Related papers: Fast computation of the matrix exponential for a T…
We show that every n-by-n matrix is generically a product of [n/2] + 1 Toeplitz matrices and always a product of at most 2n+5 Toeplitz matrices. The same result holds true if the word "Toeplitz" is replaced by "Hankel", and the generic…
Matrices with the structures of Toeplitz, Hankel, Vandermonde and Cauchy types are omnipresent in modern computation. The four classes have distinct features, but in 1990 we showed that Vandermonde and Hankel multipliers transform all these…
Computation of the large sparse matrix exponential has been an important topic in many fields, such as network and finite-element analysis. The existing scaling and squaring algorithm (SSA) is not suitable for the computation of the large…
Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an…
The Erlangian approximation of Markovian fluid queues leads to the problem of computing the matrix exponential of a subgenerator having a block-triangular, block-Toeplitz structure. To this end, we propose some algorithms which exploit the…
Matrix multiplication is a fundamental computation in many scientific disciplines. In this paper, we show that novel fast matrix multiplication algorithms can significantly outperform vendor implementations of the classical algorithm and…
In this paper, we first present an explicit expression for the inverse\emph{} of a type of matrices. As special applications, the inverse of some matrices arising from implicit time integration techniques, such as the well-known implicit…
This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…
By considering an empirical approximation, and a new class of operators that we will call walking operators, we construct, for any positive ND-toeplitz matrix, an infinite in all dimensions matrix, for which the inverse approximates the…
We propose a quadrature-based formula for computing the exponential function of matrices with a non-oscillatory integral on an infinite interval and an oscillatory integral on a finite interval. In the literature, existing quadrature-based…
The exponential of block triangular matrices arises in a wide range of scientific computing applications, including exponential integrators for solving systems of ordinary differential equations, Hamiltonian systems in control theory,…
In several applications, one must estimate a real-valued (symmetric) Toeplitz covariance matrix, typically shifted by the conjugated diagonal matrices of phase progression and phase "calibration" errors. Unlike the Hermitian Toeplitz…
In this paper, we derive a family of fast and stable algorithms for multiplying and inverting $n \times n$ Pascal matrices that run in $O(n log^2 n)$ time and are closely related to De Casteljau's algorithm for B\'ezier curve evaluation.…
Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices $T_{n}(f)$ generated by a function $f$, unfortunately, such a theory is not…
Numeric modeling of electromagnetics and acoustics frequently entails matrix-vector multiplication with block Toeplitz structure. When the corresponding block Toeplitz matrix is not highly sparse, e.g. when considering the electromagnetic…
The results on the inversion of convolution operators and Toeplitz matrices in the 1-D (one dimensional) case are classical and have numerous applications. We consider a 2-D case of Toeplitz-block Toeplitz matrices, describe a minimal…
The computation of matrix functions is a well-studied problem. Of special importance are the exponential and the logarithm of a matrix, where the latter also raises existence and uniqueness questions. This is particularly relevant in the…
A square matrix is $k$-Toeplitz if its diagonals are periodic sequences of period $k$. We find universal formulas for the determinant, the characteristic polynomial, some eigenvectors, and the entries of the inverse of any tridiagonal…
Fast matrix-by-matrix multiplication (hereafter MM) is a highly recognized research subject. The record upper bound 3 of 1968 on the exponent of the complexity MM decreased below 2.38 by 1987, applies to celebrated problems in many areas of…
Many standard conversion matrices between coefficients in classical orthogonal polynomial expansions can be decomposed using diagonally-scaled Hadamard products involving Toeplitz and Hankel matrices. This allows us to derive…