Abstract. We discuss a new method for the iterative computation of a portion of the singular values and vectors of a large sparse matrix. Similar to the Jacobi–Davidson method for the eigenvalue problem, we compute in each step a correction by (approximately) solving a correction equation. We give a few variants of this Jacobi–Davidson SVD (JDSVD) method with their theoretical properties. It is shown that the JDSVD can be seen as an accelerated (inexact) Newton scheme. We experimentally compare the method with some other iterative SVD methods. Key words. Jacobi–Davidson, singular value decomposition (SVD), singular values, singular vectors, norm, augmented matrix, correction equation, (inexact) accelerated Newton, improving singular values AMS subject classifications. 65F15 (65F35) PII. S1064827500372973

Abstract. We examine the behavior of Newton’s method in floatingpoint arithmetic, allowing for extended precision in computation of the residual, inaccurate evaluation of the Jacobian and unstable solution of the linear systems. We bound the limitingaccuracy and the smallest norm of the residual. The application that motivates this work is iterative refinement for the generalized eigenvalue problem. We show that iterative refinement by Newton’s method can be used to improve the forward and backward errors of computed eigenpairs. Key words. Newton’s method, generalized eigenvalue problem, iterative refinement, Cholesky method, backward error, forward error, roundingerror analysis, limitingaccuracy, limitingresidual AMS subject classifications. 65F15, 65F35 PII. S0895479899359837

For solving the total least squares problems, minE;f k(E; f)kF subject to (A+E)x = b+f , where A is large and sparse or structured Björck suggested a method based on Rayleigh quotient iteration. This method reduces the problem to the solution of a sequence of symmetric, positive definite linear systems of the form (A T A \Gamma ¯ oe 2 I)z = g, where ¯ oe is an approximation to the smallest singular value of (A; b). A preconditioned conjugate gradient method, using a sparse, possibly incomplete, Cholesky factor of A T A can be used for solving these systems. In this paper the method is further developed. The choice of initial approximation and termination criteria are discussed. Numerical results confirm that the method achieves rapid convergence and good accuracy for problems which are not too ill-conditioned.

This research focuses on finding a large number of eigenvalues and eigenvectors of a sparse symmetric or Hermitian matrix, for example, finding 1000 eigenpairs of a 100,000 \Theta 100,000 matrix. These eigenvalue problems are challenging because the matrix size is too large for traditional QR based algorithms and the number of desired eigenpairs is too large for most common sparse eigenvalue algorithms. In this thesis, we approach this problem in two steps. First, we identify a sound preconditioned eigenvalue procedure for computing multiple eigenpairs. Second, we improve the basic algorithm through new preconditioning schemes and spectrum transformations. Through careful analysis, we see that both the Arnoldi and Davidson methods have an appropriate structure for computing a large number of eigenpairs with preconditioning. We also study three variations of these two basic algorithms. Without preconditioning, these methods are mathematically equivalent but they differ in numerical stab...

On modern architectures, the performance of 32-bit operations is often at least twice as fast as the performance of 64-bit operations. By using a combination of 32-bit and 64-bit floating point arithmetic, the performance of many dense and sparse linear algebra algorithms can be significantly enhanced while maintaining the 64-bit accuracy of the resulting solution. The approach presented here can apply not only to conventional processors but also to other technologies such as Field Programmable Gate Arrays (FPGA), Graphical Processing Units (GPU), and the STI Cell BE processor. Results on modern processor architectures and the STI Cell BE are presented. 1