37 research outputs found
Padé and Gregory error estimates for the logarithm of block triangular matrices
In this paper we give bounds for the error arising in the approximation of the logarithm of a block triangular matrix T by Padé approximants of the function f(x)=log[(1+x)/(1-x)] and partial sums of Gregory's series. These bounds show that if the norm of all diagonal blocks of the Cayley-transform B=(T-I)(T+I)-1 is sufficiently close to zero, then both approximation methods are accurate. This will contribute for reducing the number of successive square roots of T needed in the inverse scaling and squaring procedure for the matrix logarithm.http://www.sciencedirect.com/science/article/B6TYD-4G5BJ9P-2/1/398a212a906943d2474a2cd6166c1d3
Two Taylor Algorithms for Computing the Action of the Matrix Exponential on a Vector
Ibáñez González, JJ.; Alonso Abalos, JM.; Alonso-Jordá, P.; Defez Candel, E.; Sastre, J. (2022). Two Taylor Algorithms for Computing the Action of the Matrix Exponential on a Vector. Algorithms. 15(2):1-48. https://doi.org/10.3390/a1502004814815
Faster Sparse Matrix Inversion and Rank Computation in Finite Fields
We improve the current best running time value to invert sparse matrices over
finite fields, lowering it to an expected time for the
current values of fast rectangular matrix multiplication. We achieve the same
running time for the computation of the rank and nullspace of a sparse matrix
over a finite field. This improvement relies on two key techniques. First, we
adopt the decomposition of an arbitrary matrix into block Krylov and Hankel
matrices from Eberly et al. (ISSAC 2007). Second, we show how to recover the
explicit inverse of a block Hankel matrix using low displacement rank
techniques for structured matrices and fast rectangular matrix multiplication
algorithms. We generalize our inversion method to block structured matrices
with other displacement operators and strengthen the best known upper bounds
for explicit inversion of block Toeplitz-like and block Hankel-like matrices,
as well as for explicit inversion of block Vandermonde-like matrices with
structured blocks. As a further application, we improve the complexity of
several algorithms in topological data analysis and in finite group theory
Recent Advances in Industrial and Applied Mathematics
This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress