wathen, a C++ code which compares storage schemes (full, banded, sparse triplet, sparse) and solution strategies (A\x, Linpack, conjugate gradient (CG)) for linear systems involving the Wathen matrix, which can arise when solving a problem using the finite element method (FEM).

The Wathen matrix is a typical example of a matrix that arises during finite element computations. The parameters NX and NY specify how many elements are to be set up in the X and Y directions. The number of variables N is then

        N = 3 NX NY + 2 NX + 2 NY + 1
and the full linear system will require N * N storage for the matrix.

However, the matrix is sparse, and a banded or sparse storage scheme can be used to save storage. However, even if storage is saved, a revised program may eat up too much time because MATLAB's sparse storage scheme is not efficiently used by inserting nonzero elements one at a time. Moreover, if banded storage is employed, the user must provide a suitable fast solver. Simply "translating" a banded solver from another language will probably not provide an efficient routine.

This library looks at how the complexity of the problem grows with increasing NX and NY; how the computing time increases; how the various full, banded and sparse approaches perform.


The computer code and data files made available on this web page are distributed under the MIT license


wathen is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

CG, a C++ code which implements a simple version of the conjugate gradient (CG) method for solving a system of linear equations of the form A*x=b, suitable for situations in which the matrix A is positive definite (only real, positive eigenvalues) and symmetric.

LINPACK_D, a C++ code which factors and solves linear systems using double precision real arithmetic, by Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart.

SPARSE_DISPLAY, a C++ code which can read information defining a matrix of numbers and display the sparsity pattern or location of the nonzero elements using gnuplot. This operation is already available in the built-in MATLAB "spy" command.

TEST_MAT, a C++ code which defines test matrices for which some of the determinant, eigenvalues, inverse, null vectors, P*L*U factorization or linear system solution are already known, including the Vandermonde and Wathen matrix.



  1. Nicholas Higham,
    Algorithm 694: A Collection of Test Matrices in MATLAB,
    ACM Transactions on Mathematical Software,
    Volume 17, Number 3, September 1991, pages 289-305.
  2. Andrew Wathen,
    Realistic eigenvalue bounds for the Galerkin mass matrix,
    IMA Journal of Numerical Analysis,
    Volume 7, Number 4, October 1987, pages 449-457.

Source Code:

Last revised on 18 February 2020.