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# Symmetric Eigenvalue Problems: ScaLAPACK Computational Routines

To solve a symmetric eigenproblem with ScaLAPACK, you usually need to reduce the matrix to real tridiagonal form
T
and then find the eigenvalues and eigenvectors of the tridiagonal matrix
T
. ScaLAPACK includes routines for reducing the matrix to a tridiagonal form by an orthogonal (or unitary) similarity transformation
A
=
QTQ
H
as well as for solving tridiagonal symmetric eigenvalue problems. These routines are listed in Table
"Computational Routines for Solving Symmetric Eigenproblems"
.
There are different routines for symmetric eigenproblems, depending on whether you need eigenvalues only or eigenvectors as well, and on the algorithm used (either the
QTQ
algorithm, or bisection followed by inverse iteration).
Computational Routines for Solving Symmetric Eigenproblems
Operation
Dense symmetric/Hermitian matrix
Orthogonal/unitary matrix
Symmetric tridiagonal matrix
Reduce to tridiagonal form
A
=
QTQ
H

Multiply matrix after reduction

Find all eigenvalues and eigenvectors of a tridiagonal matrix
T
by a
QTQ
method

Find selected eigenvalues of a tridiagonal matrix
T
via bisection

Find selected eigenvectors of a tridiagonal matrix
T
by inverse iteration

*
This routine is described as part of auxiliary ScaLAPACK routines.

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