Homework/Programming 6. (Due March 12)
Math. 417
This homework involves developing simple matlab functions for implementing Richardson’s iteration for computing the solution of a linear system,
(0.1)
Ax = b,
where A is a nonsingular n × n matrix and x, b ∈ Rn with b given.
Richardson’s method is a fixed point iteration with a parameter α ∈ R. Note
that (0.1) can be rewritten 0 = b − Ax. Richardson’s method comes from
the fixed point formulation,
x = x + α(b − Ax)
and is given by
xj+1 = xj + α(b − Axj ).
This method requires an initial iterate x0 . Without a priori information,
x0 = 0 ∈ Rn is often used.
Write a MATLAB m-file function implementing k steps of Richardson’s iteration with interface:
function xn=richardson(x0,k,A,b,al,tol)
MATLAB automatically provides basic linear algebra operations however
you must understand that MATLAB deals with matrices (not vectors). To
declare a column vector, you can write
x = ones(n, 1);
The “ones” function in MATLAB creates a matrix with every entry set to
one (there is also an analogous “zeros” function). As defined above, x is
actually a matrix with dimension n × 1 and so (when A is an n × n matrix)
y = A ∗ x produces another colum vector y (actually an n × 1 matrix). Note
that if you set x = ones(1, n) you would get an 1 × n matrix (a row vector)
and the operation y = A ∗ x would result in an error while y = x ∗ A would
produce another row vector (dimensioned 1 × n).
This means that if xi is a n × 1 matrix and A, an n × n matrix, you can
implement the update step in Richardsons iteration as
xip = xi + al ∗ (b − A ∗ xi).
1
2
In our examples, we shall measure the error in a weighted norm, specifically,
if x ∈ Rn ,
1/2
n
X
−1
2
kxk = n
xi
i=1
MATLAB provides a function for computing the Euclidean norm of a matrix, namely, if M is an n × m (MATLAB) matrix, the function norm(M )
computes
X
1/2
m
n X
2
norm(M ) =
M (i, j)
.
i=1 j=1
Thus, if P is the n×1 matrix representing the vector p ∈ Rn , we can compute
kpk by norm(P )/sqrt(n);. Thus, we can use
norm(xip − xi)/sqrt(n)
as our convergence indicator in the MATLAB Richardson’s code. Specifically, we run our iteration until
norm(xip − xi)/sqrt(n) < tol
The remainder of the assignment involves running your Richardson’s function on various matrices. In all cases, use a right hand side consisting of all
ones, i.e., b = ones(n, 1) and an initial iterate x0 = zeros(n, 1). There are
three possible outcomes of the iteration:
(a) The iterations converge to the unique solution of Ax = b.
(b) The iteration diverges (iterates becomes arbitrarily large).
(c) The iteration remains bounded but does not converge.
In case (a) above, stop the iteration when
norm(xip − xi)/sqrt(n) < tol := 10−5 .
Problem 1. A is a 3 × 3 diagonal matrix with diagonal entries 1, 2, 4. Run
your code for α = {−1, 1, 1/2, 1/4, 2/5} and report whether (a), (b) or (c)
holds for each iteration. When (a) holds, report the number of the iteration
when the error indicator reaches the tolerence.
Problem 2. A is a 3×3 diagonal matrix with diagonal entries 1, 2, −4. Run
your code for α = {−2, −1, 1, 2} and report whether (a), (b) or (c) holds for
each iteration. When (a) holds, report the number of the iteration when the
error indicator reaches the tolerence.
3
Problem 3. For the same A as in the previous problem, we reformulate
Ax = b as
(3.1)
A2 x = Ab
(this is called the normal equations). You can use your code to iterate on this
equation by passing down A2 as your matrix and Ab as your right hand side
vector. Run your code on the normal equations with α = {1, 1/8, 1/16} and
report whether (a), (b) or (c) holds for each iteration. When (a) holds, report
the number of the iteration when the error indicator reaches the tolerence.
Problem 4. For
2 10
A=
,
0 4
experiment with different values of α until you have found two which lead
to divergence and two which lead to convergence. In the convergent cases,
report the iteration number when the error indicator reaches the tolerence.
Hand in the results for the four problems and a copy of the matlab
code implementing the iteration.