Page 267 - Compact Numerical Methods For Computers

254 Compact numerical methods for computers
‘border’ were designed to give difficulties to a specialised algorithm for matrices of
this form in which I have been interested from time to time.

Diagonal matrix
A =i
i i
A =0 for i j.
i j
This matrix permits solutions to eigenvalue and linear-equation problems to be
computed trivially. It is included in this set because I have known several
programs to fail to run correctly when confronted with it. Sometimes programs
are unable to solve trivial problems because their designers feel they are ‘too
easy.’ Note that the ordering is ‘wrong’ for algorithms 13 and 14.
Wilkinson W+matrix
A = [n/2]+1-min(i, n-i+1) for i=1, 2, . . . , n
i i
A i,i + l =A i + 1 ,i =1 for i=1, 2, . . . , (n-1)
A = 0
i j for| j - i |>l
where [b] is the largest integer less than or equal to b. The W+matrix (Wilkinson
1965, p 308) is normally given odd order. This tridiagonal matrix then has
several pairs of close eigenvalues despite the fact that no superdiagonal element is
small. Wilkinson points out that the separation between the two largest eigen-
-2
values is of the order of (n!) so that the power method will be unable to
separate them unless n is very small.
Wilkinson W-matrix
A = [n/2]+1-i for i=1, 2, . . . , n
i i
A i,i + l =A i+ l , l for i=1, 2, . . . , (n-1)
A =0 for|j-i|>1
i j
where [b] is the largest integer less than or equal to b. For odd order, this matrix
has eigenvalues which are pairs of equal magnitude but opposite sign. The
magnitudes of these are very close to some of those of the corresponding W+
matrix.
Ones
A =1 for all i,j.
i j
This matrix is singular. It has only rank one, that is, (n-1) zero eigenvalues.
The matrices described here may all be generated by the Pascal procedure
MATRIXIN.PAS, which is on the software diskette. This procedure also allows for
keyboard entry of matrices.