Create matrix positive definide

Question

I´m developing a code in c++ using openMP to test the power of parallelism using openMp. I create the code to solve linear system using Jacobi by aproximation. Now I need to test using a big matrix.

My matrix to test is little.

$$\begin{cases}2x_1-x_2 = 1\\ x_1+2x_2=3\end{cases}$$

Now I need help because I don´t know how to create a bigger matrix like this.

Answer 1

If all you need is a large positive definite matrix, take a random matrix $A$ and take $M=A^TA$. $M$ will then be symmetric positive semidefinite, since

• $M^T = (A^TA)^T=A^T(A^T)^T=A^TA$
• For any $x$, $x^TMx =x^TA^TAx = (Ax)^T Ax\geq 0$.

If the rank of $A$ is full, meaning $Ax=0$ implies $x=0$, then $M$ will also be positive definite.

Answer 2

Well, on a $2\times 2$ system you won't see much improvements using any parallelism :-)

Since you implement the Jacobi method, I suppose you want your matrix to be sparse. Why don't you simply take, say, a finite difference discretization of a Laplacian? You can create it in MATLAB simply as follows:

% 1D Laplacian:
N  = 100; % Size of the matrix
A1 = gallery('tridiag', N); % Generate the tridiag(-1,2,-1) matrix of size N

% 2D Laplacian:
I  = speye(N); % Create the sparse NxN identity matrix
A2 = kron(I, A1) + kron(A1, I);

You can also create a 2D Laplacian by:

% 2D Laplacian, using gallery:
A2 = gallery('poisson', N);

Note that the size of $A_2$ is $N^2\times N^2$. Similarly, you can create a 3D discrete Laplacian ($N^3\times N^3$ matrix) by:

% 3D Laplacian:
A3 = kron(I,I,A1) + kron(I,A1,I) + kron(A1,I,I);

Now if you want to store a matrix, say, $A$, to a file, say, matrix.txt, you can do the following:

% Save A to a file
[i,j,a] = find(A);
matrix_data = [i,j,a];
save -ascii matrix.txt matrix_data

In the file, you find the triplets of the row and column indices and their associated matrix entry value on each line. You can read it simply in C++ and then do whatever you want to do with it.