Objective:
Lab Report 08 / Assignment # 1
Implementation of gauss elimination to find out the number of solutions by appropriate row and column operation from upper and lower triangular matrix.
Theory:
The elimination of unknowns is used to solve a pair of simultaneous equations. The procedure consisted of two steps:
1. The equations are manipulated to eliminate one of the unknowns from the equations. The result of this elimination step is that we have one equation with one unknown.
2. Consequently, this equation could be solved directly and the result back-substituted into one of the original equations to solve for the remaining unknown.
This basic approach can be extended to large sets of equations by developing a systematic scheme or algorithm to eliminate unknowns and to back-substitute.
Gauss elimination:
It is the most basic of these schemes. This section includes the systematic techniques for forward elimination and back substitution that comprise Gauss elimination. Although these techniques are ideally suited for implementation on computers, some modify cations will be required to obtain a reliable algorithm.
In particular, the computer program must avoid division by zero. The following method is called “naive” Gauss elimination because it does not avoid this problem. Subsequent sections will deal with the additional features required for an effective computer program.
The approach is designed to solve a general set of n equations:
As was the case with the solution of two equations, the technique for n equations consists of two phases: elimination of unknowns and solution through back substitution.
Forward Elimination of Unknowns.
The first phase is designed to reduce the set of equations to an upper triangular system. The initial step will be to eliminate the first unknown, x1, from the second through the nth equations. To do this, multiply by a21 /a11 eqn. (1) to give:
Now, this equation can be subtracted from Eq. (2) to give:
where the prime indicates that the elements have been changed from their original values. The procedure is then repeated for the remaining equations.
Back Substitution:
Equation can now be solved for xn :
....................6
This result can be back-substituted into the (n 2 l)th equation to solve for xn21. The procedure, which is repeated to evaluate the remaining x’s, can be represented by the following formula:
The two phases of Gauss elimination: forward elimination and back substitution. The primes indicate the number of times that the coefficients and constants have been modified.
Now, perform the following elementary row operations till it is reduced to echelon form by:
• Exchanging or swapping two rows
• Adding the certain multiple of one row to another row
• Multiplying a row by non-zero number
This procedure is repeated until the augmented matrix is reduced to following echelon form:
Thus, the solution of above system of linear equation is (a, b, c) i.e. x = a, y = b and z = c.
MATLAB Code:
>>C = [1 2 -1; 2 1 -2; -3 1 1]
>>b= [3 3 -6]'
>>A = [C b]; %Augmented Matrix
>>n= size(A,1); %number of eqns/variables
>>x = zeros(n,1); %variable matrix [x1 x2 ... xn] coulmn
>>for i=1:n-1
>> for j=i+1:n
>> m = A(j,i)/A(i,i)
>> A(j,:) = A(j,:) - m*A(i,:)
>> end
>>end
>>x(n) = A(n,n+1)/A(n,n)
>>for i=n-1:-1:1
>> summ = 0
>>for j=i+1:n
>>summ = summ + A(i,j)*x(j,:)
>>x(i,:) = (A(i,n+1) - summ)/A(i,i)
>>end
>>end
Input:
Output:
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