A Wilkinson matrix is a symmetric matrix used to test eigenvalue computations. The odd-order tridiagonal matrix 
has entries
 |
(1)
|
for 
(Wilkinson 1965, p. 308; Higham 1995, p. 67). Its eigenvalues
are distinct, but some occur in extremely close pairs. This family is different from
Wilkinson's examples of large element growth in Gaussian
elimination.
For 
,
write the eigenvalues in increasing order as 
. The plot
shows the gaps 
for 
.
High-precision arithmetic resolves the closest pairs, which may be indistinguishable
at machine precision.
