Applied Mathematics AM 117

APMA 1170 - Project on Linear Solvers

Problem

Define $h = \frac{1}{N}$ , $\mu = h^2$ , and $q[i][j] = (8 \pi^2 + 1) h^2 sin(2\pi h i) sin(2\pi h j)$ where $i,j = 0,\ldots,N-1$ . Let A be of the form given in the figure below.

**Figure:** Matrix system ${\bf A}u=q$ for a N=4
$\begin{figure} \centerline{\psfig{file=2dhelm_dismat_per.eps,width=4.0in}} \end{figure}$

Assignment

Solve the matrix system ${\bf A}u=q$ for u using the following for methods:

Serial Jacobi
Parallel Jacobi
Serial Conjugate Gradient
Serial Preconditioned Conjugate Gradient, preconditioned using incomplete Cholesky
Parallel Conjugate Gradient
Parallel Preconditioned Conjugate Gradient, preconditioned using incomplete Cholesky

Solve for both N=4 and N=20. Here N denotes the number of points used in both the x and y directions (hence the total number of grid points is N² points, which corresponds to the rank of the matrix for which we are solving). For the serial algorithms, provide a graphical plot of the solution u[i][j] at the points (hi,hj) (either a contour or surface plot). For the parallel algorithms, show the parallel speed-up using different number of processors (this will require you to use the MPI_Wtime function to time your runs).