266 research outputs found
A weakly stable algorithm for general Toeplitz systems
We show that a fast algorithm for the QR factorization of a Toeplitz or
Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A.
Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx =
A^Tb, we obtain a weakly stable method for the solution of a nonsingular
Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the
solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further
details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm
Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections
Block projections have been used, in [Eberly et al. 2006], to obtain an
efficient algorithm to find solutions for sparse systems of linear equations. A
bound of softO(n^(2.5)) machine operations is obtained assuming that the input
matrix can be multiplied by a vector with constant-sized entries in softO(n)
machine operations. Unfortunately, the correctness of this algorithm depends on
the existence of efficient block projections, and this has been conjectured. In
this paper we establish the correctness of the algorithm from [Eberly et al.
2006] by proving the existence of efficient block projections over sufficiently
large fields. We demonstrate the usefulness of these projections by deriving
improved bounds for the cost of several matrix problems, considering, in
particular, ``sparse'' matrices that can be be multiplied by a vector using
softO(n) field operations. We show how to compute the inverse of a sparse
matrix over a field F using an expected number of softO(n^(2.27)) operations in
F. A basis for the null space of a sparse matrix, and a certification of its
rank, are obtained at the same cost. An application to Kaltofen and Villard's
Baby-Steps/Giant-Steps algorithms for the determinant and Smith Form of an
integer matrix yields algorithms requiring softO(n^(2.66)) machine operations.
The derived algorithms are all probabilistic of the Las Vegas type
Representations of Toeplitz-plus-Hankel matrices using trigonometric transformations with application to fast matrix-vector multiplication
AbstractRepresentations of real Toeplitz and Toeplitz-plus-Hankel matrices are presented that involve real trigonometric transformations (DCT, DST, DHT) and diagonal matrices. These representations can be used for fast matrix-vector multiplication. In particular, it is shown that the multiplication of an n × n Toeplitz-plus-Hankel matrix by a vector requires only 4 transformations of length n plus O(n) operations
Matrix algebras and displacement decompositions
A class xi of algebras of symmetric nxn matrices, related to Toeplitz-plus-Hankel structures and including the well-known algebra H diagonalized by the Hartley transform, is investigated. The algebras of xi are then exploited in a general displacement decomposition of an arbitrary nxn matrix A. Any algebra of xi is a 1-space, i.e., it is spanned by n matrices having as first rows the vectors of the canonical basis. The notion of 1-space (which generalizes the previous notions of L1 space [Bevilacqua and Zellini, Linear and Multilinear Algebra, 25 (1989), pp.1-25] and Hessenberg algebra [Di Fiore and Zellini, Linear Algebra Appl., 229 (1995), pp.49-99]) finally leads to the identification in xi of three new (non-Hessenberg) matrix algebras close to H, which are shown to be associated with fast Hartley-type transforms. These algebras are also involved in new efficient centrosymmetric Toeplitz-plus-Hankel inversion formulas
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