boost_1_45_0/libs/math/doc/sf_and_dist/legendre.qbk - nest-learning-thermostat/5.0/boost - Git at Google

 [section:legendre Legendre (and Associated) Polynomials]

 [h4 Synopsis]

 ``
 #include <boost/math/special_functions/legendre.hpp>
 ``

    namespace boost{ namespace math{

    template <class T>
    ``__sf_result`` legendre_p(int n, T x);

    template <class T, class ``__Policy``>
    ``__sf_result`` legendre_p(int n, T x, const ``__Policy``&);

    template <class T>
    ``__sf_result`` legendre_p(int n, int m, T x);

    template <class T, class ``__Policy``>
    ``__sf_result`` legendre_p(int n, int m, T x, const ``__Policy``&);

    template <class T>
    ``__sf_result`` legendre_q(unsigned n, T x);

    template <class T, class ``__Policy``>
    ``__sf_result`` legendre_q(unsigned n, T x, const ``__Policy``&);

    template <class T1, class T2, class T3>
    ``__sf_result`` legendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1);

    template <class T1, class T2, class T3>
    ``__sf_result`` legendre_next(unsigned l, unsigned m, T1 x, T2 Pl, T3 Plm1);


    }} // namespaces

 The return type of these functions is computed using the __arg_pomotion_rules:
 note than when there is a single template argument the result is the same type
 as that argument or `double` if the template argument is an integer type.

 [optional_policy]

 [h4 Description]

    template <class T>
    ``__sf_result`` legendre_p(int l, T x);

    template <class T, class ``__Policy``>
    ``__sf_result`` legendre_p(int l, T x, const ``__Policy``&);

 Returns the Legendre Polynomial of the first kind:

 [equation legendre_0]

 Requires -1 <= x <= 1, otherwise returns the result of __domain_error.

 Negative orders are handled via the reflection formula:

 P[sub -l-1](x) = P[sub l](x)

 The following graph illustrates the behaviour of the first few
 Legendre Polynomials:

 [graph legendre_p]

    template <class T>
    ``__sf_result`` legendre_p(int l, int m, T x);

    template <class T, class ``__Policy``>
    ``__sf_result`` legendre_p(int l, int m, T x, const ``__Policy``&);

 Returns the associated Legendre polynomial of the first kind:

 [equation legendre_1]

 Requires -1 <= x <= 1, otherwise returns the result of __domain_error.

 Negative values of /l/ and /m/ are handled via the identity relations:

 [equation legendre_3]

 [caution The definition of the associated Legendre polynomial used here
 includes a leading Condon-Shortley phase term of (-1)[super m].  This
 matches the definition given by Abramowitz and Stegun (8.6.6) and that
 used by [@http://mathworld.wolfram.com/LegendrePolynomial.html Mathworld]
 and [@http://documents.wolfram.com/mathematica/functions/LegendreP
 Mathematica's LegendreP function].  However, uses in the literature
 do not always include this phase term, and strangely the specification
 for the associated Legendre function in the C++ TR1 (assoc_legendre)
 also omits it, in spite of stating that it uses Abramowitz and Stegun
 as the final arbiter on these matters.

 See:

 [@http://mathworld.wolfram.com/LegendrePolynomial.html
 Weisstein, Eric W. "Legendre Polynomial."
 From MathWorld--A Wolfram Web Resource].

 Abramowitz, M. and Stegun, I. A. (Eds.). "Legendre Functions" and
 "Orthogonal Polynomials." Ch. 22 in Chs. 8 and 22 in Handbook of
 Mathematical Functions with Formulas, Graphs, and Mathematical Tables,
 9th printing. New York: Dover, pp. 331-339 and 771-802, 1972.
  ]

    template <class T>
    ``__sf_result`` legendre_q(unsigned n, T x);

    template <class T, class ``__Policy``>
    ``__sf_result`` legendre_q(unsigned n, T x, const ``__Policy``&);

 Returns the value of the Legendre polynomial that is the second solution
 to the Legendre differential equation, for example:

 [equation legendre_2]

 Requires -1 <= x <= 1, otherwise __domain_error is called.

 The following graph illustrates the first few Legendre functions of the
 second kind:

 [graph legendre_q]

    template <class T1, class T2, class T3>
    ``__sf_result`` legendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1);

 Implements the three term recurrence relation for the Legendre
 polynomials, this function can be used to create a sequence of
 values evaluated at the same /x/, and for rising /l/.  This recurrence
 relation holds for Legendre Polynomials of both the first and second kinds.

