Principles of Mathematics in Operations Research

6.1 Solution of Ax = b^83

\\x\\ = -y> Ax = x 2 -xi = -

= 1 ana

1

-1 =* IIArll =

V2

Ml

5 x 10~^6.

T/ie relative amplification in this particular instance, ",, 1}' ~ -g-loiMJ

approximately ^-, which is a lower bound for the condition number c«2x

105.

Remark 6.1.5 As a rule of thumb (experimentally verified), a computer can
loose logc decimal places to the round-off errors in Gaussian elimination.

6.1.2 Symmetric and not positive definite

Let us now drop the positivity assumption while we keep A still symmetric.
Then, nothing is changed except

\"max |

6.1.3 Asymmetric

In this case, the ratio of eigen values cannot represent the relative amplifica-
tion.

Example 6.1.6 Let the parameter K ^> 0 be large enough.

A =

1 K

0 1

<*A~X =

1 -K

0 1

Ai = A 2 = 1.

In particular,

b = b\ = 1 =£• x = x\

Then, we have

\\b\\ = Vl + K^2 , Ab = h - h =

and b? = =>• x 2 =

\M = l;

1, Ax = X2 — X\

l|AJ

K

-1

V 1 + K^2 and

» \\AX\\ = Vl + K^2

Ah\ 1
\m\ \o\ %/i + K^2 "
The relative amplification in this particular instance is 1 + K^2. Hence, we
should have 1 C 1 + K^2 < c(A). The condition number c(A) is not just the
ratio of eigen values, which is 1; but it should have a considerably larger value
in this example, since A is not symmetric.