Dr. A.G. Sundelarnd, Dr. E.Y Breitmoser Daresbury
Laboratory, Warrington, UK WA4 4AD
EPCC, University of Edinburgh, UK
Presented by Luis Basurto
Ax=λx
Ax=λBx
A and B dense real (typically symmetric) or Hermitian
matrices
As 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.
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
1.
•
•
•
•
3.
Bisection for the Eigenvalues and inverse iteration for the
Eigenvectors.
QR algorithm
Divide and Conquer method (D&C)
Multiple Relatively Robust Representation (MR3 algorithm)
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 IBM
Provides solutions for both the standard and positive definite
generalized problem.
Solution to both standard and generalized problem
Requires LAPACK and BLAS
No 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 Smith
Developed to test efficiency and not for public use.
Solves real or Hermitian, symmetric or nonsymmetric, dense,
standard Eigenvalue problems.
Written in Fortran 77
Requires 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.