Gauss Elimination Algorithm
The Gauss Elimination Algorithm, also known as Gaussian elimination or row reduction, is a fundamental algorithm in the field of linear algebra. It is primarily used for solving systems of linear equations, inverting matrices, and determining the rank of a matrix. The algorithm consists of performing a series of elementary row operations on an augmented matrix, which is formed by combining the coefficient matrix with the constant terms of the system of equations, until it reaches a row echelon form or a reduced row echelon form. This systematic process of simplifying the matrix allows for an efficient strategy to find the unique solution, if it exists, or determine if the system has no solution or infinitely many solutions.
The main steps of the Gauss Elimination Algorithm are as follows: first, the matrix is transformed into a triangular matrix by iteratively eliminating the coefficients below the main diagonal. This is achieved through three types of row operations: swapping two rows, multiplying a row by a nonzero constant, and adding or subtracting a multiple of one row to another row. Once the matrix is in its triangular form, the algorithm proceeds to perform back-substitution, starting from the bottom row and working its way up to the top row. During this step, the variables are sequentially isolated to obtain their values, effectively solving the system of linear equations. Gaussian elimination is widely used in various fields of science and engineering, as it provides a general and reliable method for solving linear systems, which are ubiquitous in the modeling and analysis of real-world problems.
#include<stdio.h>
#include<conio.h>
void display(float a[20][20],int n)
{
int i,j;
for(i=0;i<n;i++)
{
for(j=0;j<=n;j++)
{
printf("%.2f \t",a[i][j]);
}
printf("\n");
}
}
float interchange(float m[20][20],int i,int n)
{
float tmp[20][20];
float max=fabs(m[i][i]);
int j,k=i;
for(j=i;j<n;j++)
{
if(max<fabs(m[j][i]))
{
max=fabs(m[j][i]);
k=j;
}
}
for(j=0;j<=n;j++)
{
tmp[i][j]=m[i][j];
m[i][j]=m[k][j];
m[k][j]=tmp[i][j];
}
return m[20][20];
}
float eliminate(float m[20][20],int i,int n)
{
float tmp;
int k=1,l,j;
for(j=i;j<n-1;j++)
{
tmp=-((m[i+k][i])/(m[i][i]));
for(l=0;l<=n;l++)
{
m[i+k][l]=(m[i+k][l])+(m[i][l]*tmp);
}
k++;
}
return m[20][20];
}
void main()
{
int i,j,n,k=0,l;
float m[20][20],mul,tmp[20][20],val,ans[20];
clrscr();
printf("Total No.of Equations : ");
scanf("%d",&n);
printf("\n");
for(i=0;i<n;i++)
{
printf("Enter Co-efficient Of Equations %d & Total --->>>\n",i+1);
for(j=0;j<=n;j++)
{
printf("r%d%d : ",i,j);
scanf("%f",&m[i][j]);
}
printf("\n");
}
printf(":::::::::::: Current Matrix ::::::::::::\n\n");
display(m,n);
for(i=0;i<n-1;i++)
{
printf("\n------->>>>>>>>>>>>>>>>>>>>>>>>-------- %d\n",i+1);
m[20][20]=interchange(m,i,n);
display(m,n);
printf("\n_______________________________________\n");
m[20][20]=eliminate(m,i,n);
display(m,n);
}
printf("\n\n Values are : \n");
for(i=n-1;i>=0;i--)
{
l=n-1;
mul=0;
for(j=0;j<k;j++)
{
mul=mul+m[i][l]*ans[l];
l--;
}
k++;
ans[i]=(m[i][n]-mul)/m[i][i];
printf("X%d = %.2f\n",i+1,ans[i]);
}
getch();
}
