Fast Algorithms with Preprocessing for Matrix-Vector Multiplication Problems

Abstract

In this paper the problem of complexity of multiplication of a matrix with a vector is studied for Toeplitz, Hankel, Vandermonde and Cauchy matrices and for matrices connected with them (i.e. for transpose, inverse and transpose to inverse matrices). The proposed algorithms have complexities at most O(n log 2 n) ops and in a number of cases improve the known estimates. In these algorithms, in a separate preprocessing phase, are singled out all the actions on the preparation of a given matrix, which aimed at the reduction of the complexity of the second stage of computations directly connected with the multiplication by an arbitrary vector. Incidentally, the e ective algorithms for computing the Vandermonde determinant and the determinant of a Cauchy matrix, are given.

Matrix Algebras and Displacement Decompositions

“…In this paper we further extend the results of [6,9,10,16,17,18,20,21,22,28,32] in the sense that we introduce a new class of matrix algebras L, including Hessenberg and other algebras of matrices diagonalized by means of Hartley [11,12] or Hartleytype transforms, which have not been yet considered in displacement literature. This extension requires the study of matrix algebras containing the matrix T β,β ε,ϕ displayed at the beginning of section 2.…”

“…Under the assumption that the vectors a and b are known, formula (1.2) lets one calculate the matrix-vector product (T + H) −1 f , f ∈ C n , by means of 10 fast discrete transforms reducible to 8 in case H = 0, [T −1 ] 11 = 0, matching both best limits known so far [1,10,16]. In any case, the number of transforms reduces to 6 (as in [1,10,16,21,22]) if the transforms of vectors not depending upon f are included in the preprocessing stage, where a and b are computed.…”

“…This distinction is justified when many different linear systems (T + H)x = f i have to be solved. The same point of view is assumed by Gohberg and Olshevsky in [21,22], where the complexity of the computation of Af with preprocessing on A is studied for different types of structured matrices A, including the case A = T −1 for a generic Toeplitz T . (Some results on the complexity of the preprocessing stage are also given in [21,22].)…”

“…The same point of view is assumed by Gohberg and Olshevsky in [21,22], where the complexity of the computation of Af with preprocessing on A is studied for different types of structured matrices A, including the case A = T −1 for a generic Toeplitz T . (Some results on the complexity of the preprocessing stage are also given in [21,22].) In particular, they show that the application of T −1 to the vector f can be accomplished with a cost of 6 DFTs of order n and thus generalize the analogous result obtained by Ammar and Gader in the Hermitian case [1].…”

“…These transforms are discrete Fourier transforms (DFT) in cases of formulas involving triangular Toeplitz or ε-circulant matrices [1,9,17,21,22] and are sine or cosine transforms in cases of formulas involving τ or τ ε,ϕ matrices [6,10,16,17,32], and therefore they are all associated with Hessenberg algebras [16].…”