-
Notifications
You must be signed in to change notification settings - Fork 55
/
laguerre_polynomial.html
384 lines (346 loc) · 11.4 KB
/
laguerre_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
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
<html>
<head>
<title>
LAGUERRE_POLYNOMIAL - Laguerre Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
LAGUERRE_POLYNOMIAL <br> Laguerre Polynomials
</h1>
<hr>
<p>
<b>LAGUERRE_POLYNOMIAL</b>
is a MATLAB library which
evaluates the Laguerre polynomial, the generalized Laguerre polynomials,
and the Laguerre function.
</p>
<p>
The Laguerre polynomial L(n,x) can be defined by:
<pre>
L(n,x) = exp(x)/n! * d^n/dx^n ( exp(-x) * x^n )
</pre>
where n is a nonnegative integer.
</p>
<p>
The generalized Laguerre polynomial Lm(n,m,x) can be defined by:
<pre>
Lm(n,m,x) = exp(x)/(x^m*n!) * d^n/dx^n ( exp(-x) * x^(m+n) )
</pre>
where n and m are nonnegative integers.
</p>
<p>
The Laguerre function can be defined by:
<pre>
Lf(n,alpha,x) = exp(x)/(x^alpha*n!) * d^n/dx^n ( exp(-x) * x^(alpha+n) )
</pre>
where n is a nonnegative integer and -1.0 < alpha is a real number.
</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>LAGUERRE_POLYNOMIAL</b> is available in
<a href = "../../c_src/laguerre_polynomial/laguerre_polynomial.html">a C version</a> and
<a href = "../../cpp_src/laguerre_polynomial/laguerre_polynomial.html">a C++ version</a> and
<a href = "../../f77_src/laguerre_polynomial/laguerre_polynomial.html">a FORTRAN77 version</a> and
<a href = "../../f_src/laguerre_polynomial/laguerre_polynomial.html">a FORTRAN90 version</a> and
<a href = "../../m_src/laguerre_polynomial/laguerre_polynomial.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/bernstein_polynomial/bernstein_polynomial.html">
BERNSTEIN_POLYNOMIAL</a>,
a MATLAB library which
evaluates the Bernstein polynomials,
useful for uniform approximation of functions;
</p>
<p>
<a href = "../../m_src/chebyshev_polynomial/chebyshev_polynomial.html">
CHEBYSHEV_POLYNOMIAL</a>,
a MATLAB 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 = "../../m_src/gen_laguerre_rule/gen_laguerre_rule.html">
GEN_LAGUERRE_RULE</a>,
a MATLAB program which
can compute and print a generalized Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../m_src/hermite_polynomial/hermite_polynomial.html">
HERMITE_POLYNOMIAL</a>,
a MATLAB library which
evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial,
the Hermite function, and related functions.
</p>
<p>
<a href = "../../m_src/jacobi_polynomial/jacobi_polynomial.html">
JACOBI_POLYNOMIAL</a>,
a MATLAB library which
evaluates the Jacobi polynomial and associated functions.
</p>
<p>
<a href = "../../m_src/laguerre_rule/laguerre_rule.html">
LAGUERRE_RULE</a>,
a MATLAB program which
can compute and print a Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../m_src/laguerre_test_int/laguerre_test_int.html">
LAGUERRE_TEST_INT</a>,
a MATLAB library which
defines test integrands for integration over [A,+oo).
</p>
<p>
<a href = "../../m_src/legendre_polynomial/legendre_polynomial.html">
LEGENDRE_POLYNOMIAL</a>,
a MATLAB library which
evaluates the Legendre polynomial and associated functions.
</p>
<p>
<a href = "../../m_src/lobatto_polynomial/lobatto_polynomial.html">
LOBATTO_POLYNOMIAL</a>,
a MATLAB library which
evaluates Lobatto polynomials, similar to Legendre polynomials
except that they are zero at both endpoints.
</p>
<p>
<a href = "../../m_src/polpak/polpak.html">
POLPAK</a>,
a MATLAB library which
evaluates a variety of mathematical functions.
</p>
<p>
<a href = "../../m_src/test_values/test_values.html">
TEST_VALUES</a>,
a MATLAB 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 = "imtqlx.m">imtqlx.m</a>,
diagonalizes a symmetric tridiagonal matrix.
</li>
<li>
<a href = "l_exponential_product.m">l_exponential_product.m</a>,
exponential product table for L(n,x).
</li>
<li>
<a href = "l_integral.m">l_integral.m</a>,
evaluates a monomial integral associated with L(n,x).
</li>
<li>
<a href = "l_polynomial.m">l_polynomial.m</a>,
evaluates the Laguerre polynomial L(n,x).
</li>
<li>
<a href = "l_polynomial_coefficients.m">l_polynomial_coefficients.m</a>,
coefficients of the Laguerre polynomial L(n,x).
