We have not discussed another approach to the symmetric eigenvalue problem, the Rayleigh quotient iteration. Basically, it is inverse iteration using the Rayleigh quotient as the shift. From Problem...

We have not discussed another approach to the symmetric eigenvalue problem, the Rayleigh quotient iteration. Basically, it is inverse iteration using the Rayleigh quotient as the shift. From Problem 19.17, we know that if
is a good approximation to an eigenvector of the symmetric matrix

then the Rayleigh quotient is a good approximation to the corresponding eigenvalue.

Start with an initial approximation,

₀
 to an eigenvector, compute the Rayleigh quotient, and begin inverse iteration. Unlike the inverse iteration algorithm discussed in Section
 the shift changes at each iteration to the Rayleigh quotient determined by the next approximate eigenvector. A linear system must be solved for each iteration. The Rayleigh quotient iteration computes both an eigenvector and an eigenvalue, and convergence is almost always cubic This convergence rate is very unusual and means that the number of correct digits triples for every iteration.

Let

 Let

₀

^T
, and perform two iterations of rqiter by hand.

 To obtain the starting approximation, perform two iterations of the power method starting with

₀

^T
 Using the new

₀
 perform one Rayleigh quotient iteration by hand. Start a second and comment on what happens. What eigenvalue did you approximate, and why