Numerical Methods
MATH 417 - 501/502
A. Bonito
April 7
Spring 2016
Last Name:
First Name:
Homework 8
Exercise 1
35% (MATLAB)
Implement the following conjugate-gradient algorithm in matlab:
• Inputs: A ∈ Rn×n , b, x0 ∈ Rn .
• Initialization
Compute:
r0 = b − Ax0 ,
and
α1 =
w1 = −r0 ,
(r0 )T w1
,
(w1 )T z1
z1 = Aw1
x1 = x0 + α 1 w 1 .
• Main loop
For i = 1, 2, 3, ..., compute:
ri = ri−1 − αi zi ,
(if kri k2 < tol stop),
βi =
(ri )T zi
,
(wi )T zi
and
αi+1 =
wi+1 = −ri + β i wi ,
(ri )T wi+1
,
(wi+1 )T zi+1
zi+1 = Awi+1
xi+1 = xi + αi+1 wi+1
• Outputs: xi and kri k
Consider now the following linear systems
Ax = b,
where
A = (aij )ni,j=1 ,

 2
−1
aij =

0
i=j
|i − j| = 1
otherwise
and
b = (bi )ni=1 ,
bi = 1,
x0 = (x0i )ni=1 ,
x0i = 0.
Together with your matlab code, report the number of iterations for the conjugate gradient algorithm to reach a tolerance of 10−8 as well as the tolerance reached for n = 5, 10, 20, 40, 80, 160.
Exercise 2
35% (MATLAB)
You will learn in this exercise how to employ a least-square method to denoise data. Consider the
function
g(x) = 1.7 sin(2πx) + 0.47 sin(4πx) + 0.73 sin(6πx) + 0.8 sin(8πx)
3i
for i = 0, ..., 50. The
measured for x ∈ (0, 3) at 51 equi-distributed points, this is xi = 50
measurements are polluted with random noise. This means that you are actually given (in matlab)
the vector F:
g = inline('1.7*sin(2*pi*x)+ 0.47*sin (4*pi*x) + 0.73*sin(6*pi*x) + 0.8*sin(8*pi*x)');
F = g(linspace(0,3,51)).' + .4*randn(51,1);
Write a least-square method to to find the coefficients αj , j = 1, ..., 8 in
gLS (x) =
8
X
αj sin(jπx),
j=1
so that
50
X
(Fi − gLS (xi ))2
i=1
is minimized. You can use matlab backslash routine to solve the linear solver.
Hand out your matlab code as well as two plots: one of (xi , Fi ) together with (x, gLS (x)) and the
other one of (x, g(x)) together with (x, gLS (x)).
Exercise 3
30% (BY HAND)
We want to solve the linear system Ax = b where
2 −1
A=
and
−1 2
b=
1
1
.
1. Check that A is symmetric and positive definite.
2. Perform two iterations of the conjugate gradient method starting with
3 0
2
x =
2
and verify that x2 = x is the solution of the linear system.
3. Let
1 T
y Ay − bT y.
2
Plot the isovalues of L (i.e. the set of y’s such that L(y) = c for several constants c) together
with the vectors x0 , x1 and x2 .
L(y) :=