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The Rayleigh quotient trick
If is symmetric and the vector , then the scalar
is known as
the Rayleigh quotient of p. The Rayleigh quotient is
important because it has the following property:

(4.21) 
During the Cholesky factorization of
, we have
encountered a negative pivot at the
stage of the
decomposition for some . The factorization has thus
failed ( is indefinite). It is then possible to add
to the
diagonal of
so that the leading
by submatrix of
is
singular. It's also easy to find a vector for which

(4.22) 
using the
Cholesky factors accumulated up to step . Setting for
, and backsolving:
gives the
required vector. We then obtain a lower bound on
by
forming the inner product of 4.24 with , using the
identity
and recalling that the
Rayleigh quotient is greater then
, we
can write:
This implies the bound on :
In the
algorithm, we set
Next: Termination Test.
Up: The TrustRegion subproblem
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Frank Vanden Berghen
20040419