forked from johannesgerer/jburkardt-cpp
-
Notifications
You must be signed in to change notification settings - Fork 0
/
legendre_polynomial.html
328 lines (289 loc) · 9.33 KB
/
legendre_polynomial.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
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
<html>
<head>
<title>
LEGENDRE_POLYNOMIAL - Legendre Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
LEGENDRE_POLYNOMIAL <br> Legendre Polynomials
</h1>
<hr>
<p>
<b>LEGENDRE_POLYNOMIAL</b>
is a C++ library which
evaluates the Legendre polynomial and associated functions.
</p>
<p>
The Legendre polynomial P(n,x) can be defined by:
<pre>
P(0,x) = 1
P(1,x) = x
P(n,x) = (2*n-1)/n * x * P(n-1,x) - (n-1)/n * P(n-2,x)
</pre>
where n is a nonnegative integer.
</p>
<p>
The N zeroes of P(n,x) are the abscissas used for Gauss-Legendre
quadrature of the integral of a function F(X) with weight function 1
over the interval [-1,1].
</p>
<p>
The Legendre polynomials are orthogonal under the inner product defined
as integration from -1 to 1:
<pre>
Integral ( -1 <= x <= 1 ) P(i,x) * P(j,x) dx
= 0 if i =/= j
= 2 / ( 2*i+1 ) if i = j.
</pre>
</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>LEGENDRE_POLYNOMIAL</b> is available in
<a href = "../../c_src/legendre_polynomial/legendre_polynomial.html">a C version</a> and
<a href = "../../cpp_src/legendre_polynomial/legendre_polynomial.html">a C++ version</a> and
<a href = "../../f77_src/legendre_polynomial/legendre_polynomial.html">a FORTRAN77 version</a> and
<a href = "../../f_src/legendre_polynomial/legendre_polynomial.html">a FORTRAN90 version</a> and
<a href = "../../m_src/legendre_polynomial/legendre_polynomial.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/bernstein_polynomial/bernstein_polynomial.html">
BERNSTEIN_POLYNOMIAL</a>,
a C++ library which
evaluates the Bernstein polynomials,
useful for uniform approximation of functions;
</p>
<p>
<a href = "../../cpp_src/chebyshev_polynomial/chebyshev_polynomial.html">
CHEBYSHEV_POLYNOMIAL</a>,
a C++ library which
considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x).
Functions are provided to evaluate the polynomials, determine their zeros,
produce their polynomial coefficients, produce related quadrature rules,
project other functions onto these polynomial bases, and integrate
double and triple products of the polynomials.
</p>
<p>
<a href = "../../cpp_src/hermite_polynomial/hermite_polynomial.html">
HERMITE_POLYNOMIAL</a>,
a C++ library which
evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial,
the Hermite function, and related functions.
</p>
<p>
<a href = "../../cpp_src/int_exactness_legendre/int_exactness_legendre.html">
INT_EXACTNESS_LEGENDRE</a>,
a C++ program which
tests the polynomial exactness of Gauss-Legendre quadrature rules.
</p>
<p>
<a href = "../../cpp_src/jacobi_polynomial/jacobi_polynomial.html">
JACOBI_POLYNOMIAL</a>,
a C++ library which
evaluates the Jacobi polynomial and associated functions.
</p>
<p>
<a href = "../../cpp_src/laguerre_polynomial/laguerre_polynomial.html">
LAGUERRE_POLYNOMIAL</a>,
a C++ library which
evaluates the Laguerre polynomial, the generalized Laguerre polynomial,
and the Laguerre function.
</p>
<p>
<a href = "../../cpp_src/legendre_product_polynomial/legendre_product_polynomial.html">
LEGENDRE_PRODUCT_POLYNOMIAL</a>,
a C++ library which
defines Legendre product polynomials, creating a multivariate
polynomial as the product of univariate Legendre polynomials.
</p>
<p>
<a href = "../../cpp_src/legendre_rule/legendre_rule.html">
LEGENDRE_RULE</a>,
a C++ program which
computes a 1D Gauss-Legendre quadrature rule.
