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<html>
<head>
<title>
POISSON_OPENMP - Poisson Equation, Jacobi Iteration Parallelized with OpenMP
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
POISSON_OPENMP <br> Poisson Equation, Jacobi Iteration Parallelized with OpenMP
</h1>
<hr>
<p>
<b>POISSON_OPENMP</b>
is a FORTRAN90 program which
computes an approximate solution to the Poisson equation
in a rectangular region,
using OpenMP to carry out the Jacobi iteration in parallel.
</p>
<p>
The version of Poisson's equation being solved here is
<pre>
- ( d/dx d/dx + d/dy d/dy ) U(x,y) = F(x,y)
</pre>
over the rectangle 0 <= X <= 1, 0 <= Y <= 1, with exact solution
<pre>
U(x,y) = sin ( pi * x * y )
</pre>
so that
<pre>
F(x,y) = pi^2 * ( x^2 + y^2 ) * sin ( pi * x * y )
</pre>
and with Dirichlet boundary conditions along the lines x = 0, x = 1,
y = 0 and y = 1. (The boundary conditions will actually be zero in
this case, but we write up the problem as though we didn't know that,
which makes it easy to change the problem later.)
</p>
<p>
We compute an approximate solution by discretizing the geometry,
assuming that DX = DY, and approximating the Poisson operator by
<pre>
( U(i-1,j) + U(i+1,j) + U(i,j-1) + U(i,j+1) - 4*U(i,j) ) / dx /dy
</pre>
Along with the boundary conditions at the boundary nodes, we have
a linear system for U. We can apply the Jacobi iteration to estimate
the solution to the linear system.
</p>
<p>
OpenMP is used in this example to carry
out the Jacobi iteration in parallel. Note that the Jacobi iteration can
converge very slowly, and the slowness increases as the matrix gets bigger.
Thus, if you must use the Jacobi iteration, parallelism can help you.
But you might also find, at some point, that getting a better linear
solver (even a non-parallel one!) would help you more.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and 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>POISSON_OPENMP</b> is available in
<a href = "../../c_src/poisson_openmp/poisson_openmp.html">a C version</a> and
<a href = "../../cpp_src/poisson_openmp/poisson_openmp.html">a C++ version</a> and
<a href = "../../f77_src/poisson_openmp/poisson_openmp.html">a FORTRAN77 version</a> and
<a href = "../../f_src/poisson_openmp/poisson_openmp.html">a FORTRAN90 version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/dijkstra_openmp/dijkstra_openmp.html">
DIJKSTRA_OPENMP</a>,
a FORTRAN90 program which
uses OpenMP to parallelize a simple example of Dijkstra's
minimum distance algorithm for graphs.
</p>
<p>
<a href = "../../f_src/fft_openmp/fft_openmp.html">
FFT_OPENMP</a>,
a FORTRAN90 program which
demonstrates the computation of a Fast Fourier Transform
in parallel, using OpenMP.
</p>
<p>
<a href = "../../f_src/heated_plate_openmp/heated_plate_openmp.html">
HEATED_PLATE_OPENMP</a>,
a FORTRAN90 program which
solves the steady (time independent) heat equation in a 2D
rectangular region, using OpenMP to run in parallel.
</p>
<p>
<a href = "../../f_src/hello_openmp/hello_openmp.html">
HELLO_OPENMP</a>,
a FORTRAN90 program which
prints out "Hello, world!" using the OpenMP parallel programming environment.
</p>
<p>
<a href = "../../f_src/md_openmp/md_openmp.html">
MD_OPENMP</a>,
a FORTRAN90 program which
carries out a molecular dynamics simulation using OpenMP.
<p>
<p>
<a href = "../../f_src/multitask_openmp/multitask_openmp.html">
MULTITASK_OPENMP</a>,
a FORTRAN90 program which
demonstrates how to "multitask", that is, to execute several unrelated
and distinct tasks simultaneously, using OpenMP for parallel execution.
