1. MATLAB EXAMPLES Matrix Solution Methods
58:110 Computer-Aided Engineering Spring 2005
MATLAB EXAMPLES Matrix Solution Methods
Department of Mechanical and industrial engineering
January 2005

2. Some useful functions det (A) Determinant lu(A) LU decomposition
cond(A)
Matrix condition number
inv(A)
Matrix inverse
rank(A)
Matrix rank
diag, trace
Vector and sum of the diagonal elements
tril,triu
Lower and upper triangular part of matrix
cgs, pcg
(Preconditioned) conjugate gradients iterative linear equation solver

3. LU decomposition
Here is an example to use LU decomposition for 4X4 matrix A:
>> A=[ ; ; ; ]
A =
>> [l u p]=lu(A)
l =
u =
p =
Note: p is a permutation matrix corresponding to the pivoting strategy used. If the matrix is not diagonally dominant, p will not be an identity matrix.

4. Condition number In MATLAB, Condition number is defined by:
To calculate condition number of matrix A by MATLAB built-in function:
>> cond(A)
ans =
Or by the definition, the product of 2-norm of A and inverse A:
>> norm(A,2)*norm(inv(A),2)

5. Iterative methods – Jocobi method
The following M-file shows how to use Jocobi method in MATLAB: (jocobi_example.m)
if j==i
continue
else
s=s+A(i,j)*x0(j);
end
x1(i)=(b(i)-s)/A(i,i);
if norm(x1-x0)<erp
break
x0=x1;
% show the final solution
x=x1
% show the total iteration number
n_iteration=k
% Iterative Solutions of linear euations:(1) Jocobi Method
% Linear system: A x = b
% Coefficient matrix A, right-hand side vector b
A=[ ; ; ; ];
b= [-1;0;-3;1];
% Set initial value of x to zero column vector
x0=zeros(1,4);
% Set Maximum iteration number k_max
k_max=1000;
% Set the convergence control parameter erp
erp=0.0001;
% Show the q matrix
q=diag(diag(A))
% loop for iterations
for k=1:k_max
for i=1:4
s=0.0;
for j=1:4

6. Iterative methods- Jocobi method
Running M-file in command window:
>> jocobi_r_ex
q =
x =
n_iteration = 60

7. Iterative methods - Gauss-Seidel method
In the M-file for Gauss-seidel method, the only difference is the q matrix, which replaced by: q=tril(A), the codes in the loop changed as the following:
for i=1:4
s1=0.0;
s2=0.0;
if (i==1) continue
else
for j=1:i-1
s1=s1+A(i,j)*x1(j);
end
for j=i+1:4
s2=s2+A(i,j)*x0(j);
x1(i)=(b(i)-s1-s2)/A(i,i);
Running M-file in command window
>> gauss_example
q =
x =
n_iteration = 10

8. Iterative methods - SOR method
Again, In the M-file for SOR method, the only difference is the q matrix, which replaced by: q1=tril(A)-diag(diag(A)); q2=diag(diag(A))/1.4; q=q1+q2; Note: the relaxation factor set to 1.4 for this case.
for i=1:4
s1=0.0;
s2=0.0;
if (i==1) continue
else
for j=1:i-1
s1=s1+A(i,j)*x1(j);
end
for j=i+1:4
s2=s2+A(i,j)*x0(j);
x1(i)=r*(b(i)-s1-s2)/A(i,i)+(1.0-r)*x0(i);
Run this M-file:
>> SOR_example
q =
r =
1.4000
x =
n_iteration = 12

9. Iterative methods – Conjugate gradient method
Use cgs, pcg to solve above linear equations, the tolerance is 1.0E-6 (default)
>> x=cgs(A,b, )
cgs converged at iteration 4 to a solution with relative residual 1.2e-015
x =
1.0000
>> x=pcg(A,b, )
pcg converged at iteration 4 to a solution with relative residual 3.1e-016

10. Iterative methods - Summary
Solution
Total Iteration number
Jocobi method
( , , , ) 60
Gauss –Seidel method
( , , , ) 10
SOR method
( , , , ) 12
Conjugate Gradient method
( , , , ) 4
Note: the exact solution is (-1,1,-1,1)