Vjeran Hari

A note on one-sided Jacobi algorithm

SummaryWe propose a “one-sided” or “implicit” variant of the Jacobi diagonalization algorithm for positive definite matrices. The variant is based on a previous Cholesky decomposition and currently uses essentially one square array which, on output, contains the matrix of eigenvectors thus reaching the storage economy of the symmetric QL algorithm. The current array is accessed only columnwise which makes the algorithm attractive for various parallelized and/or vectorized implementations. Even on a serial computer our algorithm shows improved efficiency, in particular if the Cholesky step is made with diagonal pivoting. On matrices of ordern=25–200 our algorithm is about 2–3.5 times slower than QL thus being almost on the halfway between the standard Jacobi and QL algorithms. The previous Cholesky decomposition can be performed with higher precision without extra time or storage costs thus offering considerable gains in accuracy with highly conditioned input matrices.

On sharp quadratic convergence bounds for the serial Jacobi methods

SummaryUsing a new technique we derive sharp quadratic convergence bounds for the serial symmetric and SVD Jacobi methods. For the symmetric Jacobi method we consider the cases of well and poorely separated eigenvalues. Our result implies the result proposed, but not correctly proved, by Van Kempen. It also extends the well-known result of Wilkinson to the case of multiple eigenvalues.

On Jacobi methods for singular value decompositions

An improvement of the Jacobi singular value decomposition algorithm is proposed. The matrix is first reduced to a triangular form. It is shown that the row-cyclic strategy preserves the triangularity. Further improvements lie in the convergence properties. It is shown that the method converges globally and a proof of the quadratic convergence is indicated as well. The numerical experiments confirm these theoretical predictions. Our method is about 2-3 times slower than the standard QR method but it almost reaches the latter if the matrix is diagonally dominant or of low rank.

Accelerating the SVD Block-Jacobi Method

Abstract.The paper discusses how to improve performance of the one-sided block-Jacobi algorithm for computing the singular value decomposition of rectangular matrices. In particular, it is shown how cosine-sine decomposition of orthogonal matrices can be used to accelerate the slowest part of the algorithm – updating the block-columns.

On Scaled Almost-Diagonal Hermitian Matrix Pairs

This paper contains estimates concerning the block structure of Hermitian matrices H and M, which make a scaled diagonally dominant definite pair. The obtained bounds are expressed in terms of relative gaps in the spectrum of the pair (H,M) and norms of certain blocks of the matrices DHD and DMD, where D is either

On the global and cubic convergence of a quasi-cyclic Jacobi method

[|\mbox{diag(H)}|]^{-{1}/{2}}

On quadratic convergence bounds for theJ-symmetric Jacobi method

or

On the global convergence of the Eberlein method for real matrices

[\mbox{diag(M)}]^{-{1}/{2}}

Applied Mathematics and Scientific Computing

. If either of the matrices H, M is diagonal, the new results assume simple and applicable form. For scaled diagonally dominant Hermitian matrices, the new estimates compare favorably with the existing ones for accurate location of the smallest eigenvalues.

Relative Residual Bounds For The Eigenvalues of a Hermitian Semidefinite Matrix

SummaryIn this paper we consider the global and the cubic convergence of a quasi-cyclic Jacobi method for the symmetric eigenvalue, problem. The method belongs to a class of quasi-cyclic methods recently proposed by W. Mascarenhas. Mascarenhas showed that the methods from his class asymptotically converge cubically per quasi-sweep (one quasi-sweep is equivalent to 1.25 cyclic sweeps) provided the eigenvalues are simple. Here we prove the global convergence of our method and derive very sharp asymptotic convergence bounds in the general case of multiple eigenvalues. We discuss the ultimate cubic convergence of the method and present several numerical examples which all well comply with the theory.