Description
1. In Problem 7.5.12, you are asked to do the following:
“Use the condition number estimator to produce a plot of κ
∗ versus n for each of the following matrix
families (you may want to consider a semilog or log-log scale for some of the plots):
(a) Tn, 4 ≤ n ≤ 20;
(b) Kn, 4 ≤ n ≤ 20;
(c) Hn, 4 ≤ n ≤ 20;
(d) An, 4 ≤ n ≤ 20.
Compare your estimates with the exact values from cond and the estimates from rcond.”
We define a new family of matrices Sn, with entries:
sij = 1/(i + j − 1) if either i=j or |i − j| = 1; and sij = 0 otherwise.
That is, the matrix is tridiagonal. Include, in your existing analyses for problem 7.5.12, the same
treatment for the Sn family of matrices (sizes 4 to 20). Comment on how your results for Sn compare
to the condition numbers/estimates of the other families of matrices.
2. Repeat Problem 7.2.11, but for S10 (defined above) and
b = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1)T
.
Make sure to print your solution vector x to the console.
Recall, in Problem 7.2.11, you are asked to do the following: “Write a code to do Gaussian elimination
with partial pivoting, and apply it to the system A5x = b, where b = (−4, −7, −6, −5, 16)T and the
solution is x = (0, 1, 2, 3, 4)T
.”
3. Repeat problem 7.4.10, but for the system Ax = b, where A = K10T10 (matrix product of K10 and
T10) and b = (−4, −1, 0, 0, 0, 0, 0, 0, 10, 40)T
. Make sure to write to the console the solution x to the
system, as well as the factors L, U (and P if you have created a permutation version of the factorization
algorithm). Verify A = (P)LU, and print this verification to the console.
Recall Problem 7.4.10: “Write an LU factorization code with partial pivoting, and apply it to the
system A5x = b, where b = (−4, −7, −6, −5, 16)T and the solution is x = (0, 1, 2, 3, 4)T
.”
4. Repeat Problem 3.9.8, for the expression −3x + 2e
−x
2
= 0 on the interval [0,3].
Recall Problem 3.9.8: “Use fixed-point iteration to find a value of x in [1,2] such that 2 sin(πx)+x = 0.”