</p>
<p>
<a href = "../../f_src/mxv_openmp/mxv_openmp.html">
MXM_OPENMP</a>,
a FORTRAN90 program which
computes a dense matrix product C=A*B,
using OpenMP for parallel execution.
<p>
<p>
<a href = "../../f_src/mxv_openmp/mxv_openmp.html">
MXV_OPENMP</a>,
a FORTRAN90 program which
compares the performance of plain vanilla Fortran and the FORTRAN90
intrinsic routine MATMUL, for the matrix multiplication problem
y=A*x, with and without parallelization by OpenMP.
<p>
<p>
<a href = "../../f_src/openmp/openmp.html">
OPENMP</a>,
FORTRAN90 programs which
illustrate the use of the OpenMP application program interface
for carrying out parallel computations in a shared memory environment.
</p>
<p>
<a href = "../../f_src/poisson_serial/poisson_serial.html">
POISSON_SERIAL</a>,
a FORTRAN90 program which
computes an approximate solution to the Poisson equation in a rectangle,
and is intended as the starting point for the creation of a parallel version.
</p>
<p>
<a href = "../../f_src/prime_openmp/prime_openmp.html">
PRIME_OPENMP</a>,
a FORTRAN90 program which
counts the number of primes between 1 and N, using OpenMP for parallel execution.
</p>
<p>
<a href = "../../f_src/quad_openmp/quad_openmp.html">
QUAD_OPENMP</a>,
a FORTRAN90 program which
approximates an integral using a quadrature rule, and carries out the
computation in parallel using OpenMP.
</p>
<p>
<a href = "../../f_src/random_openmp/random_openmp.html">
RANDOM_OPENMP</a>,
a FORTRAN90 program which
illustrates how a parallel program using OpenMP can generate multiple
distinct streams of random numbers.
</p>
<p>
<a href = "../../f_src/satisfy_openmp/satisfy_openmp.html">
SATISFY_OPENMP</a>,
a FORTRAN90 program which
demonstrates, for a particular circuit, an exhaustive search
for solutions of the circuit satisfiability problem,
using OpenMP for parallel execution.
<p>
<p>
<a href = "../../f_src/schedule_openmp/schedule_openmp.html">
SCHEDULE_OPENMP</a>,
a FORTRAN90 program which
demonstrates the default, static, and dynamic methods of "scheduling"
loop iterations in OpenMP to avoid work imbalance.
<p>
<p>
<a href = "../../f_src/sgefa_openmp/sgefa_openmp.html">
SGEFA_OPENMP</a>,
a FORTRAN90 program which
reimplements the SGEFA/SGESL linear algebra routines from
LINPACK for use with OpenMP.
</p>
<p>
<a href = "../../f_src/ziggurat_openmp/ziggurat_openmp.html">
ZIGGURAT_OPENMP</a>,
a FORTRAN90 program which
demonstrates how the ZIGGURAT library can be used to generate random numbers
in an OpenMP parallel program.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Michael Quinn,<br>
Parallel Programming in C with MPI and OpenMP,<br>
McGraw-Hill, 2004,<br>
ISBN13: 978-0071232654,<br>
LC: QA76.73.C15.Q55.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "poisson_openmp.f90">poisson_openmp.f90</a>, the source code.
</li>
<li>
<a href = "poisson_local.sh">poisson_local.sh</a>,
BASH commands to compile the source code.
</li>
<li>
<a href = "poisson_local_output.txt">poisson_local_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for POISSON_OPENMP.
</li>
<li>
<b>R8MAT_RMS</b> returns the root mean square of data stored as an R8MAT.
</li>
<li>
<b>RHS</b> initializes the right hand side "vector".
</li>
<li>
<b>SWEEP</b> carries out several steps of the Jacobi iteration.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>U_EXACT</b> evaluates the exact solution.
</li>
<li>
<b>UXXYY_EXACT</b> evaluates ( d/dx d/dx + d/dy d/dy ) of the exact solution.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 13 December 2011.
</i>
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