Householder's method for symmetric matrices

J. H. WILKINSON: Householder's method for symmetric matrices 359
working with this lower triangle of elements, the details of the transformation
being overwritten on the same triangle. From equations (6) and (10) it is clear
that we will retain full information on each transformation if we store the com-
ponents of the u i and :J: a, ~. Each non-zero element of each u i is overwritten
on the ali element from which it is derived. This is particularly convenient
because apart from the elements ui, l_ 1 each uii is equal to the corresponding
current a~i.
The storage of the elements of each u i leaves no space available for the
(1, l--t) element of each Ai+ 1. These elements are, of course, the off-diagonal
elements of the final tridiagonal matrix. They are accordingly stored in the
linear array b of order n. An array q is also used to store the components of
the successive Pi and qi- The vector qi is overwritten on Pi-
The off-diagonal elements of A~_I are obtained finally as the linear array b.
A copy of the diagonal elements of A~_ 1 is made to give the linear array c. This
is convenient since tridiagonal matrices often arise as primary data and it is
obviously desirable to present such matrices as linear arrays rather than as a
square array.
5.2. Backtrans/ormation
The matrix z containing the eigenvectors of the tridiagonal matrix is over-
written by the corresponding matrix of the eigenvectors of the original matrix.
The matrix A is equal to the matrix resulting from a Householder tridiagonali-
sation, but the process uses only its lower triangle.
6. Discussion of numerical properties
In determining the (1 -- t) element of the vector w i it is important to get exact
orthogonality of the transformation even when cancellation occurs, therefore
normalized addition must be used there.
7. Examples of the use of the procedure
householder tridiagonalisation may be the first step in the determination
of the eigenvalues and eigenvectors of the matrix A followed by a determination
of the eigenvalues by the procedure tridibisection and possibly by a determination
of the eigenvectors of the resulting tridiagonal matrix by the procedure tridi-
inverse iteration. In the last case the procedure backtrans/ormation is to be
applied.
householder tridiagonalisation may also be the preparation for the appli-
cation of the QD or QR algorithm.
8. Test results
The procedures were tested on the Z 22 and Siemens 2002 in the Institute
for Applied Mathematics at Mainz University and a second test was performed
in the Institute for Applied Mathematics of the EidgenSssische Technische
Hochschule at Zfirich.
We give two examples, the second being of a matrix for which the correspond-
ing tridiagonal matrix decomposes into the direct sum of two smaller tridiagonal
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