Complete the program given below that implements the Gauss-Seidel iterative meth

Question

Complete the program given below that implements the Gauss-Seidel iterative method for a general system of n equations with n unknowns ( n should be an input variable). Limit the maximum number of equations that can be solved at any one time to 100. Your code should be able to solve the matrix equation Ax ? b , and print out the vector x . The program:

? Reads in the size of the system to be solved ( n ).

? Then, it reads in all the elements of the system matrix A and of the right-hand-side

vector b.

? Next, it tests whether the sufficient convergence criterion for the method is met.

? Next, it solves the system by calling a function that implements the Gauss-Seidel method.

You need to complete the function. Recall that the method is iterative and will not

always converge; therefore, you should include a maximum number of allowed iterations.

? Test your code with the following system:

/* code to solve a nxn system using the Gauss-Seidel method */

void gauss_seidel (double a[][MAX_DIM], double b[], double x[], int n); void main ()
{

");
}

printf ("Enter the dimension of the system to be solved: scanf ("%d",&n);

for (j = 0; j < n; j++) {
printf ("Enter a[%d][%d] element of the system

}
printf ("Enter b[%d] of the right-hand-side vector: ",i); scanf ("%lf",&b[i]);
printf (" ");

violation_rows[violation_counter]= i; violation_counter = violation_counter + 1;

printf ("Found diagonal element equal to zero; rearrange equations; exiting ... ");

exit (0); } }

if (violation_counter > 0) {
printf ("The Gauss-Seidel convergence criterion is violated

in %d rows out of %d ", violation_counter, n); printf ("Specifically, it was violated in rows: ");
for (i = 0; i < violation_counter; i++)

printf ("Enter 1 if you want to continue; any other number to abort: ");

exit(1);
printf ("Check results carefully ");

}
/* initialize the solution vector -- initial guesses */

for (i = 0; i < n; i++) {
printf ("Enter an initial guess for x[%d] of the

}
/* function to solve a system using Gauss-Seidel */

void gauss_seidel (double a[][MAX_DIM], double b[], double x[], int n) {

Explanation / Answer

Please use the below piece of code to solve a system using Gauss-Seidel method:

#include <stdio.h>
#include <math.h>
#include <stdlib.h>

#define MAX_DIM 100
#define MAX_ITER 500
#define TOLERANCE 1.e-6

void gauss_seidel (double a[][MAX_DIM], double b[], double x[], int n);

void main ()

{
   int i, j, n;
   int violation_counter, answer;
   int violation_rows[MAX_DIM];
   double sum;
   double a[MAX_DIM][MAX_DIM];
   double b[MAX_DIM],x[MAX_DIM];

   /* read in data */

   n = MAX_DIM + 1;
   while (n > MAX_DIM) {
       printf ("Enter the dimension of the system to be solved: ");
       scanf ("%d",&n);
   }

   printf (" ");
   for (i = 0; i < n; i++) {
       for (j = 0; j < n; j++) {
           printf ("Enter the (%d,%d) element of the system matrix: ",i+1,j+1);
           scanf ("%lf",&a[i][j]);
       }
       printf ("Enter element (%d) of the right-hand-side vector: ",i+1);
       scanf ("%lf",&b[i]);
       printf (" ");
   }

   /* test the convergence criterion */

   violation_counter = 0;
   for (i = 0; i < n; i++) {
       sum = 0.0;
       for (j = 0; j < n; j++)
           if (i != j)
               sum = sum + fabs(a[i][j]);
       if (fabs(a[i][j]) < sum) {
           violation_rows[violation_counter] = i;
           violation_counter = violation_counter + 1;
       }
       if (a[i][j] == 0.0) {
           printf ("Found diagonal element equal to zero; rearrange equations; exiting ... ");
           exit (0);
       }
   }

   if (violation_counter > 0) {
       printf ("The Gauss_Seidel convergence criterion is violated in %d rows out of %d ", violation_counter, n);
       printf ("Specifically, it was violated in rows: ");
       for (i = 0; i < violation_counter; i++)
           printf ("%d ",violation_rows[i]+1);
       printf (" ");
       printf ("Enter 1 if you want to continue; any other number to abort: ");
       scanf ("%d",&answer);
       if (answer != 1)
           exit(1);
       printf ("Check results carefully ");
   }

   /* Initialize the solution vector -- initial guesses */

   for (i = 0; i < n; i++) {
       printf ("Enter an initial guess for element (%d) of the solution vector: ",i+1);
       scanf ("%lf",&x[i]);
   }

   /* solve the system */

   gauss_seidel (a,b,x,n);

   /* output solution */

   for (i = 0; i < n; i++)
       printf ("x[%d]=%f ",i+1,x[i]);
   printf (" ");
}

/* function to solve a system using Gauss-Seidel */

void gauss_seidel (double a[][MAX_DIM], double b[], double x[], int n) {

double maxerror = 1.0;
double iteration_error;
double e, sum, temp;
int i, j;

while (maxerror > 1.e-6) {
   iteration_error = 0.0;
  
   for (i = 0; i < n; i++) {
       sum = 0.0;
       for (j = 0; j < n; j++) {
           if (i != j)
               sum = sum + (a[i][j] * x[j]);
       }
   }
  
   temp = (a[i][n] - sum)/a[i][i];
   e = fabs((temp - x[i]) / x[i]);
   x[i]= temp;
  
   if (e > iteration_error)
           iteration_error = e;
   }
   maxerror = iteration_error;
}

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