McGill University
School of Computer Science

Computer Science 308-767: Parallel Programming

Gaussian Elimination on Shared Memory :

Gaussian elimination is a classical method for solving a matrix of linear equations of the form Ax = b. By performing elementary row operations, Gaussian elimination transforms the square matrix A into an equivalent upper-triangular matrix. Following transformation of A into upper triangular form, a back substitution phase solves for x. A high level overview of Gaussian elimination and back substitution can be found on Mathworld.

In this assignment, you will develop two parallel linear solvers that use Gaussian elimination with partial pivoting (partial pivoting is used to enhance numerical stability) to transform a dense n x n matrix into an upper-triangular one and then solve the resulting linear system by back substitution. Below is pseudocode written using Fortran 90 array notation for a sequential implementation of Gaussian elimination with partial pivoting.

    inputs: a(n,n), b(n)
    outputs: a(n,n), b(n) in echelon form

    do j=1,n-1
      ksave = maxloc(abs(a(j:n,j)))
      k = ksave(1) + j-1
      swap a(j,:) and a(k,:)
      swap b(j) and b(k)
      do k=j+1, n
        m = a(k,j)/a(j,j)
        a(k,:) = a(k,:) - m*a(j,:)
        b(k) = b(k) - m*b(j)
You will use two different shared-memory programming models to write a parallel linear solver based on Gaussian elimination with partial pivoting. Each parallel solver implementation will accept an arguments n - the size of a matrix. The number of threads may be specified as an argument, through an environment variable, or as an argument to a launching script. Your programs will allocate an n x n matrix, fill it with random floating point numbers ranging from 1 to 1000, apply Gaussian elimination with partial pivoting to transform the matrix into an upper triangular one, and then apply back substitution to compute x. To check your answer, compute the square root of the sum of the squares of the residual vector (this sum is known as the L2-norm) computed as Ax-b. Print the value of the L2-norm of the residual. (It should be very small.)

The back substitution and verification steps need not be parallelized. Have your program separately time the Gaussian elimination phase by reading the real-time clock before and afterward and printing the difference.

The formal components of the assignment are listed below:

Submitting your Assignment

Your assignment should be submitted in two parts. Email me a message with a text attachment containing the program and a second attachment containing your writeup (in either Word or pdf format).

Grading Criteria

Due Date

Thursday, March 8