MATLAB Lab for Math 526 Project 2
Iterative Methods and Conjugate Gradient
Department of Mathematics
Overview
Discuss iterative methods to solve Ax=b, when Ais n×nand invertible.
Write a Matlab script that implements the Conjugate Gradient method (CG).
Find the condition number of a matrix.
Activities
Read the Additional Notes page to understand the CG method.
•xkthe approximate solution to Ax=b. It is the projection of the true solution
x=A−1bonto Kk.
•rkthe residual vector. It is equal to b−Axk, the error in our approximation;
however in the algorithm, we compute it with a different formula. The key step
in CG is that rkis made orthogonal to Kk.
•pkthe conjugate gradient vector. It’s described on the Additional Notes page.
Pseudocode for the conjugate gradient:
% Initialize values
x=0;
r0=b;
p0=b;
for k >= 1
αk= (rT
k−1rk−1)/(pT
k−1Apk−1)% step length
xk=xk−1+αkpk−1% approximate solution
rk=rk−1−αkApk−1% error
βk= (rT
krk)/(rT
k−1rk−1)% improvement
pk=rk+βkpk−1% search direction
% Terminate if ||rk|| < or if k=n.
Assignments
1. Let = 10−8be your termination criterion.
Write and submit a Matlab script to implement the conjugate gradient algorithm
to solve Ax=bwith the following modifications:
(i) Do one multiplication by Aper iteration (pseudocode does two).
(ii) Compute only three dot products per iteration (pseudocode does five count-
ing the norm of rto check to see if you have converged).
(iii) Store only the three vectors xk,rk, and pkto pass to the next iteration (you
will also need to store Apkfor use within each iteration).
1 April 15, 2009
