Matrix computations

Regularized Matrix Computations

We review the basic results on: (1) the singular value decomposition (SVD); (2) sensitivity and conditioning of solutions of linear systems of equations; (3) regularization; and (4) iterative solution of linear systems of equations. These are applied to the specific problem of computing a matrix null vector. I. SINGULAR VALUE DECOMPOSITION A. Basics The singular value decomposition (SVD) of any...

Parallel Matrix Computations

In this article we develop some algorithms and tools for solving matrix problems on parallel processing computers. Operations are synchronized through data-flow alone, which makes global synchronization unnecessary and enables the algorithms to be implemented on machines with very simple operating systems and communication protocols. As examples, zve present algorithms that form the main module...

Fast Matrix Polynomial Computations

Pour une matrice polynomiale P(x) de degr e d dans Mn;n(Kx]), o u K est un corps commutatif, une r eduction a la forme normale d'Hermite peut ^ etre calcul ee en O ~ (M(nd)) op erations arithm etiques, si M(n) est le co^ ut du produit de deux matrices n n sur K. De plus, une telle r eduction peut ^ etre obtenue en temps parall ele O(log +1 (nd)) en utilisant O(L(nd)) processeurs, si le probl em...

Visualizing Sparse Matrix Computations

This paper describes ideas and tools that help with the visualization of sparse matrix computations. The paper uses the Sparse Matrix Manipulation System, an environment for handling sparse matrices of all types in a vy flexible manner via ASCII file interfaces. Two commands in this environment are ShowMalrix and ShowTree. The first illustrates the pattern of nonzeroes of a sparse matrix. The s...

Lecture Notes of Matrix Computations