtained from the vectorized algorithm. This fast algorithm is quite similar to that suggested in [11] where the Schur -type algorithm was used as a basis for constructing the fast algorithm.…”

“…This fast algorithm is quite similar to that suggested in [11] where the Schur -type algorithm was used as a basis for constructing the fast algorithm. The present paper suggests a modification of the algorithm from [22] whose number of complex multiplications and complex additions-subtractions drops up to fnlog^n and %n\og\n, respectively. In the real case, the computational costs are halved.…”

“…For comparison we recall that the Cooley-Tukey algorithm requires the same number of additions and ^nlog 2 n multiplications. Note that the similar modification can be suggested for Hankel linear systems (the construction of the fast algorithm to solve them can be found in [22]). Note also the fact that problems of solving Toeplitz and Hankel linear systems are to be considered in general as different ones as the assumptions imposed on the coefficient matrices are different.…”