Homework, Material, and Software Manual for Math 4610
Routine Name: gaussSeidelMP
Author: Tanner Wheeler
Language: C++. This can be compiled using the GNU compiler. On the command line when in the file containing the code use the command
g++ -fopenmp ./<fileNameHere>.cpp
This will create a a.out file. To run the program use the command
./a
Description/Purpose: This method will run the Gauss-Seidel method, using parallelization, on a matrix that is symmetric and positive definite.
Input: Given the equation Ax = b. A two dimensional vector array is the first input representing A. It has dimensions nxn. The second is a vector array of length n representing b. The third input is a vector array of length n representing a guess of x. The fourth is an integer of the maximum number of times the method should iterate. The last input is a small double number for the amount of error allowed for the answer.
Output: This will output the solution x that satisfies the equation Ax=b. It will be a vector array of length n.
Usage/Example:
In our main function we want to create a two dimension vector array a, vector array b, vector array x0 as a guess, use 15 for maxiter, and .0001 for tol.
#include<vector>
#include<iostream>
int main(void)
{
int n = 3;
std::vector<double> b(n);
std::vector<double> x(n);
std::vector<std::vector<double>> a(n, std::vector<double>(n))
a[0][0] = 7.0
a[0][1] = 3.0
a[0][2] = 1.0
a[1][0] = 3.0
a[1][1] = 10.0
a[1][2] = 2.0
a[2][0] = 1.0
a[2][1] = 2.0
a[2][2] = 15.0
b[0] = 11.0
b[1] = 15.0
b[2] = 18.0
for (int i = 0; i < x.size(); i++)
{
x[i] = 0.0; // Our guess will be a vector of zeros
}
std::cout << gaussSeidelMP(a, b, 15, .0001) << std::endl;
return 0;
}
You have created
a = [[7.0, 3.0, 1.0],
[3.0, 10.0, 2.0],
[1.0, 2.0, 15.0]]
b = [11.0, 15.0, 18.0]
Our output for the gaussSeidelMP method will be
[1.0, 1.0, 1.0]
Implementation/Code: The following is the code for gaussSeidelMP(a, b, x0, maxiter, tol)
#include <vector>
std::vector<double> gaussSeidelMP(std::vector<std::vector<double>> a, std::vector<double> b, std::vector<double> x0, int maxiter, double tol)
{
int iter = 0;
int n = b.size();
double error = 10 * tol;
std::vector<double> x1 = x0;
while (error > tol && iter < maxiter)
{
{
int i,j;
double sum;
#pragma omp parallel for shared(x1, b, a, x0, n) private(i, j, sum)
for (i = 0; i < n; i++)
{
sum = 0.0f;
for (j = i + 1; j < n; j++)
sum += a[i][j] * x0[j];
x1[i] = sum;
}
}
{
int i,j;
double sum;
#pragma omp parallel for shared(x1, b, a, x0, n) private(i, j, sum)
for (i = 0; i < n; i++)
{
sum = 0.0f;
for (j = 0; j < i; j++)
sum += a[i][j];
x1[i] = (b[i] - x1[i]) / (a[i][i] + sum);
}
}
iter++;
double sum = 0.0f;
{
int j;
#pragma omp parallel private(j)
{
#pragma omp for
for (j = 0; j < n; j++)
{
sum += (x1[j] - x0[j]) * (x1[j] - x0[j]);
}
}
}
error = sqrt(sum);
{
int i;
#pragma omp parallel private(i)
{
#pragma omp for
for (i = 0; i < n; i++)
{
x0[i] = x1[i];
}
}
}
}
return x0;
}
Last Modified: December 2018