SSL II USER'S GUIDE - Lahey Computer Systems

CHAPTER 4
EIGENVALUES AND EIGENVECTORS
4.1 OUTLINE
Eigenvalue problems can be organized as show in Table
4.1 according to the type of problem (Ax =λx, Ax =λBx)
and the shape (dense, band), type (real, complex), and
form (symmetric, unsymmetric) of the matrices. The
reader should refer to the appropriate section specified
in the table.
Table 4.1 Organization of eigenvalue problem
Shape of
matrix
Dense
matrix
Band
matrix
Type of
problem
Matrix type and form
Ax =λx Real matrix
Complex matrix
Real symmetric matrix
Hermitian matrix
Explanation
section
4.2
4.3
4.4
4.5
Ax =λBx Real symmetric matrix 4.7
Ax =λx Real symmetric band
matrix
4.6
Ax =λBx Real symmetric band
matrix
4.8
Note:
Refer to the section on a real symmetric matrix concerning a
real symmetric tridiagonal matrix.
Table 4.2 Subroutines used for standard eigenproblem of a real matrix
4.2 EIGENVALUES AND
EIGENVECTORS OF A REAL MATRIX
A standard sequenses of procedures are shown here
when SSL II routines are used to solve the eigenvalue
problems.
SSL II provides the following:
− Standard routines by which the entire procedures for
obtaining eigenvalues and eigenvectors of real
matrices may be performed at one time.
− Component routines performing component functions.
For details, see Table 4.2.
User problems are classified as follows :
• Obtaining all eigenvalues
• Obtaining all eigenvalues and corresponding
eigenvectors (or selected eigenvectors)
In the first and second items that follow, the use of
component routines and standard routines is explained
by describing their procedures. Further comments on
processing are in the third item.
When obtaining eigenvalues and eigenvectors of a real
matrix, the choice of calling the various component
routines or calling the standard routine is up to the user.
However, the standard routine, which is easier to use, is
recommended to be called.
Level Function Subroutine name
Standard routines Eigenvalues and eigenvectors of a real matrix
EIG1
(B21-11-0101)
Balancing of a real matrix
BLNC
(B21-11-0202)
Reduction of a real matrix to a Hessenberg matrix
HES1
(B21-11-0302)
Component
Obtaining the eigenvalues of a real Hessenberg
matrix
HSQR
(B21-11-0402)
routines
Obtaining the eigenvectors of a real Hessenberg
matrix
HVEC
(B21-11-0502)
Back transformation to and normalization of the
HBK1
eigenvectors of a real matrix
(B21-11-0602)
Normalization of the eigenvectors of a real matrix
NRML
(B21-11-0702)
35