Sparse Matrices

Q: One can use LinearSolve to solve a system of linear equations. However, I work with very large matrices having only five nonzero diagonals. Are there efficient methods for dealing with such matrices?

A: Daniel Lichtblau (danl@wolfram.com) answers: If you require only machine arithmetic, this can be done effectively using the built-in sparse linear solver in Version 5. Here is an example.

Create a diagonally-dominant random sparse -diagonal matrix using SparseArray.

With and , we obtain a sparse array with nonzero elements.

Use Normal to obtain the ordinary array corresponding to a sparse array object.

Produce a pentadiagonal sparse matrix of dimension 2000.

Solve for a random right-hand side.

Compare with the time required for an ordinary (nonsparse) matrix.

Check that the solutions agree using (for matrices, gives the maximum singular value of ).