In this post we will be doing a few problems on Gauss-Elimination. These problems/exercises were given in my Numerical Analysis class.

These will help in polishing one’s skills in solving different kind of systems, and working with different kinds of matrices, and in the process bring out some inherent problems/intricacies with the Gaussian Elimination procedure.

### Exercise 1:

Solve a system of equations, given by:

where is a square Hilbert matrix whose elements are given as:

where

and

Notice how the results change when you change the precision from float to double.

### CODE:

/**gaussElim Exercise***/ #include<stdio.h> #include<math.h> /******* Function that performs Gauss-Elimination and returns the Upper triangular matrix and solution of equations: There are two options to do this in C. 1. Pass the augmented matrix (a) as the parameter, and calculate and store the upperTriangular(Gauss-Eliminated Matrix) in it. 2. Use malloc and make the function of pointer type and return the pointer. This program uses the first option. ********/ void gaussEliminationLS(int m, int n, double a[m][n], double x[n-1][1]){ int i,j,k; for(i=0;i<m-1;i++){ //Partial Pivoting for(k=i+1;k<m;k++){ //If diagonal element(absolute vallue) is smaller than any of the terms below it if(fabs(a[i][i])<fabs(a[k][i])){ //Swap the rows for(j=0;j<n;j++){ double temp; temp=a[i][j]; a[i][j]=a[k][j]; a[k][j]=temp; } } } //Begin Gauss Elimination for(k=i+1;k<m;k++){ double term=a[k][i]/ a[i][i]; for(j=0;j<n;j++){ a[k][j]=a[k][j]-term*a[i][j]; } } } //Begin Back-substitution for(i=m-1;i>=0;i--){ x[i][0]=a[i][n-1]; for(j=i+1;j<n-1;j++){ x[i][0]=x[i][0]-a[i][j]*x[j][0]; } x[i][0]=x[i][0]/a[i][i]; } } /******* Function that generates the Hilbert matrix Parameters: order (n) ,matrix[n][n] *******/ void genMatrix(int n, double matrix[n][n]){ int i,j; //Initialize Coefficients for(i=0;i<n;i++){ for(j=0;j<n;j++){ matrix[i][j]=(double)1.0/((i+1)+(j+1)-1); } } } /******* Function that generates the Augmented Hilbert matrix Parameters: order (n) ,matrix[n][n+1] *******/ void genAugMatrix(int n, double matrix[n][n+1]){ int i,j; //Initialize Coefficients for(i=0;i<n;i++){ for(j=0;j<n;j++){ matrix[i][j]=(double)1.0/((i+1)+(j+1)-1); } } //Initialize RHS part for(i=0;i<n;i++){ matrix[i][n]=1; } } /******* Function that prints the elements of a matrix row-wise Parameters: rows(m),columns(n),matrix[m][n] *******/ void printMatrix(int m, int n, double matrix[m][n]){ int i,j; for(i=0;i<m;i++){ for(j=0;j<n;j++){ printf("%lf\t",matrix[i][j]); } printf("\n"); } } /******* Function that copies the elements of a matrix to another matrix Parameters: rows(m),columns(n),matrix1[m][n] , matrix2[m][n] *******/ void copyMatrix(int m, int n, double matrix1[m][n], double matrix2[m][n]){ int i,j; for(i=0;i<m;i++){ for(j=0;j<n;j++){ matrix2[i][j]=matrix1[i][j]; } } } /******* Function that calculates the product of two matrices: There are two options to do this in C. 1. Pass a matrix (prod) as the parameter, and calculate and store the product in it. 2. Use malloc and make the function of pointer type and return the pointer. This program uses the first option. ********/ void matProduct(int m, int n, int n1,double a[m][n1], double b[n1][n], double prod[m][n]){ int i,j,k; for(i=0;i<m;i++){ for(j=0;j<n;j++){ prod[i][j]=0; for(k=0;k<n1;k++){ prod[i][j]=prod[i][j]+a[i][k]*b[k][j]; } } } } int main(){ int n,i,j; printf("Enter the order:\n(n)\n"); scanf("%d",&n); //Declare a matrix to store the augmented Hilbert matrix for the problem double a[n][n+1]; //Declare another matrix to store the resultant matrix obtained after Gauss Elimination double U[n][n+1]; //Declare an array to store the solution of equations double x[n][1]; genAugMatrix(n,a); printf("The auto-generated augmented Hilbert matrix for the problem is:\n\n"); printMatrix(n,n+1,a); copyMatrix(n,n+1,a,U); //Perform Gauss Elimination gaussEliminationLS(n,n+1,U,x); printf("\nThe Upper Triangular matrix after Gauss Eliminiation is:\n\n"); printMatrix(n,n+1,U); printf("\nThe solution of linear equations is:\n\n"); for(i=0;i<n;i++){ printf("x[%d]=\t%lf\n",i+1,x[i][0]); } //Now we will verify if the answer is correct by multiplying X (solution) with the Hilbert matrix and see if we get 1. double B[n][1]; //matrix to store product: A.X=B //Declare a matrix to store the un-augmented(square) Hilbert matrix for the problem double a1[n][n]; genMatrix(n,a1); matProduct(n,1,n,a1,x,B); //Print the product to verify printf("\nThe product of matrix A.X=B:\n\n"); printMatrix(n,1,B); }

### OUTPUT:

### Android Apps:

I’ve also created a few Android apps that perform various matrix operations and can come in handy to those taking a course on Numerical Methods.

Download: https://play.google.com/store/apps/details?id=com.bragitoff.numericalmethods

Download: https://play.google.com/store/apps/details?id=com.bragitoff.matrixcalculator