- Problem 3.2
(a) with condition number 6.17 in norm 2

(b) with condition number 10.98 in norm 2

(c) with condition number 6.98 in norm 2

(d) with condition number 2 in norm 2

- Problem 3.10
(a)

(b) To solve , write , so that . Solve by backward , solve by forward

(c)

I do not see obvious connection

(d)

The reason we prefer LU to UL is purely conventional, there is no mathematical reason beyound this choice.

- Problem 3.12
(a)

Use page 91 to calculate inverse of a matrix, use 1 norm of the matrix, and assume that .

We get

Then the condition number of a matrix is

for

and

for .

Then we conclude that the matrix is ill conditioned for very large and very small values of .

(b) If the residual is small, but nonzero, the error will be small for small condition number. If condition number is large, the error will be large. So if we take , the error will be large even for a small residual.

Use and , where is a numerical solution, to show that . So we have and

(c) Using we get , so for large the residual is going to be very large with small error.