Principles of Mathematics in Operations Research

6.2 Problems 89

The second stage is similar: x consists of the last n — 2 entries in the second

column, z is the first unit coordinate vector of matching length, and H 2 is of

order n — 2:

Uo =

100 0

010 0

00

0 0 H 2

00

= U2^1 , and U^iU^AUiM 0 * * * *

0 0***

00***

Following a similar approach, one may operate on the upper right corner

of A simultaneously to generate a tridiagonal matrix at the end. This process

is the main motivation of the QR algorithm.

Problems

6.1. Show that for orthogonal matrices ||Q|| = c(Q) — 1. Orthogonal matrices

and their multipliers (aQ) are only perfect condition matrices.

6.2. Apply the QR algorithm for

A =

0.5000 -1.1180 0 0 0

-1.1180 91.2000 -80.0697 0 0

0 -80.0697 81.0789 4.1906 0

0 0 4.1906 2.5913 0.2242

0 0 0 0.2242 0.1257 -

0 0 0 0 -0.0100

6.3. Let A{n) 6 Rnxn, A(n) = (a^-

, wnere aij — i,j_^.

(a) Take A{2).

1. Let 6/ =
   1.0
   0.5
   and bu =
   1.5
   1.0
   . Calculate the relative

0

0

0

0

-0.0100

0.0041

error.

2. Find a good upper bound for the relative error obtained after perturbing
   the right hand side.


3. Find the relative error of perturbing A(2) by AA( 2 ). Take

h

1.0

0.5

as the right hand side.

4. Find a good upper bound for the relative error obtained after perturbing
   .4(2).
   (b) Take ^4(3)TA(3) and find its condition number and compare with the
   condition number of A(3).
   (c) Take A(4) and calculate its condition number after finding the eigen values
   using the QR algorithm.