The equation for a string on an elastic foundation is
where
and
are positive constants. The problem for finding the natural frequencies of the string can be reduced to solving
Ax
= λx, where
Note that
A
is an
×
symmetric tri-diagonal matrix, and
Also, by solving the string equation one finds that the exact values for the λ’s are
(a) Explain why
A
is positive definite.
(b) Use inverse orthogonal iteration to compute the five smallest eigenvalues of
A. To do this take
= 250 and
=
= 1. Also, compute the relative error as in Table 4.16.
(c) The exact eigenvalues of
A
are λ
i
=
+ 2 cos(
), for
= 1, 2,...,, where
=
/(
+ 1). Taking
= 250 and
=
= 1, as in Figure 4.4, plot the ratios λ
i
−1/λ
i
and use this to explain why inverse orthogonal iteration should converge relatively quickly when computing the smaller eigenvalues.