Page 264 - Compact Numerical Methods For Computers

Conjugate gradients method in linear algebra 251
Example 19.4. Negative definite property of Fröberg’s matrix
In example 19.1 the coefficient matrix arising in the linear equations ‘turns out to
be negative definite’. In practice, to determine this property the eigenvalues of the
matrix could be computed. Algorithm 25 is quite convenient in this respect, since
a matrix A having a positive minimal eigenvalue is positive definite. Conversely,
if the smallest eigenvalue of (-A) is positive, A is negative definite. The minimum
eigenvalues of Fröberg coefficient matrices of various orders were therefore
computed. (The matrices were multiplied by -1.)

2
Rayleigh quotient Matrix products Gradient norm
T
Order of (-A) needed g g
4 0·350144 5 1·80074E-13
10 7·44406E-2 11 2·08522E-10
50 3·48733E-3 26 1·9187E-10
100 8·89398E-4 49 7·23679E-9

These calculations were performed on a Data General NOVA in 23-bit binary
arithmetic.
Note that because the Fröberg matrices are tridiagonal, other techniques may
be preferable in this specific instance (Wilkinson and Reinsch 1971).