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<html>
<head>
<title>
WATHEN - Assemble, Factor, Solve a Finite Element System
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
WATHEN <br> Assemble, Factor, Solve a Finite Element System
</h1>
<hr>
<p>
<b>WATHEN</b>
is a C++ 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).
</p>
<p>
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
<pre>
N = 3 NX NY + 2 NX + 2 NY + 1
</pre>
and the full linear system will require N * N storage for the matrix.
</p>
<p>
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.
</p>
<p>
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.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>WATHEN</b> is available in
<a href = "../../c_src/wathen/wathen.html">a C version</a> and
<a href = "../../cpp_src/wathen/wathen.html">a C++ version</a> and
<a href = "../../f77_src/wathen/wathen.html">a FORTRAN77 version</a> and
<a href = "../../f_src/wathen/wathen.html">a FORTRAN90 version</a> and
<a href = "../../m_src/wathen/wathen.html">a MATLAB version</a> and
<a href = "../../py_src/wathen/wathen.html">a Python version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/cg/cg.html">
CG</a>,
a C++ 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.
</p>
<p>
<a href = "../../cpp_src/linpack_d/linpack_d.html">
LINPACK_D</a>,
a C++ library which
factors and solves linear systems using double precision real arithmetic,
by Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart.
</p>
<p>
<a href = "../../cpp_src/sparse_display/sparse_display.html">
SPARSE_DISPLAY</a>,
a C++ library 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.
</p>
<p>
<a href = "../../cpp_src/test_mat/test_mat.html">
TEST_MAT</a>,
a C++ 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.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Nicholas Higham,<br>
Algorithm 694:
A Collection of Test Matrices in MATLAB,<br>
ACM Transactions on Mathematical Software,<br>
Volume 17, Number 3, September 1991, pages 289-305.
</li>
<li>
Andrew Wathen,<br>
Realistic eigenvalue bounds for the Galerkin mass matrix,<br>
IMA Journal of Numerical Analysis,<br>
Volume 7, Number 4, October 1987, pages 449-457.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "wathen.cpp">wathen.cpp</a>, the source code.
</li>
<li>
<a href = "wathen.hpp">wathen.hpp</a>, the source code.
</li>
<li>
<a href = "wathen.sh">wathen.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "wathen_prb.cpp">wathen_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "wathen_prb.sh">wathen_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "wathen_prb_output.txt">wathen_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>BANDWIDTH</b> returns the bandwidth of a matrix.
</li>
<li>
<b>CG_FULL</b> uses the conjugate gradient method for a full storage matrix.
</li>
<li>
<b>CG_ST</b> uses the conjugate gradient method for a sparse triplet storage matrix.
</li>
<li>
<b>DAXPY</b> computes constant times a vector plus a vector.
</li>
<li>
<b>DDOT</b> forms the dot product of two vectors.
</li>
<li>
<b>DGBFA</b> factors a real band matrix by elimination.
</li>
<li>
<b>DGBSL</b> solves a real banded system factored by DGBCO or DGBFA.
</li>
<li>
<b>DGEFA</b> factors a real general matrix.
</li>
<li>
<b>DGESL</b> solves a real general linear system A * X = B.
</li>
<li>
<b>DSCAL</b> scales a vector by a constant.
</li>
<li>
<b>IDAMAX</b> indexes the array element of maximum absolute value.
</li>
<li>
<b>NONZEROS</b> counts the nonzeros in a matrix.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8GE_CG</b> uses the conjugate gradient method on an R8GE system.
</li>
<li>
<b>R8GE_MXV</b> multiplies an R8GE matrix by an R8VEC.
</li>
<li>
<b>R8MAT_UNIFORM_01</b> fills an R8MAT with unit pseudorandom numbers.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
</li>
<li>
<b>ST_MV</b> multiplies a sparse triple matrix times a vector.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>WATHEN_BANDWIDTH</b> returns the bandwidth of the WATHEN matrix.
</li>
<li>
<b>WATHEN_BANDED</b> returns the Wathen matrix, using banded storage.
</li>
<li>
<b>WATHEN_FULL</b> returns the Wathen matrix, using full storage.
</li>
<li>
<b>WATHEN_ORDER</b> returns the order of the WATHEN matrix.
</li>
<li>
<b>WATHEN_ST:</b> Wathen matrix stored in sparse triplet format.
</li>
<li>
<b>WATHEN_ST_SIZE:</b> Size of Wathen matrix stored in sparse triplet format.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 08 June 2014.
</i>
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