CS 323E Homework 10 Due: April 22 2003
1. The MATLAB command hilb(n) generates an n×nHilbert matrix, which we denote
by Hn. Try n= 3,10,20 in the following problems:
1) (4 points)
Solve:
Hnxn=bn
for xn, where bn=Hn∗ones(n,1).
Use the MATLAB command “\” to solve the above system. (See help mldivide).
b) (2 points)
How close is xnto the exact solution? Comment.
c) (4 points)
Explain the accuracy of xn. Use the command cond to get the condition number of Hn.
2. (5 points)
Does the MATLAB command “\” do pivoting? Give an example to justify your answer.
3. (Use pen & paper). Let
A=10−16 10−17
−10−16 10−17
a) (2 points)
Compute the determinant of A.
b) (5 points)
Compute κ1(A) = kAk1· kA−1k1.
c) (2 points)
Is Anearly singular? Comment.
d) (1 point)
Does the small magnitude of the determinant imply that Ais nearly singular?
4. The MATLAB command pascal(n) generates an n×nPascal matrix, which we denote
by Pn. Try n= 16 in the following.
a) (1 point)
Using MATLAB, find the determinant of Pn.
b) (1 point)
Using MATLAB, find the condition number of Pn.
c) (3 points)
Is Pnclose to singularity? Comment.