# WATHEN Assemble, Factor, Solve a Finite Element System

WATHEN is a MATLAB library 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.

### Languages:

WATHEN is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

CG, a MATLAB library 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 MATLAB library which factors and solves linear systems using double precision real arithmetic, by Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart.

SPARSE, MATLAB programs which illustrate the use of MATLAB's sparse matrix utilities;

TEST_MAT, a MATLAB library 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.

### Reference:

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:

• bandwidth.m, returns the lower, diagonal and upper bandwidths of a matrix.
• cg_gb.m, solves A*x=b using the conjugate gradient method, with A a positive definite symmetric matrix using General Banded (GB) storage.
• cg_ge.m, solves A*x=b using the conjugate gradient method, with A a positive definite symmetric matrix using general (GE) storage.
• cg_sparse.m, solves A*x=b using the conjugate gradient method, with A a positive definite symmetric matrix using sparse storage.
• cg_st.m, solves A*x=b using the conjugate gradient method, with A a positive definite symmetric matrix using sparse triplet (ST) storage.
• daxpy.m, adds a multiple of one vector to another.
• dgbfa.m, factors a banded matrix.
• dgbsl.m, solves a linear system whose matrix has been factored by dgbfa.
• idamax.m, returns the maximum element in an integer vector.
• mv_gb.m, multiplies a banded matrix times a vector.
• mv_st.m, multiplies a sparse triplet matrix times a vector.
• r8_uniform_01.m, returns a unit random R8.
• r8mat_uniform_01.m, returns a unit random R8MAT.
• r8vec_print.m, prints an R8VEC.
• r8vec_uniform_01.m, returns a unit random R8VEC.
• st_to_ge.m, converts a sparse triplet (ST) matrix to general (GE) format.
• timestamp.m, prints the YMDHMS date as a timestamp.
• wathen_bandwidth.m, returns the bandwidth of the Wathen matrix.
• wathen_davis.m, sets up the Wathen matrix using sparse storage, as recommended by Tim Davis.
• wathen_gb.m, sets up the Wathen matrix using General Banded (GB) storage.
• wathen_ge.m, sets up the Wathen matrix using general (GE) storage.
• wathen_order.m, returns the number of unknowns associated with a given Wathen matrix.
• wathen_sparse.m, sets up the Wathen matrix using sparse storage.
• wathen_st.m, sets up the Wathen matrix using sparse triplet storage.
• wathen_st_size.m, returns the size of the Wathen matrix when using sparse triplet storage.

### Examples and Tests:

You can go up one level to the MATLAB source codes.