Advanced Numerical Methods in Applied Sciences
Dublin Core
Title
Advanced Numerical Methods in Applied Sciences
Subject
Numerical Methods
Description
The generalized Schur algorithm (GSA) allows computing well-known matrix decompositions,
such as QR and LU factorizations [1]. In particular, if the involved matrix is structured, i.e., Toeplitz, block-Toeplitz or Sylvester, the GSA computes the R factor of the QR factorization with complexity of one order of magnitude less than that of the classical QR algorithm [2], since it relies only on the knowledge of the so-called generators [2] associated to the given matrix, rather than on the knowledge of the matrix itself. The stability properties of the GSA are described in [3–5], where it is proven that the algorithm is weakly stable provided the involved hyperbolic rotations are performed in a stable way. In this manuscript, we first show that, besides the efficiency properties, the GSA provides new theoretical insights on the bounds of the entries of the R factor of the QR factorization of some structured matrices
such as QR and LU factorizations [1]. In particular, if the involved matrix is structured, i.e., Toeplitz, block-Toeplitz or Sylvester, the GSA computes the R factor of the QR factorization with complexity of one order of magnitude less than that of the classical QR algorithm [2], since it relies only on the knowledge of the so-called generators [2] associated to the given matrix, rather than on the knowledge of the matrix itself. The stability properties of the GSA are described in [3–5], where it is proven that the algorithm is weakly stable provided the involved hyperbolic rotations are performed in a stable way. In this manuscript, we first show that, besides the efficiency properties, the GSA provides new theoretical insights on the bounds of the entries of the R factor of the QR factorization of some structured matrices
Creator
Brugnano, Luigi, - Iavernaro, Felice
Source
https://www.mdpi.com/books/pdfview/book/1360
Publisher
MDPI - Multidisciplinary Digital Publishing Institute
Date
2019
Contributor
Baihaqi
Rights
Creative Commons
Format
Ebooks
Language
English
Type
Textbooks
Files
Collection
Citation
Brugnano, Luigi, - Iavernaro, Felice , “Advanced Numerical Methods in Applied Sciences,” Open Educational Resource (OER) - USK Library, accessed April 24, 2025, http://202.4.186.74:8004/oer/items/show/2965.