Getting Started

• All matlab files you need can be found in /u/rvdg/class/CS383C.F04/QRalg/matlab/

• You may want to copy these over into a directory of yours

• Start up matlab

The Power Method

• The first demonstration centers around the Power Method

• Two M-script files are involved:– PowerMethod.m

• This is the main driver

– ShowPowerMethod.m• This is a utility routine that prints out interesting

stuff

Enter matrix size

Hit “Return”

A random diagonal matrix is created

A random orthogonalmatrix is created

A randommatrix is created

Hit “Return”

Current estimate of lambda(1)

Hit “Return”

Actual lambda(1)

Keep an eye on thisThe first element should become 1 (since q should eventually be the direction of the first columnof U).The other elements should become 0 (since q should eventually be orthogonalTo the other columns of U.)

Notice that the component of q in the direction of uj should decrease in every iteration by a factor roughlyequal to | j|/|1|Notice that the jth component of UT * q equals the length of the component of q in the direction of uj. By looking at this ratio, we aretracking by what factor these components are decreasing in each step.

Here I report the ratios | j|/|1| as the i-th element of this vector

I hit return a few times

!!!!! Starting to look like we want!

I hit return a few more times

!!!!! Starting to look like we want!

Finally reply “0” (or anything elsebut a return)

Subspace Iteration

• The second demonstration centers around the relation between the Power Method and Subspace Iteration

Hit “return” a few times to createa random matrix, etc.

The Power Method and Subspace Iteration are now both tracked

A few iterations later

Again, these are the factors bywhich we would predict thatcomponents in the directions of uj would decrease(see similar discussion forpower method.)

QR algorithm

• The third demonstration centers around the relation between Subspace Iteration and the QR algorithm

Recall the QR algorithm:A0 = AFor i=0,…

Ai – I = QRAi+1 = R Q + I

The difference here simply comes from a different numberof digits being printed

For now, just use shift rho = 0by hitting return every time

A few iterations later…

A few more iterations later…

All off-diagonal elements arestarting to become small

QR algorithm

• The fourth demonstration centers around the effects of choosing shifts

Notice MUCH faster convergenceto zero!

After only two iterations!

Let’s start over, printing out moredigits.

A few iterations later…

Now let’s pick a differentshift

Now these elements are converging faster