</p>
<p>
<a href = "../../cpp_src/lobatto_polynomial/lobatto_polynomial.html">
LOBATTO_POLYNOMIAL</a>,
a C++ library which
evaluates Lobatto polynomials, similar to Legendre polynomials
except that they are zero at both endpoints.
</p>
<p>
<a href = "../../cpp_src/polpak/polpak.html">
POLPAK</a>,
a C++ library which
evaluates a variety of mathematical functions.
</p>
<p>
<a href = "../../cpp_src/test_values/test_values.html">
TEST_VALUES</a>,
a C++ library which
supplies test values of various mathematical functions.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Theodore Chihara,<br>
An Introduction to Orthogonal Polynomials,<br>
Gordon and Breach, 1978,<br>
ISBN: 0677041500,<br>
LC: QA404.5 C44.
</li>
<li>
Walter Gautschi,<br>
Orthogonal Polynomials: Computation and Approximation,<br>
Oxford, 2004,<br>
ISBN: 0-19-850672-4,<br>
LC: QA404.5 G3555.
</li>
<li>
Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,<br>
NIST Handbook of Mathematical Functions,<br>
Cambridge University Press, 2010,<br>
ISBN: 978-0521192255,<br>
LC: QA331.N57.
</li>
<li>
Gabor Szego,<br>
Orthogonal Polynomials,<br>
American Mathematical Society, 1992,<br>
ISBN: 0821810235,<br>
LC: QA3.A5.v23.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "legendre_polynomial.cpp">legendre_polynomial.cpp</a>, the source code.
</li>
<li>
<a href = "legendre_polynomial.hpp">legendre_polynomial.hpp</a>, the include file.
</li>
<li>
<a href = "legendre_polynomial.sh">legendre_polynomial.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "legendre_polynomial_prb.cpp">legendre_polynomial_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "legendre_polynomial_prb.sh">legendre_polynomial_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "legendre_polynomial_prb_output.txt">legendre_polynomial_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>IMTQLX</b> diagonalizes a symmetric tridiagonal matrix.
</li>
<li>
<b>P_EXPONENTIAL_PRODUCT:</b> exponential products for P(n,x).
</li>
<li>
<b>P_INTEGRAL</b> evaluates a monomial integral associated with P(n,x).
</li>
<li>
<b>P_POLYNOMIAL</b> evaluates the Legendre polynomials P(n,x).
</li>
<li>
<b>P_POLYNOMIAL_COEFFICIENTS:</b> coefficients of Legendre polynomials P(n,x).
</li>
<li>
<b>P_POLYNOMIAL_PRIME</b> evaluates the derivative of Legendre polynomials P'(n,x).
</li>
<li>
<b>P_POLYNOMIAL_VALUES</b> returns values of the Legendre polynomials P(n,x).
</li>
<li>
<b>P_POLYNOMIAL_ZEROS:</b> zeros of Legendre function P(n,x).
</li>
<li>
<b>P_POWER_PRODUCT:</b> power products for Legendre polynomial P(n,x).
</li>
<li>
<b>P_QUADRATURE_RULE:</b> quadrature for Legendre function P(n,x).
</li>
<li>
<b>PM_POLYNOMIAL</b> evaluates the Legendre polynomials Pm(n,m,x).
</li>
<li>
<b>PM_POLYNOMIAL_VALUES</b> returns values of Legendre polynomials Pm(n,m,x).
</li>
<li>
<b>LEGENDRE_ASSOCIATED_NORMALIZED_VALUES:</b> normalized associated Legendre.
</li>
<li>
<b>LEGENDRE_FUNCTION_Q_VALUES</b> returns values of the Legendre Q function.
</li>
<li>
<b>LEGENDRE_ASSOCIATED_NORMALIZED:</b> normalized associated Legendre functions.
</li>
<li>
<b>LEGENDRE_FUNCTION_Q</b> evaluates the Legendre Q functions.
</li>
<li>
<b>R8MAT_PRINT</b> prints an R8MAT.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC2_PRINT</b> prints an R8VEC2.
</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 14 March 2012.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>