 [equation legendre_4]

 For example we could produce a vector of the first 10 polynomial
 values using:

    double x = 0.5;  // Abscissa value
    vector<double> v;
    v.push_back(legendre_p(0, x)).push_back(legendre_p(1, x));
    for(unsigned l = 1; l < 10; ++l)
       v.push_back(legendre_next(l, x, v[l], v[l-1]));

 Formally the arguments are:

 [variablelist
 [[l][The degree of the last polynomial calculated.]]
 [[x][The abscissa value]]
 [[Pl][The value of the polynomial evaluated at degree /l/.]]
 [[Plm1][The value of the polynomial evaluated at degree /l-1/.]]
 ]

    template <class T1, class T2, class T3>
    ``__sf_result`` legendre_next(unsigned l, unsigned m, T1 x, T2 Pl, T3 Plm1);

 Implements the three term recurrence relation for the Associated Legendre
 polynomials, this function can be used to create a sequence of
 values evaluated at the same /x/, and for rising /l/.

 [equation legendre_5]

 For example we could produce a vector of the first m+10 polynomial
 values using:

    double x = 0.5;  // Abscissa value
    int m = 10;      // order
    vector<double> v;
    v.push_back(legendre_p(m, m, x)).push_back(legendre_p(1 + m, m, x));
    for(unsigned l = 1 + m; l < m + 10; ++l)
       v.push_back(legendre_next(l, m, x, v[l], v[l-1]));

 Formally the arguments are:

 [variablelist
 [[l][The degree of the last polynomial calculated.]]
 [[m][The order of the Associated Polynomial.]]
 [[x][The abscissa value]]
 [[Pl][The value of the polynomial evaluated at degree /l/.]]
 [[Plm1][The value of the polynomial evaluated at degree /l-1/.]]
 ]

 [h4 Accuracy]

 The following table shows peak errors (in units of epsilon)
 for various domains of input arguments.
 Note that only results for the widest floating point type on the system are
 given as narrower types have __zero_error.

 [table Peak Errors In the Legendre P Function
 [[Significand Size] [Platform and Compiler] [Errors in range

 0 < l < 20]  [Errors in range

 20 < l < 120]]
 [[53] [Win32, Visual C++ 8] [Peak=211 Mean=20]  [Peak=300 Mean=33]]
 [[64] [SUSE Linux IA32, g++ 4.1] [Peak=70 Mean=10]  [Peak=700 Mean=60]]
 [[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=70 Mean=10]  [Peak=700 Mean=60]]
 [[113] [HPUX IA64, aCC A.06.06] [Peak=35 Mean=6]  [Peak=292 Mean=41]]
 ]

 [table Peak Errors In the Associated Legendre P Function
 [[Significand Size] [Platform and Compiler] [Errors in range

 0 < l < 20] ]
 [[53] [Win32, Visual C++ 8] [Peak=1200 Mean=7]]
 [[64] [SUSE Linux IA32, g++ 4.1] [Peak=80 Mean=5]]
 [[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=80 Mean=5] ]
 [[113] [HPUX IA64, aCC A.06.06] [Peak=42 Mean=4] ]
 ]

 [table Peak Errors In the Legendre Q Function
 [[Significand Size] [Platform and Compiler] [Errors in range

 0 < l < 20]  [Errors in range

 20 < l < 120]]
 [[53] [Win32, Visual C++ 8] [Peak=50 Mean=7]  [Peak=4600 Mean=370]]
 [[64] [SUSE Linux IA32, g++ 4.1] [Peak=51 Mean=8]  [Peak=6000 Mean=480]]
 [[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=51 Mean=8]  [Peak=6000 Mean=480]]
 [[113] [HPUX IA64, aCC A.06.06] [Peak=90 Mean=10]  [Peak=1700 Mean=140]]
 ]

 Note that the worst errors occur when the order increases, values greater than
 ~120 are very unlikely to produce sensible results, especially in the associated
 polynomial case when the degree is also large.  Further the relative errors
 are likely to grow arbitrarily large when the function is very close to a root.

 No comparisons to other libraries are shown here: there appears to be only one
 viable implementation method for these functions, the comparisons to other
 libraries that have been run show identical error rates to those given here.

 [h4 Testing]

 A mixture of spot tests of values calculated using functions.wolfram.com,
 and randomly generated test data are
 used: the test data was computed using
 [@http://shoup.net/ntl/doc/RR.txt NTL::RR] at 1000-bit precision.

 [h4 Implementation]

 These functions are implemented using the stable three term
 recurrence relations.  These relations guarentee low absolute error
 but cannot guarentee low relative error near one of the roots of the
 polynomials.

