An extended set of FORTRAN basic linear algebra subprograms
ACM Transactions on Mathematical Software (TOMS)
A set of level 3 basic linear algebra subprograms
ACM Transactions on Mathematical Software (TOMS)
Accurate singular values of bidiagonal matrices
SIAM Journal on Scientific and Statistical Computing
Using MPI: portable parallel programming with the message-passing interface
Using MPI: portable parallel programming with the message-passing interface
A Jacobi--Davidson Iteration Method for Linear EigenvalueProblems
SIAM Journal on Matrix Analysis and Applications
Practical experience in the numerical dangers of heterogeneous computing
ACM Transactions on Mathematical Software (TOMS)
Using PLAPACK: parallel linear algebra package
Using PLAPACK: parallel linear algebra package
Computing an Eigenvector with Inverse Iteration
SIAM Review
The symmetric eigenvalue problem
The symmetric eigenvalue problem
A new O (N(2)) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem
A new O (N(2)) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem
Unconstrained energy functionals for electronic structure calculations
Journal of Computational Physics
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
SIAM Journal on Scientific Computing
Parallel empirical pseudopotential electronic structure calculations for million atom systems
Journal of Computational Physics
Basic Linear Algebra Subprograms for Fortran Usage
ACM Transactions on Mathematical Software (TOMS)
For tridiagonals T replace T with LDLt
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. III: linear algebra
PLAPACK: parallel linear algebra package design overview
SC '97 Proceedings of the 1997 ACM/IEEE conference on Supercomputing
MPI: The Complete Reference
An updated set of basic linear algebra subprograms (BLAS)
ACM Transactions on Mathematical Software (TOMS)
SIAM Journal on Scientific Computing
LAPACK Working Note 94: A User''s Guide to the BLACS v1.0
LAPACK Working Note 94: A User''s Guide to the BLACS v1.0
A Proposal for a Set of Parallel Basic Linear Algebra Subprograms
A Proposal for a Set of Parallel Basic Linear Algebra Subprograms
LAPACK Working Note 37: Two Dimensional Basic Linear Algebra Communication Subprograms
LAPACK Working Note 37: Two Dimensional Basic Linear Algebra Communication Subprograms
Computing Approximate Eigenpairs of Symmetric Block Tridiagonal Matrices
SIAM Journal on Scientific Computing
Orthogonal Eigenvectors and Relative Gaps
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific Computing
Glued Matrices and the MRRR Algorithm
SIAM Journal on Scientific Computing
The design and implementation of the MRRR algorithm
ACM Transactions on Mathematical Software (TOMS)
A parallel symmetric block-tridiagonal divide-and-conquer algorithm
ACM Transactions on Mathematical Software (TOMS)
Algorithm 880: A testing infrastructure for symmetric tridiagonal eigensolvers
ACM Transactions on Mathematical Software (TOMS)
Performance and Accuracy of LAPACK's Symmetric Tridiagonal Eigensolvers
SIAM Journal on Scientific Computing
The Spectrum of a Glued Matrix
SIAM Journal on Matrix Analysis and Applications
Implementation and tuning of a parallel symmetric Toeplitz eigensolver
Journal of Parallel and Distributed Computing
MR3-SMP: A symmetric tridiagonal eigensolver for multi-core architectures
Parallel Computing
The algorithm of multiple relatively robust representations for multi-core processors
PARA'10 Proceedings of the 10th international conference on Applied Parallel and Scientific Computing - Volume Part I
Divide and Conquer on Hybrid GPU-Accelerated Multicore Systems
SIAM Journal on Scientific Computing
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The (sequential) algorithm of Multiple Relatively Robust Representations, MRRR, is a more efficient variant of inverse iteration that does not require reorthogonalization. It solves the eigenproblem of an unreduced symmetric tridiagonal matrix T ∈ Rn × n at O(n2) cost. The computed normalized eigenvectors are numerically orthogonal in the sense that the dot product between different vectors is O (n ε), where ε refers to the relative machine precision. This article describes the design of ScaLAPACK's parallel MRRR algorithm. One emphasis is on the critical role of the representation tree in achieving both adequate accuracy and parallel scalability. A second point concerns the favorable properties of this code: subset computation, the use of static memory, and scalability. Unlike ScaLAPACK's Divide & Conquer and QR, MRRR can compute subsets of eigenpairs at reduced cost. And in contrast to inverse iterations which can fail, it is guaranteed to produce a satisfactory answer while maintaining memory scalability. ParEig, the parallel MRRR algorithm for PLAPACK, uses dynamic memory allocation. This is avoided by our code at marginal additional cost. We also use a different representation tree criterion that allows for more accurate computation of the eigenvectors but can make parallelization more difficult.