</li>
<li>
<a href = "l_polynomial_values.m">l_polynomial_values.m</a>,
some values of the Laguerre polynomial L(n,x).
</li>
<li>
<a href = "l_polynomial_zeros.m">l_polynomial_zeros.m</a>,
zeros of the Laguerre polynomial L(n,x).
</li>
<li>
<a href = "l_power_product.m">l_power_product.m</a>,
power product table for L(n,x).
</li>
<li>
<a href = "l_quadrature_rule.m">l_quadrature_rule.m</a>,
Gauss-Laguerre quadrature based on L(n,x).
</li>
<li>
<a href = "lf_integral.m">lf_integral.m</a>,
evaluates a monomial integral associated with Lf(n,alpha,x).
</li>
<li>
<a href = "lf_function.m">lf_function.m</a>,
evaluates the Laguerre function Lf(n,alpha,x).
</li>
<li>
<a href = "lf_function_values.m">lf_function_values.m</a>,
returns values of the Laguerre function Lf(n,alpha,x).
</li>
<li>
<a href = "lf_function_zeros.m">lf_function_zeros.m</a>,
returns the zeros of Lf(n,alpha,x).
</li>
<li>
<a href = "lf_quadrature_rule.m">lf_quadrature_rule.m</a>,
Gauss-Laguerre quadrature rule for Lf(n,alpha,x);
</li>
<li>
<a href = "lm_integral.m">lm_integral.m</a>,
evaluates a monomial integral associated with Lm(n,m,x).
</li>
<li>
<a href = "lm_polynomial.m">lm_polynomial.m</a>,
evaluates Laguerre polynomials Lm(n,m,x).
</li>
<li>
<a href = "lm_polynomial_coefficients.m">lm_polynomial_coefficients.m</a>,
coefficients of Laguerre polynomial Lm(n,m,x).
</li>
<li>
<a href = "lm_polynomial_values.m">lm_polynomial_values.m</a>,
returns values of Laguerre polynomials Lm(n,m,x).
</li>
<li>
<a href = "lm_polynomial_zeros.m">lm_polynomial_zeros.m</a>,
returns the zeros for Lm(n,m,x).
</li>
<li>
<a href = "lm_quadrature_rule.m">lm_quadrature_rule.m</a>,
Gauss-Laguerre quadrature rule for Lm(n,m,x);
</li>
<li>
<a href = "r8_factorial.m">r8_factorial.m</a>,
computes the factorial of N.
</li>
<li>
<a href = "r8_gamma.m">r8_gamma.m</a>,
evaluates Gamma(X) for an R8..
</li>
<li>
<a href = "r8_sign.m">r8_sign.m</a>,
returns the sign of an R8.
</li>
<li>
<a href = "r8mat_print.m">r8mat_print.m</a>,
prints an R8MAT.
</li>
<li>
<a href = "r8mat_print_some.m">r8mat_print_some.m</a>,
prints some of an R8MAT.
</li>
<li>
<a href = "r8vec_print.m">r8vec_print.m</a>,
prints an R8VEC.
</li>
<li>
<a href = "r8vec2_print.m">r8vec2_print.m</a>,
prints a pair of R8VEC's.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "laguerre_polynomial_test.m">laguerre_polynomial_test.m</a>,
calls all the tests.
</li>
<li>
<a href = "laguerre_polynomial_test01.m">laguerre_polynomial_test01.m</a>,
tests L_POLYNOMIAL.
</li>
<li>
<a href = "laguerre_polynomial_test02.m">laguerre_polynomial_test02.m</a>,
tests L_POLYNOMIAL_COEFFICIENTS.
</li>
<li>
<a href = "laguerre_polynomial_test03.m">laguerre_polynomial_test03.m</a>,
tests L_POLYNOMIAL_ZEROS.
</li>
<li>
<a href = "laguerre_polynomial_test04.m">laguerre_polynomial_test04.m</a>,
tests L_QUADRATURE_RULE.
</li>
<li>
<a href = "laguerre_polynomial_test05.m">laguerre_polynomial_test05.m</a>,
tests LM_POLYNOMIAL.
</li>
<li>
<a href = "laguerre_polynomial_test06.m">laguerre_polynomial_test06.m</a>,
tests LM_POLYNOMIAL_COEFFICIENTS.
</li>
<li>
<a href = "laguerre_polynomial_test07.m">laguerre_polynomial_test07.m</a>,
tests L_EXPONENTIAL_PRODUCT.
</li>
<li>
<a href = "laguerre_polynomial_test08.m">laguerre_polynomial_test08.m</a>,
tests L_POWER_PRODUCT.
</li>
<li>
<a href = "laguerre_polynomial_test_output.txt">laguerre_polynomial_test_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 09 March 2012.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>