 [endsect][/section:beta_function The Beta Function]
 [/
   Copyright 2006 John Maddock and Paul A. Bristow.
   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).
 ]
	[section:legendre Legendre (and Associated) Polynomials]

	[h4 Synopsis]

	``
	#include <boost/math/special_functions/legendre.hpp>
	``

	namespace boost{ namespace math{

	template <class T>
	``__sf_result`` legendre_p(int n, T x);

	template <class T, class ``__Policy``>
	``__sf_result`` legendre_p(int n, T x, const ``__Policy``&);

	template <class T>
	``__sf_result`` legendre_p(int n, int m, T x);

	template <class T, class ``__Policy``>
	``__sf_result`` legendre_p(int n, int m, T x, const ``__Policy``&);

	template <class T>
	``__sf_result`` legendre_q(unsigned n, T x);

	template <class T, class ``__Policy``>
	``__sf_result`` legendre_q(unsigned n, T x, const ``__Policy``&);

	template <class T1, class T2, class T3>
	``__sf_result`` legendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1);

	template <class T1, class T2, class T3>
	``__sf_result`` legendre_next(unsigned l, unsigned m, T1 x, T2 Pl, T3 Plm1);


	}} // namespaces

	The return type of these functions is computed using the __arg_pomotion_rules:
	note than when there is a single template argument the result is the same type
	as that argument or `double` if the template argument is an integer type.

	[optional_policy]

	[h4 Description]

	template <class T>
	``__sf_result`` legendre_p(int l, T x);

	template <class T, class ``__Policy``>
	``__sf_result`` legendre_p(int l, T x, const ``__Policy``&);

	Returns the Legendre Polynomial of the first kind:

	[equation legendre_0]

	Requires -1 <= x <= 1, otherwise returns the result of __domain_error.

	Negative orders are handled via the reflection formula:

	P[sub -l-1](x) = P[sub l](x)

	The following graph illustrates the behaviour of the first few
	Legendre Polynomials:

	[graph legendre_p]

	template <class T>
	``__sf_result`` legendre_p(int l, int m, T x);

	template <class T, class ``__Policy``>
	``__sf_result`` legendre_p(int l, int m, T x, const ``__Policy``&);

	Returns the associated Legendre polynomial of the first kind:

	[equation legendre_1]

	Requires -1 <= x <= 1, otherwise returns the result of __domain_error.

	Negative values of /l/ and /m/ are handled via the identity relations:

	[equation legendre_3]

	[caution The definition of the associated Legendre polynomial used here
	includes a leading Condon-Shortley phase term of (-1)[super m]. This
	matches the definition given by Abramowitz and Stegun (8.6.6) and that
	used by [@http://mathworld.wolfram.com/LegendrePolynomial.html Mathworld]
	and [@http://documents.wolfram.com/mathematica/functions/LegendreP
	Mathematica's LegendreP function]. However, uses in the literature
	do not always include this phase term, and strangely the specification
	for the associated Legendre function in the C++ TR1 (assoc_legendre)
	also omits it, in spite of stating that it uses Abramowitz and Stegun
	as the final arbiter on these matters.

	See:

	[@http://mathworld.wolfram.com/LegendrePolynomial.html
	Weisstein, Eric W. "Legendre Polynomial."
	From MathWorld--A Wolfram Web Resource].

	Abramowitz, M. and Stegun, I. A. (Eds.). "Legendre Functions" and
	"Orthogonal Polynomials." Ch. 22 in Chs. 8 and 22 in Handbook of
	Mathematical Functions with Formulas, Graphs, and Mathematical Tables,
	9th printing. New York: Dover, pp. 331-339 and 771-802, 1972.
	]

	template <class T>
	``__sf_result`` legendre_q(unsigned n, T x);

	template <class T, class ``__Policy``>
	``__sf_result`` legendre_q(unsigned n, T x, const ``__Policy``&);

	Returns the value of the Legendre polynomial that is the second solution
	to the Legendre differential equation, for example:

	[equation legendre_2]

	Requires -1 <= x <= 1, otherwise __domain_error is called.

	The following graph illustrates the first few Legendre functions of the
	second kind:

	[graph legendre_q]

	template <class T1, class T2, class T3>
	``__sf_result`` legendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1);

	Implements the three term recurrence relation for the Legendre
	polynomials, this function can be used to create a sequence of
	values evaluated at the same /x/, and for rising /l/. This recurrence
	relation holds for Legendre Polynomials of both the first and second kinds.

