Page 249 - MATLAB an introduction with applications

234 ——— MATLAB: An Introduction with Applications

[L,U] = lu(A);
% solution of y
y = L\b;
%final solution x
x = U\y;
fprintf(‘Solution of the equations is\n’);
disp(x)
Output is as follows:
Solution of the equations is
1.0000
2.0000
–1.0000
0.0000

Check with MATLAB built-in function:

>> A=[1 1 1 –1;1 –1 –1 2;4 4 1 1;2 1 2 –2];
b = [2;0;11;2];
>> x = A\b
x =
1.0000
2.0000
–1.0000
0.0000

Example E4.15: Using the Gauss-Seidel method, solve the system of equations given below:
x + 2y + z = 0
3x + y –z = 0
x – y + 4z =3
Solution:
The Gauss-Seidel method is a technique used to solve a linear system of equations. In solving equations
AX = b, first the matrix A is written as: A = D + L + U where the matrices D, L, and U represent the diagonal,
negative strictly lower triangular, and negative strictly upper triangular parts of the coefficient matrix A.
Then the solution is given for every iteration counter k as:

(k)
–1
X (k + 1) = (D + L) (–U X + b) Gauss-Seidel Method
–1
(k)
X (k + 1) = D (–(L + U) X + b) Jacobi Method
Disadvantages:
1. The matrix (D + L) is not always invertible. One must check that the values on the diagonal are non-
zero before starting the iteration because it can lead to unpredictable results.