Math 6800
D. Schwendeman Problem Set 4
Due:
Thursday, 10/13/11
1. Text exercise 10.1, p.76.
2. Text exercise 10.4, p.76.
3. Let f(x)=1/(1 −x2). Find κ(f) for an infinitesimal perturbation about x. For
what values of xis fill-conditioned? Explain your result briefly. Repeat for the function
g(x) = sin x/(1 −x2).
4. Text exercise 15.1, parts (a) through (e), p.112.
5. Text exercise 16.1, p.199. Note that part (b) of the problem is described on page 200.
(Hint: My approach to this problem was to first consider the stability of Qa =b, where Q
is unitary and aand bare vectors. Also, even though the problem says that Qis not to be
perturbed, I found it convenient to consider the computed bin the form ˜
b= (Q+δQ)a.)
6. This question concerns conditioning, stability, and accuracy, and is to be done using
matlab (unless stated otherwise). To begin let’s generate a 10 ×10 matrix A. Use the
following steps in matlab
A=diag(10^(0:-1:-9));
[Q0,R0]=qr(rand(10));
A=Q0*A*Q0’;
(a) Note that Ais symmetric and positive definite. What are the singular values of A?
What is κ(A) in the 2-norm? (You should be able to answer this without using matlab.)
(b) Now let’s generate a linear system Ax =bwhose solution xwe know. Use the commands
x=ones(10,1);
b=A*x;
Let us consider various ways to solve this system. First, use matlab’s R=chol(A) to compute
the upper triangular matrix Rin the Cholesky factorization A=R∗R. Solve for a computed
xcusing the commands
y=R’\b;
xc=R\y;
Compute kxc−xkand kR∗R−Akin the 2-norm using matlab and explain your results
in terms of conditioning and backward stability. Do you expect the computed Rto be
accurate, i.e. its relative error to be O(mach)? Explain.
(c) Solve the linear system constructed in part (b) but now using a QR-factorization and
matlab’s qr(A) command. Compute kxc−xkand kQR −Akand explain your results.
