This problem considers solving Ax = b. The × symmetric matrix A has elements ii = ( + 1)) +( − ) + 1 and ij = + for . Also the exact solution is = (1, 1, , 1)T , and so calculate b using the...

This problem considers solving
Ax
=
b. The

×

symmetric matrix
A
has elements


_ii

=
(
+ 1)) +(
−
) + 1 and


_ij

=

+

for


. Also the exact solution is

= (1, 1, , 1)
^T

, and so calculate
b
using the formula
b
=
Ax.

(a) Write out the matrix in the case of when

= 2,

= 3, and

= 4, and explain why they are all positive definite. It is possible to prove that
A
is positive definite for all values of

(you do not need to show this).

(b) Taking

= 1000 and using the SDM, plot the error, iteration error, and the relative residual error as a function of the iteration number (as is done in Figure 8.19). Note that the relative residual is||r||||b||.

(c) Taking
= 1000 and using the CGM, plot the error, iteration error, and the relative residual error as a function of the iteration number (as is done in Figure 8.19).