forked from johannesgerer/jburkardt-f
-
Notifications
You must be signed in to change notification settings - Fork 1
/
hermite_polynomial.html
365 lines (324 loc) · 10.3 KB
/
hermite_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
<html>
<head>
<title>
HERMITE_POLYNOMIAL - Hermite Polynomials
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
HERMITE_POLYNOMIAL <br> Hermite Polynomials
</h1>
<hr>
<p>
<b>HERMITE_POLYNOMIAL</b>
is a FORTRAN90 library which
evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial,
the Hermite function, and related functions.
</p>
<p>
The physicist's Hermite polynomial H(n,x) can be defined by:
<pre>
H(n,x) = (-1)^n exp(x^2/2) * d^n/dx^n ( exp(-x^2/2) )
</pre>
</p>
<p>
The normalized physicist's Hermite polynomial Hn(n,x) is scaled so that
<pre>
Integral ( -oo < X < +oo ) exp ( - X^2 ) * Hn(M,X) Hn(N,X) dX = delta ( N, M )
</pre>
</p>
<p>
The probabilist's Hermite polynomial He(n,x) is related to H(n,x) by:
<pre>
He(n,x) = H(n,x/sqrt(2)) / sqrt ( 2^n )
</pre>
</p>
<p>
The normalized probabilist's Hermite polynomial Hen(n,x) is scaled so that
<pre>
Integral ( -oo < X < +oo ) exp ( - 0.5*X^2 ) * Hen(M,X) Hen(N,X) dX = delta ( N, M )
</pre>
</p>
<p>
The Hermite function Hf(n,x) is related to H(n,x) by:
<pre>
Hf(n,x) = H(n,x) * exp(-x^2/2) / sqrt ( 2^n * n! * sqrt ( pi ) )
</pre>
</p>
<p>
The Hermite function Hf(n,x) is scaled so that:
<pre>
Integral ( -oo < X < +oo ) Hf(M,X) Hf(N,X) dX = delta ( N, M )
</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>HERMITE_POLYNOMIAL</b> is available in
<a href = "../../cpp_src/hermite_polynomial/hermite_polynomial.html">a C++ version</a> and
<a href = "../../f_src/hermite_polynomial/hermite_polynomial.html">a FORTRAN90 version</a> and
<a href = "../../m_src/hermite_polynomial/hermite_polynomial.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/chebyshev_polynomial/chebyshev_polynomial.html">
CHEBYSHEV_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the Chebyshev polynomial and associated functions.
</p>
<p>
<a href = "../../f_src/gen_hermite_rule/gen_hermite_rule.html">
GEN_HERMITE_RULE</a>,
a FORTRAN90 program which
can compute and print a generalized Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../f_src/hermite_rule/hermite_rule.html">
HERMITE_RULE</a>,
a FORTRAN90 program which
can compute and print a Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../f_src/jacobi_polynomial/jacobi_polynomial.html">
JACOBI_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the Jacobi polynomial and associated functions.
</p>
<p>
<a href = "../../f_src/laguerre_polynomial/laguerre_polynomial.html">
LAGUERRE_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the Laguerre polynomial, the generalized Laguerre polynomials,
and the Laguerre function.
</p>
<p>
<a href = "../../f_src/legendre_polynomial/legendre_polynomial.html">
LEGENDRE_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the Legendre polynomial and associated functions.
</p>
<p>
<a href = "../../f_src/pce_burgers/pce_burgers.html">
PCE_BURGERS</a>,
a FORTRAN90 program which
defines and solves a version of the time-dependent viscous Burgers equation,
with uncertain viscosity, using a polynomial chaos expansion in terms
of Hermite polynomials,
by Gianluca Iaccarino.
</p>
<p>
<a href = "../../f_src/pce_ode_hermite/pce_ode_hermite.html">
PCE_ODE_HERMITE</a>,
a FORTRAN90 program which
sets up a simple scalar ODE for exponential decay with an uncertain
decay rate, using a polynomial chaos expansion in terms of Hermite polynomials.
</p>
<p>
<a href = "../../f_src/polpak/polpak.html">
POLPAK</a>,
a FORTRAN90 library which
evaluates a variety of mathematical functions.
