## Description

1. Consider the two mathmatically equivalent formulas

(i) a = x

2 − y

2

(ii) a = (x + y) ∗ (x − y).

(a) Compute both formulas in 2 digit decimal arithmetic using x = 11, y = 10. Which formula

gives the best answer? Ignore any limits on the exponent.

(b) Compute both formulas in 3-bit binary arithmetic using x = 3/2 (dec) and y = 1. Again,

which formula gives the most accurate answer? Ignore any limits on the exponent.

(c) In your favorite programming language and platform, find two numbers x, y such that

the two formulas above give different answers. Which answer is more accurate. Report

the precision used, the machine epsilon (unit round-off) and the general description of your

machine. Can you find two numbers for which one of the answers has no accuracy whatsoever,

but the other is almost OK? Note: in Matlab all arithmetic is in double precision, but you can

force single precision by using the single function: a = single(5/4) forces all arithmeitc

involving a to be on single precision.

2. Consider the system

Ax ≡

−0.001 1.001

0.001 −0.001 x1

x2

=

1

0

≡ b

whose solution is x1 = x2 = 1 and the system

(A + ∆A)y = b + ∆b

where ∆A = ε|A|, ∆b = ε|b|. Here |A| means take the absolute value of all the elements

individually. In the following we let ε = 10−4

.

(a) Compute κ∞(A). Compute the actual value of kx − yk∞/kxk∞ and its estimate obtained

from using the (standard) condition number κ∞ (Theorem 2 in notes).

Which is quite far from the actual error.

(b) Now repeat the above, but this time use ∆b = ε

0

1

.

3. (a) Show that the following matrix is singular

A =

1 2 −1

2 1 1

−1 1 −2

(b) What is the range or column space of A ? What is its null space? Give a basis for each

subspace.

(c) Consider the matrix B obtained from A by adding η = 0.001 to the entry (1,3) (So

B = A + ηe1e

T

3

). Without computing the inverse of B, show that kB−1k1 ≥ 3, 000.

(d) Find a lower bound for the condition number κ1(B).

Continued overleaf . . .

4. Consider the n × n matrix

An =

1

−2 1

−2 1

.

.

.

.

.

.

.

.

.

.

.

.

−2 1

What is A−1

n

? [Hint: Write An = I − En and use expansion I + E + E2 + · · · .]

Calculate the condition numbers κ1(An) and κ∞(An). Verify your results with matlab for

the case n = 10.

2