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FAST PARALLELIZABLE METHODS FOR COMPUTING INVARIANT SUBSPACES OF HERMITIAN MATRICES *
"... We propose a quadratically convergent algorithm for computing the invariant subspaces of an Hermitian matrix. Each iteration of the algorithm consists of one matrixmatrix multiplication and one QR decomposition. We present an accurate convergence analysis of the algorithm without using the big O no ..."
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We propose a quadratically convergent algorithm for computing the invariant subspaces of an Hermitian matrix. Each iteration of the algorithm consists of one matrixmatrix multiplication and one QR decomposition. We present an accurate convergence analysis of the algorithm without using the big O notation. We also propose a general framework based on implicit rational transformations which allows us to make connections with several existing algorithms and to derive classes of extensions to our basic algorithm with faster convergence rates. Several numerical examples are given which compare some aspects of the existing algorithms and the new algorithms.
A Parallel Implementation of Symmetric Band Reduction Using PLAPACK
, 1996
"... Successive band reduction (SBR) is a twophase approach for reducing a full symmetric matrix to tridiagonal (or narrow banded) form. In its simplest case, it consists of a fulltoband reduction followed by a bandto tridiagonal reduction. Its richness in BLAS3 operations makes it potentially more ..."
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Successive band reduction (SBR) is a twophase approach for reducing a full symmetric matrix to tridiagonal (or narrow banded) form. In its simplest case, it consists of a fulltoband reduction followed by a bandto tridiagonal reduction. Its richness in BLAS3 operations makes it potentially more efficient on highperformance architectures than the traditional tridiagonalization method. However, a scalable, portable, generalpurpose parallel implementation of SBR is still not available. In this article, we review some existing parallel tridiagonalization routines and describe the implementation of a fulltoband reduction routine using PLAPACK as a first step toward a parallel SBR toolbox. The PLAPACKbased routine turns out to be simple and efficient and, unlike the other existing packages, does not suffer restrictions on physical data layout or algorithmic block size. 1 Introduction Reducing a full, dense symmetric matrix to tridiagonal form is one of the key steps in computing eig...