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1F1: start working on the docs.
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After Width: | Height: | Size: 30 KiB |
@@ -147,35 +147,123 @@ Otherwise we use the defining series.
|
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|
||||
[h4 Description]
|
||||
|
||||
The function `hypergeometric_1F1(a, b, x)` returns the non-singular solution to [@https://en.wikipedia.org/wiki/Confluent_hypergeometric_function Kummer's equation] which for |x| < 1 has the hypergeometric series expansion
|
||||
The function `hypergeometric_1F1(a, b, x)` returns the non-singular solution to [@https://en.wikipedia.org/wiki/Confluent_hypergeometric_function Kummer's equation]
|
||||
|
||||
sum[sub n=0]^[super [infin]] a[super (n)]/b[super (n)] x[super n]/n!
|
||||
[/\large $$z \frac{d^2 w}{d z^2} + (b-z) \frac{dw}{dz} - aw = 0 $$]
|
||||
[$../equations/hypergeometric_1f1_2.svg]
|
||||
|
||||
where /a/[super (/n/)] denotes rising factorial.
|
||||
which for |x| < 1 has the hypergeometric series expansion
|
||||
|
||||
[$../equations/hypergeometric_1f1_1.svg]
|
||||
|
||||
where (a)[sub n] denotes rising factorial.
|
||||
This function has the same definition as Mathematica's `Hypergeometric1F1[a, b, x]` and Maple's `KummerM(a, b, x)`.
|
||||
|
||||
[h4 Known Issues]
|
||||
|
||||
Unfortunately this function is still very much the subject of active research, and a full range of methods capable of calculating the function
|
||||
over it's entire domain are not yet available. We have worked extremely hard to ensure that problem domains result in an exception being thrown
|
||||
(an __evaluation_error) rather than return a garbage result. Domains that are still unsolved include:
|
||||
|
||||
[table
|
||||
[[domain][comment][example]]
|
||||
[[[sub 1]F[sub 1](-a, -b, -z)][a, b and z all large.][[sub 1]F[sub 1](-814723.75, -13586.87890625, -15.87335205078125)]]
|
||||
[[[sub 1]F[sub 1](-a, -b, z)][a < b, b > z, this is really the same domain as above.][ ]]
|
||||
[[[sub 1]F[sub 1](a, -b, z)][a, b and z all large.][[sub 1]F[sub 1](9057.91796875, -1252.51318359375, 15.87335205078125)]]
|
||||
[[[sub 1]F[sub 1](a[sub tiny], b, -z)][This domain is mostly handled by A&S 13.3.6 (see implementation notes below), but there
|
||||
are some values where either that series is non-convergent (most particularly for b < 0)
|
||||
or where the series become divergent after a few terms limiting the precision that can be achieved.][[sub 1]F[sub 1](-5.9981750131794866e-15, 0.499999999999994, -240.42092034220695)]]
|
||||
]
|
||||
|
||||
[h4 Implementation]
|
||||
|
||||
The implementation combines naive series, asymptotics, Pade approximations, and recurrence relations as necessary to obtain optimal performance.
|
||||
The implementation implementation for this function is really remarkably complex, and new methods for valuating the function are still a matter of active research,
|
||||
readers will have to refer to the code for the transitions between methods, as we can only give an overview here.
|
||||
|
||||
The main series representation
|
||||
|
||||
[$../equations/hypergeometric_1f1_1.svg]
|
||||
|
||||
is used only when:
|
||||
|
||||
* We are sure that it is convergent and not subject to excessive cancellation, and
|
||||
* There is no other better method available.
|
||||
|
||||
The implementation of this series is complicated by the fact that for `b < 0` then what appears to be a fully converged series can in fact suddenly jump back
|
||||
into life again as `b` crosses the origin.
|
||||
|
||||
A&S 13.3.6 gives:
|
||||
|
||||
[/\large $$_1F_1(a, b; z) = e^{ \frac{z}{2} } \Gamma(b-a- \frac{1}{2} ) ( \frac{z}{4})^{a-b+ \frac{1}{2}} \sum_{n=0}^{\infty} \frac{(2b-2a-1)_n(b-2a)_n(b-a-\frac{1}{2}+n)}{n!(b)_n} (-1)^n I_{b-a-\frac{1}{2}+n}(\frac{z}{2})$$]
|
||||
[$../equations/hypergeometric_1f1_3.svg]
|
||||
|
||||
Which is particularly useful for ['a [cong] b] and ['z > 0], or ['|a| << 1], and ['z < 0].
|
||||
|
||||
Then we have Tricomi's approximation (given in simplified form in A&S 13.3.7):
|
||||
|
||||
[/\large $$_1F_1(a, b; z) = \Gamma(b) e^{ \frac{1}{2} z} \sum_{n=0}^{\infty} 2^{-n}z^n A_n(a,b) E_{b-1+n}(z(\frac{b}{2}-a)) $$]
|
||||
[$../equations/hypergeometric_1f1_4.svg]
|
||||
|
||||
With:
|
||||
|
||||
[/\large $$A_0(a,b) = 1, A_1(a,b) = 0, A_2(a,b) = \frac{b}{2}, A_3(a,b)= - \frac{1}{3}(b-2a) $$]
|
||||
[$../equations/hypergeometric_1f1_5.svg]
|
||||
|
||||
and
|
||||
|
||||
[/\large $$(n+1)A_{n+1} = (n+b-1)A_{n-1} - (b-2a)A_{n-2} \quad;\quad n \geq 2 $$]
|
||||
[$../equations/hypergeometric_1f1_6.svg]
|
||||
|
||||
With ['E[sub v](x)] defined as:
|
||||
|
||||
[/
|
||||
\begin{equation*}
|
||||
\begin{split}
|
||||
E_v(z) & = z^{-\frac{1}{2}v} J_v(2 \sqrt{z})\\
|
||||
E_v(-z) & = z^{-\frac{1}{2}v} I_v(2 \sqrt{z})\\
|
||||
E_v(0) & = \frac{1}{\Gamma(v + 1)}
|
||||
\end{split}
|
||||
\end{equation*}]
|
||||
[$../equations/hypergeometric_1f1_7.svg]
|
||||
|
||||
This approximation is especially effective when ['a < 0] and somewhat effective when ['b < 0].
