forked from johannesgerer/jburkardt-cpp
-
Notifications
You must be signed in to change notification settings - Fork 0
/
rkf45.html
259 lines (227 loc) · 6.69 KB
/
rkf45.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
<html>
<head>
<title>
RKF45 - Runge-Kutta-Fehlberg ODE Solver
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
RKF45 <br> Runge-Kutta-Fehlberg ODE Solver
</h1>
<hr>
<p>
<b>RKF45</b>
is a C++ library which
implements the Watt and Shampine RKF45 ODE solver.
</p>
<p>
The RKF45 ODE solver is a Runge-Kutta-Fehlberg algorithm for
solving an ordinary differential equation, with automatic error
estimation using rules of order 4 and 5.
</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>RKF45</b> is available in
<a href = "../../c_src/rkf45/rkf45.html">a C version</a> and
<a href = "../../cpp_src/rkf45/rkf45.html">a C++ version</a> and
<a href = "../../f77_src/rkf45/rkf45.html">a FORTRAN77 version</a> and
<a href = "../../f_src/rkf45/rkf45.html">a FORTRAN90 version</a> and
<a href = "../../m_src/rkf45/rkf45.html">a MATLAB version</a> and
<a href = "../../py_src/rkf45/rkf45.html">a PYTHON version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/gsl/gsl.html">
GSL</a>,
a C++ library which
includes routines for solving differential equations.
</p>
<p>
<a href = "../../f_src/nms/nms.html">
NMS</a>,
a FORTRAN90 library which
includes the DDRIV package of ODE solvers.
</p>
<p>
<a href = "../../cpp_src/ode/ode.html">
ODE</a>,
a C++ library which
solves a system of ordinary differential equations,
by Shampine and Gordon.
</p>
<p>
<a href = "../../f77_src/odepack/odepack.html">
ODEPACK</a>,
a FORTRAN77 library which
contains nine ODE solvers, including LSODE, LSODES, LSODA,
LSODAR, LSODPK, LSODKR, LSODI, LSOIBT, and LSODIS,
by Alan Hindmarsh.
</p>
<p>
<a href = "../../cpp_src/rk4/rk4.html">
RK4</a>,
a C++ library which
applies the fourth order Runge-Kutta algorithm to estimate the
solution of an ordinary differential equation at the next time step.
</p>
<p>
<a href = "../../f_src/test_ode/test_ode.html">
TEST_ODE</a>,
a FORTRAN90 library which
contains routines which define some test problems for ODE solvers.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Erwin Fehlberg,<br>
Low-order Classical Runge-Kutta Formulas with Stepsize Control,<br>
NASA Technical Report R-315, 1969.
</li>
<li>
Lawrence Shampine, Herman Watts, S Davenport,<br>
Solving Non-stiff Ordinary Differential Equations -
The State of the Art,<br>
SIAM Review,<br>
Volume 18, pages 376-411, 1976.
</li>
<li>
The source code for Shampine and Watt's original FORTRAN77
routine is available at
<a href = "http://www.netlib.org/ode/">
http://www.netlib.org/ode/</a>
the NETLIB ODE web site.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "rkf45.cpp">rkf45.cpp</a>, the source code;
</li>
<li>
<a href = "rkf45.hpp">rkf45.hpp</a>, the include file;
</li>
<li>
<a href = "rkf45.sh">rkf45.sh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>RKF45_PRB</b> includes a number of examples of how to use RKF45.
<ul>
<li>
<a href = "rkf45_prb.cpp">rkf45_prb.cpp</a>, a sample calling program;
</li>
<li>
<a href = "rkf45_prb.sh">rkf45_prb.sh</a>,
commands to compile and run the sample calling program;
</li>
<li>
<a href = "rkf45_prb_output.txt">rkf45_prb_output.txt</a>,
the output file;
</li>
</ul>
</p>
<p>
<b>RKF45_PRB2</b> includes an example in which the ODE includes parameters
ALPHA, BETA, and GAMMA, which the user wants to set at run time. The example
shows one way in which these values can be shared with the derivative routine.
<ul>
<li>
<a href = "rkf45_prb2.cpp">rkf45_prb2.cpp</a>, a sample calling program;
</li>
<li>
<a href = "rkf45_prb2.sh">rkf45_prb2.sh</a>,
commands to compile and run the sample calling program;
</li>
<li>
<a href = "rkf45_prb2_output.txt">rkf45_prb2_output.txt</a>,
the output file;
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>R4_ABS</b> returns the absolute value of an R4.
</li>
<li>
<b>R4_EPSILON</b> returns the round off unit for floating arithmetic.
</li>
<li>
<b>R4_FEHL</b> takes one Fehlberg fourth-fifth order step.
</li>
<li>
<b>R4_MAX</b> returns the maximum of two real values.
</li>
<li>
<b>R4_MIN</b> returns the minimum of two real values.
</li>
<li>
<b>R4_RKF45</b> carries out the Runge-Kutta-Fehlberg method.
</li>
<li>
<b>R4_SIGN</b> returns the sign of a real number.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of a double precision number.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 round off unit.
</li>
<li>
<b>R8_FEHL</b> takes one Fehlberg fourth-fifth order step.
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8s.
</li>
<li>
<b>R8_MIN</b> returns the minimum of two R8s.
</li>
<li>
<b>R8_RKF45</b> carries out the Runge-Kutta-Fehlberg method.
</li>
<li>
<b>R8_SIGN</b> returns the sign of a real number.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</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 20 November 2007.
</i>
<!-- John Burkardt -->
</body>
</html>