This paper is the result of contrived efforts to break the barrier between numerical accuracy and run time efficiency in computing the fundamental decomposition of numerical linear algebra – the singular value decomposition (SVD) of a general dense matrix. It is an unfortunate fact that the numerically most accurate one–sided Jacobi SVD algorithm is several times slower than generally less accu...

We consider the problem of computing reachability probabilities: given a Markov chain, an initial state of the Markov chain, and a set of goal states of the Markov chain, what is the probability of reaching any of the goal states from the initial state? This problem can be reduced to solving a linear equation A · x = b for x, where A is a matrix and b is a vector. We consider two iterative meth...

In this article, we compare different iterative techniques enhanced by the CBFM, which are used to analyze finite arrays of disjoint antenna elements. These based on stationary-type methods (Jacobi, Gauss–Seidel, and macroblock Jacobi), nonstationary GMRES, hybrid alternating GMRES-Jacobi (AGJ) method that combines these two types. each iteration, reduced CBFM system is constructed previous ite...

In this work we solve the problem of the minimization of the spectral norm of the SOR operator associated with a block two-cyclic consistently ordered matrix A ∈ C, assuming that the corresponding Jacobi matrix has eigenvalues μ ∈ [−β, β] ∪ [−ıα, ıα], with β ∈ [0, 1), α ∈ [0,+∞) and ı = √ −1. Previous results obtained by other researchers are extended.

Abstract This paper studies the inverse eigenvalue problem for an arrow-shaped generalised Jacobi matrix, inverting matrices through two eigen-pairs. In paper, existence and uniqueness of solution to are discussed, mathematical expressions as well a numerical example given. Finally, theorem its matrix is established by derivation.

The serial Jacobi algorithm (either one-sided or two-sided) for the computation of a singular value decomposition (SVD) of a general matrix has excellent numerical properties and parallelization potential, but it is considered to be the slowest method for computing the SVD. Even its parallelization with some parallel cyclic (static) ordering of subproblems does not lead to much improvement when...

It is known that interval Newton methods can verify existence and uniqueness of solutions of a nonlinear system of equations near points where the Jacobi matrix of the system is not ill-conditioned. Recently, we have shown how to verify existence and uniqueness, up to multiplicity, for solutions at which the Jacobi matrix is singular. We do this by efficient computation of the topological index...

This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and convergence of Padé approximants appears as the strong resolvent convergence of finite matrix approximations to a Jacobi matrix. As a bonus of this, we o...

Solving a system of linear and nonlinear equations lies at the heart of many scientific and engineering applications such as circuit simulation, applications in electric power networks, and structural analysis. The exponentially increasing complexity of these computing applications and the high cost of supercomputing force us to explore affordable high performance computing platforms. The ultim...

A coupled pair of harmonic equations is solved by the application of Chebyshev acceleration to the Jacobi, Gauss-Seidel, and related iterative methods, where the Jacobi iteration matrix has purely imaginary (or zero) eigenvalues. Comparison is made with a block SOR method used to solve the same problem. Introduction. In [4], we proposed a general block SOR method for solving the biharmonic equa...