C

of the perturbed system satis¬es the equation,

t §¦¥

d C B C ¢ B

— — ¨

¤

¢ ¤¦

¢ ¢

CB B

3

¤

¢ ¤)

¢¦ ¦

Let . Then,

C

B C #B— % C B C ¢ — B

— B 3

¤

¢ ¤

¢ ¢ ¢ ¦

¢

C

W¤

3

¥

¢ ¦

¢

B 0 C ¢ B B C ¢

—C 3W¤

¤

¢ ¥

¢ ¢ ¦

&

C

c

B 7

¤¦

¢ ¢

As an immediate result, the function is differentiable at and its derivative is

C

given by

B

¢ CB £

¢

t §¦¥

C

B0&

¡ ¨

#

7 W¤

3

(

§¦

¦ ¦

¢

C

& ¢¨ ¢

1

B

¢£ ¦

The size of the derivative of is an indication of the size of the variation that the solu-

1 ¤3

¢1

3

C

B1

¤¦

¢

tion undergoes when the data, i.e., the pair is perturbed in the direction .

3 ¤ © ¢ 1¢

C ¢

In absolute terms, a small variation will cause the solution to vary by roughly

C 0

B¦ ¢ 3 ¤B &

7

¦¢ ¥

¢ ¦

. The relative variation is such that

BC ¤

3

¤¦

¢ ¦

C B —

0

b

—

¢ ¤

¢

¦C ¢

# &

¦

¦

Using the fact that in the above equation yields

#

¦ 3 £B1¦ ¤

¢

t §¦¥

0

¢

— ¢ —

B ¨

2

¤

¢

¦C

# &

C

which relates the relative variation in the solution to the relative sizes of the perturbations.

The quantity

C § 0

B

&

uY u d¥ £ ¡ ¨ ¢¤¥

n ¦¢¥

¤¤§ ¥ §¢

¡

£ ¡© ¥ ¡ © ¥ ¡ ¥ © ¥ ¥

¥ ¨

1i )1Y

is called the condition number of the linear system (1.65) with respect to the norm . The

#

£

condition number is relative to a norm. When using the standard norms , ,

¡

§

B

it is customary to label with the same label as the associated norm. Thus,

C

# 0 # C B #

&