Purpose¶

Attempts to solve the system of linear equations $$Ax = b$$ using the conjugate gradient method where A is a symmetric sparse matrix.

Format¶

x = spConjGradSol(a, b, epsilon, maxit)
Parameters
• a (sparse matrix) – NxN symmetric sparse matrix.

• b (vector) – Nx1 dense vector.

• epsilon (scalar) – Method tolerance: If epsilon is set to 0, the default tolerance is set to 1e-6.

• maxit (scalar) – Maximum number of iterations. If maxit is set to 0, the default setting is 300 iterations.

Returns

x (Nx1 dense vector) – dense vector, the solution to the system of linear equations $$Ax = b$$.

Examples¶

nz = {   0.000   2845.607     0.000     0.000     0.000,
2845.607  10911.430     0.000     0.000     0.000,
0.000      0.000  3646.798  2736.338 -2674.440,
0.000      0.000  2736.338  7041.526 -3758.528,
0.000      0.000 -2674.440 -3758.528  7457.899 };

sparse matrix a;

// Set 'a' to be a sparse matrix with the same contents as
// the dense matrix 'nz'
a = densetosp(nz, 0);

// Create our right-hand-side
b = { 10.349,
-3.117,
4.240,
0.013,
2.115 };

// Setting the third and fourth arguments to 0 employs the
// default tolerance maxit settings
x = spConjGradSol(a, b, 0, 0);

newb = a*x;


The results from the above code are:

      -0.01504075
0.00363683
x  =   0.00203504
-0.00033936
0.00084234

10.34900000
-3.11700000
newb = 4.24000000
0.01300000
2.11500000


Remarks¶

If convergence is not reached within the maximum number of iterations allowed, the function will either terminate the program with an error message or return an error code which can be tested for with the scalerr() function. This depends on the trap state as follows:

 trap 1 return error code: 60 trap 0 terminate with error message: Unable to converge in allowed number of iterations.

If matrix A is not symmetric or well conditioned use the / operator to perform the solve. For a nonsymmetric, but well conditioned matrix A, use spBiconjGradSol().

Functions spBiconjGradSol()