Question

1. Write a program in Matlab that solves linear systems of equations using Gauss elimination with partial pivoting. Make sure that you use variables that are explicit, and make sure to include comment lines (each subroutine should have at least a sentence stating what it does). Make sure that your program checks for valid inputs in matrix and vectors dimensionality.

   • Using your code, solve the systems of equations in problems 9.11, 9.12, and 9.13

9.11

9.12

9.13

2x1-6x2-X3 =-38 6x1 + 2x2 + 2x3 = 2 3x1 + 4x2 + x3 = 1

Answer 1

Consider the following system of linear equations:

A1x + B1y + C1z = D1 . . . . . . . ( 1 )
A2x + B2y + C2z = D2 . . . . . . . . ( 2 )
A3x + B3y + C3z = D3 . . . . . . . . . . ( 3)

In order to apply Gauss elimination method, we need to express the above three linear equations in matrix form as given below:

A = | A1    B1    C1 |
         |A2   B2   C2   |
         | A3 B3   C3 |

B=    D1
         D2
         D3    
Arrange matrices A and B in the following form (augmented matrix):

         | A1    B1    C1 D1|
         |A2   B2   C2    D2|
         | A3 B3   C3   D3|

As per that given matrix is

9.11)
   2    -6     -1
-3     -1     7
-8     1      -2

-38
-34
-20

9.12)
2         1         -1
5        2          2
3       1         1

1
-4
5

9.13)

1           1         -1
6           2          2
-3         4          1

-3
2
1

program:

function C = gauss_elimination(A,B) % defining the function
A= [ 2 -6 -1; -3 -1 7; -8 1 -2] % Inputting the value of coefficient matrix 9.11
B = [-38; -34; -20] % % Inputting the value of coefficient matrix 9.11
    i = 1; % loop variable
    X = [ A B ];
    [ nX mX ] = size( X); % determining the size of matrix
    while i <= nX % start of loop
        if X(i,i) == 0 % checking if the diagonal elements are zero or not
            disp('Diagonal element zero') % displaying the result if there exists zero
            return
        end
        X = elimination(X,i,i); % proceeding forward if diagonal elements are non-zero
        i = i +1;
    end  
    C = X(:,mX);

function X = elimination(X,i,j)
% Pivoting (i,j) element of matrix X and eliminating other column
% elements to zero
[ nX mX ] = size( X);
a = X(i,j);
X(i,:) = X(i,:)/a;
for k = 1:nX % loop to find triangular form
    if k == i
        continue
    end
    X(k,:) = X(k,:) - X(i,:)*X(k,j); % final result
end