Computing generalized inverses using LU factorization of matrix product

An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4} -inverses and the Moore-Penrose inverse of a given rational matrix A is established. Classes A(2, 3)s and A(2, 4)s are characterized in terms of matrix products (R*A)+R* and T*(AT*)+, where R and T are rational matrices with appropriate dimensions and corresponding rank. The proposed algorithm is based on these general representations and the Cholesky factorization of symmetric positive matrices. The algorithm is implemented in programming languages MATHEMATICA and DELPHI, and illustrated via examples. Numerical results of the algorithm, corresponding to the Moore-Penrose inverse, are compared with corresponding results obtained by several known methods for computing the Moore-Penrose inverse

A Computational Framework for the Mixing Times in the QBD Processes with Infinitely-Many Levels

In this paper, we develop some matrix Poisson's equations satisfied by the mean and variance of the mixing time in an irreducible positive-recurrent discrete-time Markov chain with infinitely-many levels, and provide a computational framework for the solution to the matrix Poisson's equations by means of the UL-type of $RG$-factorization as well as the generalized inverses. In an important special case: the level-dependent QBD processes, we provide a detailed computation for the mean and variance of the mixing time. Based on this, we give new highlight on computation of the mixing time in the block-structured Markov chains with infinitely-many levels through the matrix-analytic method

Sparse approximate inverse preconditioners on high performance GPU platforms

Simulation with models based on partial differential equations often requires the solution of (sequences of) large and sparse algebraic linear systems. In multidimensional domains, preconditioned Krylov iterative solvers are often appropriate for these duties. Therefore, the search for efficient preconditioners for Krylov subspace methods is a crucial theme. Recent developments, especially in computing hardware, have renewed the interest in approximate inverse preconditioners in factorized form, because their application during the solution process can be more efficient. We present here some experiences focused on the approximate inverse preconditioners proposed by Benzi and Tůma from 1996 and the sparsification and inversion proposed by van Duin in 1999. Computational costs, reorderings and implementation issues are considered both on conventional and innovative computing architectures like Graphics Programming Units (GPUs)

Minimizing Communication for Eigenproblems and the Singular Value Decomposition

Algorithms have two costs: arithmetic and communication. The latter represents the cost of moving data, either between levels of a memory hierarchy, or between processors over a network. Communication often dominates arithmetic and represents a rapidly increasing proportion of the total cost, so we seek algorithms that minimize communication. In \cite{BDHS10} lower bounds were presented on the amount of communication required for essentially all $O(n^3)$-like algorithms for linear algebra, including eigenvalue problems and the SVD. Conventional algorithms, including those currently implemented in (Sca)LAPACK, perform asymptotically more communication than these lower bounds require. In this paper we present parallel and sequential eigenvalue algorithms (for pencils, nonsymmetric matrices, and symmetric matrices) and SVD algorithms that do attain these lower bounds, and analyze their convergence and communication costs.Comment: 43 pages, 11 figure

Fast Algorithms for Displacement and Low-Rank Structured Matrices

This tutorial provides an introduction to the development of fast matrix algorithms based on the notions of displacement and various low-rank structures

Application-tailored Linear Algebra Algorithms: A search-based Approach

In this paper, we tackle the problem of automatically generating algorithms for linear algebra operations by taking advantage of problem-specific knowledge. In most situations, users possess much more information about the problem at hand than what current libraries and computing environments accept; evidence shows that if properly exploited, such information leads to uncommon/unexpected speedups. We introduce a knowledge-aware linear algebra compiler that allows users to input matrix equations together with properties about the operands and the problem itself; for instance, they can specify that the equation is part of a sequence, and how successive instances are related to one another. The compiler exploits all this information to guide the generation of algorithms, to limit the size of the search space, and to avoid redundant computations. We applied the compiler to equations arising as part of sensitivity and genome studies; the algorithms produced exhibit, respectively, 100- and 1000-fold speedups

Computation of generalized inverses by using the LDL∗ decomposition

AbstractAn efficient algorithm, based on the LDL∗ factorization, for computing {1,2,3} and {1,2,4} inverses and the Moore–Penrose inverse of a given rational matrix A, is developed. We consider matrix products A∗A and AA∗ and corresponding LDL∗ factorizations in order to compute the generalized inverse of A. By considering the matrix products (R∗A)†R∗ and T∗(AT∗)†, where R and T are arbitrary rational matrices with appropriate dimensions and ranks, we characterize classes A{1,2,3} and A{1,2,4}. Some evaluation times for our algorithm are compared with corresponding times for several known algorithms for computing the Moore–Penrose inverse