|
||||
|
||||
For /a/ and /b/ both large and positive and somewhat similar in size then an expantion in terms of gamma functions can be used:
|
||||
|
||||
[/\large $$_1F_1(a, b; x) = \frac{1}{B(a, b-a)} e^x \sum_{k=0}^{\infty} \frac{(1-a)_k}{k!} \frac{\gamma(b-a+k, x)}{x^{b-a+k}} $$]
|
||||
[$../equations/hypergeometric_1f1_8.svg]
|
||||
|
||||
For /z/ large we have the asymptotic expansion:
|
||||
|
||||
[/\large $$_1F_1(a, b; x) \approx \frac{e^x x^{a-b}}{\Gamma(a)} \sum_{k=0}^{\infty} \frac{(1-a)_k(b-a)_k}{k! x^k} $$]
|
||||
[$../equations/hypergeometric_1f1_9.svg]
|
||||
|
||||
Which is a special case of the gamma function expansion above.
|
||||
|
||||
|
||||
|
||||
|
||||
[endsect]
|
||||
[section:hypergeometric_refs Hypergeometric References]
|
||||
|
||||
* Beals, Richard, and Roderick Wong. ['Special functions: a graduate text.] Vol. 126. Cambridge University Press, 2010.
|
||||
# Beals, Richard, and Roderick Wong. ['Special functions: a graduate text.] Vol. 126. Cambridge University Press, 2010.
|
||||
# Pearson, John W., Sheehan Olver, and Mason A. Porter. ['Numerical methods for the computation of the confluent and Gauss hypergeometric functions.] Numerical Algorithms 74.3 (2017): 821-866.
|
||||
# Luke, Yudell L. ['Algorithms for Rational Approximations for a Confluent Hypergeometric Function] II. MISSOURI UNIV KANSAS CITY DEPT OF MATHEMATICS, 1976.
|
||||
# Dereziński, Jan. ['Hypergeometric type functions and their symmetries.] Annales Henri Poincaré. Vol. 15. No. 8. Springer Basel, 2014.
|
||||
# Keith E. Muller ['Computing the confluent hypergeometric function, M(a, b, x)]. Numer. Math. 90: 179–196 (2001).
|
||||
# Carlo Morosi, Livio Pizzocchero. ['On the expansion of the Kummer function in terms of incomplete Gamma functions.] Arch. Inequal. Appl. 2 (2004), 49-72.
|
||||
# Jose Luis Lopez, Nico M. Temme. ['Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions]. Numerische Mathematik, August 2010.
|
||||
# Yudell L. Luke. ['Algorithms for Rational Approximations for A Confluent Hypergeometric Function II]. US Airforce report, 1976.
|
||||
# John W. Pearson, Sheehan Olver, and Mason A. Porter. ['Numerical Methods for the Computation of the Confluent and Gauss Hypergeometric Functions]. Numerical Algorithms, March 2017, Volume 74, Issue 3, pp 821–866.
|
||||
# Javier Sesma. ['The Temme’s sum rule for con%uent hypergeometric functions revisited]. Journal of Computational and Applied Mathematics 163 (2004) 429–431.
|
||||
# Javier Segura, Nico M. Temme. ['Numerically satisfactory solutions of Kummer recurrence relations]. Numer. Math. (2008) 111:109–119.
|
||||
# Alfredo Deano AND Javier Segura. ['Transitory Minimal Solutions Of Hypergeometric Recursions And Pseudoconvergence Of Associated Continued Fractions]. Mathematics Of Computation, Volume 76, Number 258, April 2007.
|
||||
|
||||
* Pearson, John W., Sheehan Olver, and Mason A. Porter. ['Numerical methods for the computation of the confluent and Gauss hypergeometric functions.] Numerical Algorithms 74.3 (2017): 821-866.
|
||||
|
||||
* Luke, Yudell L. ['Algorithms for Rational Approximations for a Confluent Hypergeometric Function] II. MISSOURI UNIV KANSAS CITY DEPT OF MATHEMATICS, 1976.
|
||||
|
||||
* Dereziński, Jan. ['Hypergeometric type functions and their symmetries.] Annales Henri Poincaré. Vol. 15. No. 8. Springer Basel, 2014.
|
||||
|
||||
[endsect]
|
||||
[endsect]
|
||||
|
||||
[/
|
||||
Copyright 2017 John Maddock.
|
||||
[/ Copyright 2017 John Maddock.
|
||||
Distributed under the Boost Software License, Version 1.0.
|
||||
(See accompanying file LICENSE_1_0.txt or copy at
|
||||
http://www.boost.org/LICENSE_1_0.txt).
|
||||
|
||||
Reference in New Issue
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