The QR Method and Linear Algebra

The QR Method:

The Power Method as well as Inverse Power Method each give us only one ew–ev pair. While both of these methods are able to be modified to give more ew’s and ev’s, there is a better method for obtaining all the ew’s called the QR method. This is the foundation of all modern ew software, including Matlab, so we summarize it briefly here.

The QR method utilizes the fact that any square matrix has QR decomposition. Specifically for any A there are matrices Q and R such the A = QR where Q has the property:

Q−1 = Q′

and R is upper triangular. A matrix Q with the property that its transpose equivalent its inverse is called an orthogonal matrix for the reason that its column vectors are mutually orthogonal.

The QR method comprises of iterating following steps:

- Transform A into a tridiagonal matrix H.
- decompose H in QR.
- multiply Q and R mutually in reverse order to form a new H.

The diagonal of H will converge to the eigenvalues.

We facts of what creates this method converge are beyond the scope of this book. Nevertheless we note the following theory behind it for those with more familiarity with linear algrebra. First the Hessian matrix H is acquired from A by a series of similarity transformation therefore it has the same ew’s as A. Secondly if we signify by H0, H1, H2, . . ., the sequence of matrices produced by the iteration, then:

Hi+1 = RiQi= Qi−1QiRiQi = Q′iHiQi.

Therefore each Hi+1 is a related to Hi by an (orthogonal) similarity transformation and so they have the same ew’s as A.
There is a built-in QR decomposition in Mat lab which is called with the command [Q R] = qr(A).

Therefore the following program implements QR method until it converges:

function E = myqrmethod(A)
[m n] = size(A);
if m ~= n
warning(’The input matrix is not square.’)
H = hess(A);
E = diag(H);
change = 1;
steps = 0;
while change > 0
Eold = E;
[Q R] = qr(H);
H = R*Q;
E = diag(H);
change = norm(E - Eold);
steps = steps +1;

As you are able to see the main steps of the program are very simple. The actually hard calculations are contained in the built-in command qr(A).

Run this program as well as compare the results with Mat lab’s built in command:

>format long
>format compact
> A = hilb(5)
> Eqr = myqrmtheod(A)
> Eml =eig(A)

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