Math 6800
D. Schwendeman Problem Set 6
Due:
Monday, 11/8/10
1. Text exercise 23.2, p.177.
2. Consider the linear system Ax =b, where Ais a symmetric, positive definite matrix.
(a) Write a matlab function, R=myCholesky(A) say, following algorithm 23.1 in the text.
Test your function using A=Z’*Z, where Z=triu(rand(12,12)). Compute the following
quantities: norm(R’*R-A)/norm(A) and norm(R-Z)/norm(Z). Comment on the results.
(b) Write a second matlab function, x=mySolve(R,b), that solves the linear system given
the Cholesky factorization of A. Test your function using b=Ay, where Ais the matrix
constructed above and y=ones(12,1). Compute norm(x-y)/norm(y) and comment on the
results.
3. Text exercise 24.1, p.188.
4. Text exercise 25.1, p.194.
5. Consider the following matlab experiments:
(a) Let Abe a 7 ×7 matrix generated using rand(7,7), and find its eigenvalues using
matlab’s eig function. Now consider the basic steps of the QR algorithm: (i) compute the
QR-factorization of A, (ii) set A=RQ. Apply these two steps iteratively until Areduces
to an “eigenvalue revealing form.” (You may use matlab’s qr function to determine the
Q,Rfactors in step (i) of the iteration.) Determine the eigenvalues from this form (without
using matlab’s eig function) and compare them to the ones found originally.
(b) Let Bbe a 7 ×7 matrix generated using the following commands B=diag(rand(7,1));
[Q2,R2]=qr(rand(7,7)); B=Q2’*B*Q2. Repeat the iteration described in part (a) for the
matrix Bin order to determine its eigenvalues.
