Page 266 - Compact Numerical Methods For Computers

Appendix 1

NINE TEST MATRICES

In order to test programs for the algebraic eigenproblem and linear equations, it is
useful to have a set of easily generated matrices whose properties are known. The
following nine real symmetric matrices can be used for this purpose.

Hilbert segment of order n
A = l / (i+ j-1).
i j
This matrix is notorious for its logarithmically distributed eigenvalues. While it
can be shown in theory to be positive definite, in practice it is so ill conditioned
that most eigenvalue or linear-equation algorithms fail for some value of n<20.
Ding Dong matrix
A =0·5/(n-i-j+1·5).
i j
The name and matrix were invented by Dr F N Ris of IBM, Thomas J Watson
Research Centre, while he and the author were both students at Oxford. This
Cauchy matrix has few trailing zeros in any elements, so is always represented
inexactly in the machine. However, it is very stable under inversion by elimination
methods. Its eigenvalues have the property of clustering near ±p/2.
Moler matrix
A =i
i i
A =min(i,j)-2 for i j.
i j
Professor Cleve Moler devised this simple matrix. It has the very simple Choleski
decomposition given in example 7.1, so is positive definite. Nevertheless, it has
one small eigenvalue and often upsets elimination methods for solving linear-
equation systems.
Frank matrix
A =min(i,j).
i j
A reasonably well behaved matrix.
Bordered matrix
A =1
i i
1-i
A =A n i =2 for i n
i n
A =0 otherwise.
i j
The matrix has (n-2) eigenvalues at 1. Wilkinson (1965, pp 94-7) gives some
discussion of this property. The high degree of degeneracy and the form of the
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