</p>
<p>
<a href = "../../f_src/test_values/test_values.html">
TEST_VALUES</a>,
a FORTRAN90 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 = "hermite_polynomial.f90">hermite_polynomial.f90</a>, the source code.
</li>
<li>
<a href = "hermite_polynomial.sh">hermite_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 = "hermite_polynomial_prb.f90">hermite_polynomial_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "hermite_polynomial_prb.sh">hermite_polynomial_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "hermite_polynomial_prb_output.txt">hermite_polynomial_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>H_INTEGRAL</b> evaluates a monomial physicist's Hermite integral for H(n,x).
</li>
<li>
<b>H_POLYNOMIAL</b> evaluates the physicist's Hermite polynomial H(n,x).
</li>
<li>
<b>H_POLYNOMIAL_COEFFICIENTS:</b> coeffs of physicist's Hermite polynomial H(n,x).
</li>
<li>
<b>H_POLYNOMIAL_VALUES:</b> values of the physicist's Hermite polynomial H(n,x).
</li>
<li>
<b>H_POLYNOMIAL_ZEROS:</b> zeros of the physicist's Hermite polynomial H(n,x).
</li>
<li>
<b>H_QUADRATURE_RULE:</b> quadrature for physicist's Hermite polynomial H(n,x).
</li>
<li>
<b>H_TRIPLE_PRODUCT_INTEGRAL:</b> integral of H(i,x)*H(j,x)*H(k,x)*e^(-x^2/2).
</li>
<li>
<b>HE_INTEGRAL</b> evaluates a monomial probabilist's Hermite integral for He(n,x).
</li>
<li>
<b>HE_POLYNOMIAL</b> evaluates the probabilist's Hermite polynomial He(n,x).
</li>
<li>
<b>HE_POLYNOMIAL_COEFFICIENTS:</b> coeffs probabilist's Hermite polynomial He(n,x).
</li>
<li>
<b>HE_POLYNOMIAL_VALUES:</b> values of the probabilist's Hermite polynomial He(n,x).
</li>
<li>
<b>HE_POLYNOMIAL_ZEROS:</b> zeros of the probabilist's Hermite polynomial He(n,x).
</li>
<li>
<b>HE_QUADRATURE_RULE:</b> quadrature for probabilist's Hermite polynomial He(n,x).
</li>
<li>
<b>HEN_EXPONENTIAL_PRODUCT:</b> probabilist Hermite exponential product, Hen(n,x).
</li>
<li>
<b>HEN_POLYNOMIAL:</b> evaluate normalized probabilist's Hermite poly Hen(n,x).
</li>
<li>
<b>HEN_POWER_PRODUCT:</b> power products, normalized probabilist's Hermite Hen(n,x).
</li>
<li>
<b>HF_EXPONENTIAL_PRODUCT:</b> exponential products, Hermite function Hf(n,x).
</li>
<li>
<b>HF_FUNCTION</b> evaluates the Hermite function Hf(n,x).
</li>
<li>
<b>HF_FUNCTION_VALUES:</b> values of the Hermite function Hf(n,x).
</li>
<li>
<b>HF_POWER_PRODUCT:</b> power products for Hermite function Hf(n,x).
</li>
<li>
<b>HF_QUADRATURE_RULE:</b> quadrature for Hermite function Hf(n,x).
</li>
<li>
<b>HN_EXPONENTIAL_PRODUCT:</b> exponential products for Hn(n,x).
</li>
<li>
<b>HN_POLYNOMIAL</b> evaluates normalized physicist's Hermite polynomials Hn(n,x).
</li>
<li>
<b>HN_POWER_PRODUCT:</b> power products for normalized physicist's Hermite Hn(n,x).
</li>
<li>
<b>IMTQLX</b> diagonalizes a symmetric tridiagonal matrix.
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial of N.
</li>
<li>
<b>R8_FACTORIAL2</b> computes the double factorial function.
</li>
<li>
<b>R8_SIGN</b> returns the sign of an R8.
</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 = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 16 March 2012.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>