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160 lines
4.9 KiB
Plaintext
160 lines
4.9 KiB
Plaintext
[section:laguerre Laguerre (and Associated) Polynomials]
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/laguerre.hpp>
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``
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namespace boost{ namespace math{
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template <class T>
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``__sf_result`` laguerre(unsigned n, T x);
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template <class T, class ``__Policy``>
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``__sf_result`` laguerre(unsigned n, T x, const ``__Policy``&);
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template <class T>
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``__sf_result`` laguerre(unsigned n, unsigned m, T x);
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template <class T, class ``__Policy``>
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``__sf_result`` laguerre(unsigned n, unsigned m, T x, const ``__Policy``&);
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template <class T1, class T2, class T3>
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``__sf_result`` laguerre_next(unsigned n, T1 x, T2 Ln, T3 Lnm1);
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template <class T1, class T2, class T3>
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``__sf_result`` laguerre_next(unsigned n, unsigned m, T1 x, T2 Ln, T3 Lnm1);
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}} // namespaces
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[h4 Description]
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The return type of these functions is computed using the __arg_promotion_rules:
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note than when there is a single template argument the result is the same type
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as that argument or `double` if the template argument is an integer type.
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[optional_policy]
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template <class T>
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``__sf_result`` laguerre(unsigned n, T x);
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template <class T, class ``__Policy``>
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``__sf_result`` laguerre(unsigned n, T x, const ``__Policy``&);
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Returns the value of the Laguerre Polynomial of order /n/ at point /x/:
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[equation laguerre_0]
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The following graph illustrates the behaviour of the first few
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Laguerre Polynomials:
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[graph laguerre]
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template <class T>
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``__sf_result`` laguerre(unsigned n, unsigned m, T x);
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template <class T, class ``__Policy``>
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``__sf_result`` laguerre(unsigned n, unsigned m, T x, const ``__Policy``&);
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Returns the Associated Laguerre polynomial of degree /n/
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and order /m/ at point /x/:
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[equation laguerre_1]
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template <class T1, class T2, class T3>
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``__sf_result`` laguerre_next(unsigned n, T1 x, T2 Ln, T3 Lnm1);
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Implements the three term recurrence relation for the Laguerre
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polynomials, this function can be used to create a sequence of
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values evaluated at the same /x/, and for rising /n/.
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[equation laguerre_2]
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For example we could produce a vector of the first 10 polynomial
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values using:
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double x = 0.5; // Abscissa value
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vector<double> v;
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v.push_back(laguerre(0, x)).push_back(laguerre(1, x));
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for(unsigned l = 1; l < 10; ++l)
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v.push_back(laguerre_next(l, x, v[l], v[l-1]));
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Formally the arguments are:
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[variablelist
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[[n][The degree /n/ of the last polynomial calculated.]]
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[[x][The abscissa value]]
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[[Ln][The value of the polynomial evaluated at degree /n/.]]
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[[Lnm1][The value of the polynomial evaluated at degree /n-1/.]]
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]
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template <class T1, class T2, class T3>
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``__sf_result`` laguerre_next(unsigned n, unsigned m, T1 x, T2 Ln, T3 Lnm1);
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Implements the three term recurrence relation for the Associated Laguerre
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polynomials, this function can be used to create a sequence of
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values evaluated at the same /x/, and for rising degree /n/.
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[equation laguerre_3]
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For example we could produce a vector of the first 10 polynomial
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values using:
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double x = 0.5; // Abscissa value
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int m = 10; // order
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vector<double> v;
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v.push_back(laguerre(0, m, x)).push_back(laguerre(1, m, x));
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for(unsigned l = 1; l < 10; ++l)
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v.push_back(laguerre_next(l, m, x, v[l], v[l-1]));
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Formally the arguments are:
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[variablelist
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[[n][The degree of the last polynomial calculated.]]
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[[m][The order of the Associated Polynomial.]]
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[[x][The abscissa value.]]
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[[Ln][The value of the polynomial evaluated at degree /n/.]]
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[[Lnm1][The value of the polynomial evaluated at degree /n-1/.]]
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]
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[h4 Accuracy]
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The following table shows peak errors (in units of epsilon)
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for various domains of input arguments.
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Note that only results for the widest floating point type on the system are
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given as narrower types have __zero_error.
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[table_laguerre_n_x_]
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[table_laguerre_n_m_x_]
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Note that the worst errors occur when the degree increases, values greater than
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~120 are very unlikely to produce sensible results, especially in the associated
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polynomial case when the order is also large. Further the relative errors
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are likely to grow arbitrarily large when the function is very close to a root.
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[h4 Testing]
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A mixture of spot tests of values calculated using functions.wolfram.com,
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and randomly generated test data are
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used: the test data was computed using
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[@http://shoup.net/ntl/doc/RR.txt NTL::RR] at 1000-bit precision.
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[h4 Implementation]
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These functions are implemented using the stable three term
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recurrence relations. These relations guarantee low absolute error
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but cannot guarantee low relative error near one of the roots of the
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polynomials.
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[endsect][/section:beta_function The Beta Function]
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[/
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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