This problem considers solving a matrix equation using the Crout factorization.
(a) The algorithm for finding a Doolittle factorization of
A, assuming pivoting is not needed, is given in Table 3.1. Find a similar algorithm for the Crout factorization.
(b) Use your algorithm from part (a) to solve the matrix equation in the case of when
A
has diagonal entries
ii
= 2 and off-diagonal entries
ij
= 1. Also,
= 1000 and take
z
=
Ax, where
x
= (1, 1, ··· , 1)
T
. It is only necessary to report the values of
(1) and
(2) (to 16 digits). Also, report the computed value of ||r||
∞
and ||e||
∞
.
(c) Using the same matrix as in part (b), use your algorithm to solve the matrix equation when
x
= (1, −1, 1, −1, ··· , −1)
T
. It is only necessary to report the values of
(1) and
(2) (to 16 digits). Moreover, you must give a compelling explanation of why you believe you answer is correct (within the limits of double precision).