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Some refs.:
A. Ostrowski, On the convergence of the Rayleigh quotient iteration for
the computation of the characteristic roots and vectors 
I,  Arch. Rational Mech. Anal. 1,   (1958) 233 -- 241.
II, Arch. Rational Mech. Anal. 2(1) (1958) 423 -- 428.
IV, Arch. Rational Mech. Anal. 3    (1959) 341 -- 347.

S.H. Crandall, Proc. Royal Soc. 207, 416-423, 1951.
Iterative relaxation procedures for eigenvalue problems,

B.N. Parlett, The Rayleigh quotient and some of its generalizations 
for nonormal matrices
Math. Comp. 28(127), 679-693, 1974.

S. Batterson & J. Smillie,
Rayleigh Quotient Iteration for Nonsymmetric Matrices 
Mathematics of Computation, Vol. 55, No. 191 (Jul., 1990), pp. 169-178.

From BN. Parlett, "The Symmetric Eigenvalue Problem"
Theorem 4.9.1 of SEP.  Let (x_k) be the Rayleigh sequence generated by 
any unit vector xo.  As k-> infinity,
1. (rho_k) converges and either
2. (rho_k, x_k) -> (lambda,z) cubically, where Az=z lambda, or
3.  x_2k  -> x+ and x_2k+1  -> x-, where x+ and x_ are the bisectors 
of a pair of eigenvectors whose eigenvalues have mean 
rho = lim_k rho_k.
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 May  4 11:50:59 MDT 2007

> When using Rayleigh quotient iteration, how do you know
> what vector to start with in order to converge to the
> largest eigenvalue? Wouldn't it be better to use inverse
> iteration, using the largest Gershgorin estimate as mu
> since this would definitely converge to the closest (i.e.,
> the largest) eigenvalue?

Sure, inverse iteration is guaranteed to converge, but it only converges
quadratically (i.e. correct digits doubled at each step. RQ converges
cubically, so in principle it is the fastest way to do the job.
I would try a step (or two) of inverse iteration to get a fix on the
eigenvector, then switch to RQ to speed up. It may require some effort
to get it to converge to the right place. It is an open question: is it
always possible to tell RQ where to converge, or does it have favorites?
I do not know the answer, perhaps consulting the literature may shed some
light:
 (1) O. Axelsson, Iterative Solution Methods, p.621 (comments on the lack
    of guarantees for convergence to prescribed values)
 (2) B.N. Parlett, The symmetric eigenvalue problem, (1980) (I don't have
     this one)
 (3) J.H. Wilkinson, The Algebraic Eigenvalue Problem, (1965) has lots
     of great insights.
  As a result, there is indeed a research component to this project!

> Also, I am not exactly sure what
> you mean in bullet 2 on page 1 of the project description.
> Do you want us to use LU or QR to apply the inverse of
> (A-lambda*I) in the Rayleigh quotient iteration?

Either; if you do LU it must employ partial pivoting (it is faster if
these were full matrices), QR might be competitive for tridiagonal.

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On Fri, 6 Apr 2007, ****** wrote:

>             On HW question 24.3, what does ". . . entries from the standard
> normal distribution . . ."?

The Normal Distribution has probability distribution function (PDF)

y = (1/(sigma^2*sqrt(2*pi)))*exp(-(x-mu)^2/(2*sigma^2))

with mean mu, standard deviation sigma.
 The standard normal distribution has mu = 0, sigma = 1.
To get an array of normally distributed numbers you need the statistics
toolbox (available on campus machines); then use the command
y = normrnd(0,1,10,10) to generate a 10X10 array with entries from the
standard normal distribution


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