	[equation legendre_4]

	For example we could produce a vector of the first 10 polynomial
	values using:

	double x = 0.5; // Abscissa value
	vector<double> v;
	v.push_back(legendre_p(0, x)).push_back(legendre_p(1, x));
	for(unsigned l = 1; l < 10; ++l)
	v.push_back(legendre_next(l, x, v[l], v[l-1]));

	Formally the arguments are:

	[variablelist
	[[l][The degree of the last polynomial calculated.]]
	[[x][The abscissa value]]
	[[Pl][The value of the polynomial evaluated at degree /l/.]]
	[[Plm1][The value of the polynomial evaluated at degree /l-1/.]]
	]

	template <class T1, class T2, class T3>
	``__sf_result`` legendre_next(unsigned l, unsigned m, T1 x, T2 Pl, T3 Plm1);

	Implements the three term recurrence relation for the Associated Legendre
	polynomials, this function can be used to create a sequence of
	values evaluated at the same /x/, and for rising /l/.

	[equation legendre_5]

	For example we could produce a vector of the first m+10 polynomial
	values using:

	double x = 0.5; // Abscissa value
	int m = 10; // order
	vector<double> v;
	v.push_back(legendre_p(m, m, x)).push_back(legendre_p(1 + m, m, x));
	for(unsigned l = 1 + m; l < m + 10; ++l)
	v.push_back(legendre_next(l, m, x, v[l], v[l-1]));

	Formally the arguments are:

	[variablelist
	[[l][The degree of the last polynomial calculated.]]
	[[m][The order of the Associated Polynomial.]]
	[[x][The abscissa value]]
	[[Pl][The value of the polynomial evaluated at degree /l/.]]
	[[Plm1][The value of the polynomial evaluated at degree /l-1/.]]
	]

	[h4 Accuracy]

	The following table shows peak errors (in units of epsilon)
	for various domains of input arguments.
	Note that only results for the widest floating point type on the system are
	given as narrower types have __zero_error.

	[table Peak Errors In the Legendre P Function
	[[Significand Size] [Platform and Compiler] [Errors in range

	0 < l < 20] [Errors in range

	20 < l < 120]]
	[[53] [Win32, Visual C++ 8] [Peak=211 Mean=20] [Peak=300 Mean=33]]
	[[64] [SUSE Linux IA32, g++ 4.1] [Peak=70 Mean=10] [Peak=700 Mean=60]]
	[[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=70 Mean=10] [Peak=700 Mean=60]]
	[[113] [HPUX IA64, aCC A.06.06] [Peak=35 Mean=6] [Peak=292 Mean=41]]
	]

	[table Peak Errors In the Associated Legendre P Function
	[[Significand Size] [Platform and Compiler] [Errors in range

	0 < l < 20] ]
	[[53] [Win32, Visual C++ 8] [Peak=1200 Mean=7]]
	[[64] [SUSE Linux IA32, g++ 4.1] [Peak=80 Mean=5]]
	[[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=80 Mean=5] ]
	[[113] [HPUX IA64, aCC A.06.06] [Peak=42 Mean=4] ]
	]

	[table Peak Errors In the Legendre Q Function
	[[Significand Size] [Platform and Compiler] [Errors in range

	0 < l < 20] [Errors in range

	20 < l < 120]]
	[[53] [Win32, Visual C++ 8] [Peak=50 Mean=7] [Peak=4600 Mean=370]]
	[[64] [SUSE Linux IA32, g++ 4.1] [Peak=51 Mean=8] [Peak=6000 Mean=480]]
	[[64] [Red Hat Linux IA64, g++ 3.4.4] [Peak=51 Mean=8] [Peak=6000 Mean=480]]
	[[113] [HPUX IA64, aCC A.06.06] [Peak=90 Mean=10] [Peak=1700 Mean=140]]
	]

	Note that the worst errors occur when the order increases, values greater than
	~120 are very unlikely to produce sensible results, especially in the associated
	polynomial case when the degree is also large. Further the relative errors
	are likely to grow arbitrarily large when the function is very close to a root.

	No comparisons to other libraries are shown here: there appears to be only one
	viable implementation method for these functions, the comparisons to other
	libraries that have been run show identical error rates to those given here.

	[h4 Testing]

	A mixture of spot tests of values calculated using functions.wolfram.com,
	and randomly generated test data are
	used: the test data was computed using
	[@http://shoup.net/ntl/doc/RR.txt NTL::RR] at 1000-bit precision.

	[h4 Implementation]

	These functions are implemented using the stable three term
	recurrence relations. These relations guarentee low absolute error
	but cannot guarentee low relative error near one of the roots of the
	polynomials.

	[endsect][/section:beta_function The Beta Function]
	[/
	Copyright 2006 John Maddock and Paul A. Bristow.
	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).
	]