Exercise Sheet 10 - Bergische Universität Wuppertal

Bergische Universität Wuppertal
Faculty C - Department of Mathematics
Prof. Dr. M. Bolten
Dipl. Math. N. Božović
Numerical Linear Algebra
WS 2011/2012
Exercise Sheet 10
Exercise 1 (4 points)
Gaussian elimination reduces a full matrix A to an upper-triangular matrix U. The
key question for stability is whether amplification of the entries takes place during
this reduction. In particular, let the growth factor for A be defined as the ratio
ρ = max i,j |u ij |
max i,j |a ij | . (1)
If ρ is of order 1, not much growth has taken place, and the elimination process is
stable. If ρ is bigger than this, we must expect instability.
Write a MATLAB program that computes the Gaussian elimination with partial
pivoting for given A ∈ R m×m and b ∈ R m . Furthermore, have your program compute
the growth factor ρ defined in (1).
Here “partial pivoting” means that in each step k of the elimination process, the
largest entry in the first column of the submatrix Ã(k:m,k:m) is chosen as pivot. Note:
Largest entry means the largest magnitude.
Exercise 2 (4 points)
a) Show that for Gaussian elimination with partial pivoting, applied to any matrix
A ∈ C m×m , the growth factor (1) satisfies ρ ≤ 2 m−1 .
b) Experiment with solving 60 × 60 systems of equations Ax = b by Gaussian
elimination with partial pivoting, with A having the following form:
⎡
⎤
1 1
−1 1 1
A =
⎢−1 −1 1 1
⎥
⎣−1 −1 −1 1 1⎦ .
−1 −1 −1 −1 1
Do you observe that the results are useless because of the growth factor of
order 2 60 At your first attempt you may not observe this, because the integer
entries of A may prevent any rounding errors from occurring. If so, find a way
to modify your problem slightly so that the growth factor is the same or nearly
so and catastrophic rounding errors really do take place.
– Continues on next page –

Exercise 3 (4 points)
The idea of this exercise is to carry out an experiment analoguous to the one described
in the lecture (householder_stability), but for the SVD instead of QR
factorization.
a) Write a MATLAB program that constructs a 50 × 50 matrix A=U*S*V’, where
U and V are random orthogonal matrices (i.e. the Q of a QR-decomposition of a
random matrix) and S is a diagonal matrix whose diagonal entries are random
uniformly distributed numbers in [0, 1], sorted into nonincreasing order. Have
your program compute [U2,S2,V2]=svd(A) and the norms of U-U2, V-V2, S-
S2 and A-U2*S2*V2’. Do this for five matrices A and comment on the results.
(Hint: Plots of diag(U2’*U2) and diag(V2’*V2) may be informative).
b) Fix the signs in your computed SVD so that the difficulties of (a) go away.
Run the program again for five random matrices and comment on the various
norms. Do they have a connection with cond(A)
c) Replace the diagonal entries of S by their sixth powers and repeat (b). Do you
see significant differences between the results of this exercise and those of the
experiment for QR factorization
Hand in by: Tue., 10.01.2012, in the exercise