Dr. A.G. Sundelarnd, Dr. E.Y Breitmoser Daresbury Laboratory, Warrington, UK WA4

4ADEPCC, University of Edinburgh, UK

Presented by Luis Basurto

Ax=λx

Ax=λBxA and B dense real (typically symmetric) or

Hermitian matricesAs B is Hermitian positive definite, we can

always express B in terms of Cholesky decomposition, specifically B=LL*, where L is a

lower triangular matrix.

1. Reduction of the matrix to tri-diagonal form, typically using the Housedef Reduction.

2. Solutions of the matrix tri-diagonal Eigenproblem via one of the folliwng methods• Bisection for the Eigenvalues and inverse iteration for the

Eigenvectors.• QR algorithm• Divide and Conquer method (D&C)• Multiple Relatively Robust Representation (MR3

algorithm)

3. Back substitution to find Eigenvectors for the full problem.

•Jacobi method•Symmetric Invariant Subspace Decomposition Algorithm (SYSDA)

•Complexity•Overheads•Parallelisation Issues•The Nonsymmetric Eigenvalue problem

Memory requirements O(n2)Large workspace requirement for orthogonal Eigenvectors

Provided by IBMProvides solutions for both the standard and positive definite generalized problem.

Solution to both standard and generalized problemRequires LAPACK and BLASNo longer under active development

Solves standard dense symmetric, real and Hermitian Eigenproblems.Requires BLAS and MPI.Is an iterative method.

Solves for dense symmetric standard problem.Written in Fortran and C.Requires BLAS and MPI.MR3 algorithm requires O(n2) operations and O(n) workspace.

Solves for large symmetric and nonsymmetric, generalized Eigenproblems.Written in Fortran 77.Requires BLACS ans MPI.Software available for both serial and parallel version.

Solves for dense symmetric Eigensystems.Requires LAPACK, BLAS and MPI.Written in C, no Fortran interface available.

Eigensolver for symmetric Hermitian for the standard problem.Size can be from n=2 to 8192.Algorithm is a hybrid of Jacobi methods.

Solves real, dense symmetric matrices.Focuses on the reduction to tri-diagonal form and back-transformation of Eigenvectors.Developed by Hendrickson, Jessup and SmithDeveloped to test efficiency and not for public use.

Solves real or Hermitian, symmetric or nonsymmetric, dense, standard Eigenvalue problems.Written in Fortran 77Requires BLACS, BLAS and MPI.No longer available, now included in NAG parallel library.

ScaLAPACK and PESSL available for all users.PeIGS, BFG, PLAPACK and PARPACK on request.PRSIM and PJAC not currently available.NAG (PINEAPL) not available.HJS not available, never intended for public use.

All parallel Eigensolvers presented are for dense systems, except PARPACK which is only suitable for sparse systems.ScaLAPACK solves many kinds of dense Eigenvalues problems.PESSL can be used also, as it is optimised for IBM.Progess should be monitored for the new MR3 algorithm in PLAPACK.