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Legendre (and Associated) Polynomials

Synopsis
#include <boost/math/special_functions/legendre.hpp>
namespace boost{ namespace math{

template <class T>
calculated-result-type legendre_p(int n, T x);

template <class T, class Policy>
calculated-result-type legendre_p(int n, T x, const Policy&);

template <class T>
calculated-result-type legendre_p_prime(int n, T x);

template <class T, class Policy>
calculated-result-type legendre_p_prime(int n, T x, const Policy&);

template <class T, class Policy>
std::vector<T> legendre_p_zeros(int l, const Policy&);

template <class T>
std::vector<T> legendre_p_zeros(int l);

template <class T>
calculated-result-type legendre_p(int n, int m, T x);

template <class T, class Policy>
calculated-result-type legendre_p(int n, int m, T x, const Policy&);

template <class T>
calculated-result-type legendre_q(unsigned n, T x);

template <class T, class Policy>
calculated-result-type legendre_q(unsigned n, T x, const Policy&);

template <class T1, class T2, class T3>
calculated-result-type legendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1);

template <class T1, class T2, class T3>
calculated-result-type legendre_next(unsigned l, unsigned m, T1 x, T2 Pl, T3 Plm1);


}} // namespaces

The return type of these functions is computed using the result type calculation 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.

The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more details.

Description
template <class T>
calculated-result-type legendre_p(int l, T x);

template <class T, class Policy>
calculated-result-type legendre_p(int l, T x, const Policy&);

Returns the Legendre Polynomial of the first kind:

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

Negative orders are handled via the reflection formula:

P-l-1(x) = Pl(x)

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

template <class T>
calculated-result-type legendre_p_prime(int n, T x);

template <class T, class Policy>
calculated-result-type legendre_p_prime(int n, T x, const Policy&);

Returns the derivatives of the Legendre polynomials.

template <class T, class Policy>
std::vector<T> legendre_p_zeros(int l, const Policy&);

template <class T>
std::vector<T> legendre_p_zeros(int l);

The zeros of the Legendre polynomials are calculated by Newton's method using an initial guess given by Tricomi with root bracketing provided by Szego.

Since the Legendre polynomials are alternatively even and odd, only the non-negative zeros are returned. For the odd Legendre polynomials, the first zero is always zero. The rest of the zeros are returned in increasing order.

Note that the argument to the routine is an integer, and the output is a floating-point type. Hence the template argument is mandatory. The time to extract a single root is linear in l (this is scaling to evaluate the Legendre polynomials), so recovering all roots is 𝑶(l2). Algorithms with linear scaling exist for recovering all roots, but requires tooling not currently built into boost.math. This implementation proceeds under the assumption that calculating zeros of these functions will not be a bottleneck for any workflow.

template <class T>
calculated-result-type legendre_p(int l, int m, T x);

template <class T, class Policy>
calculated-result-type legendre_p(int l, int m, T x, const Policy&);

Returns the associated Legendre polynomial of the first kind:

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

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

[Caution] Caution

The definition of the associated Legendre polynomial used here includes a leading Condon-Shortley phase term of (-1)m. This matches the definition given by Abramowitz and Stegun (8.6.6) and that used by Mathworld and 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:

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>
calculated-result-type legendre_q(unsigned n, T x);

template <class T, class Policy>
calculated-result-type 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:

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

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

template <class T1, class T2, class T3>
calculated-result-type 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.

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));
v.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]));
// Double check values:
for(unsigned l = 1; l < 10; ++l)
   assert(v[l] == legendre_p(l, x));

Formally the arguments are:

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>
calculated-result-type 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.

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));
v.push_back(legendre_p(1 + m, m, x));
for(unsigned l = 1; l < 10; ++l)
   v.push_back(legendre_next(l + 10, m, x, v[l], v[l-1]));
// Double check values:
for(unsigned l = 1; l < 10; ++l)
   assert(v[l] == legendre_p(10 + l, m, x));

Formally the arguments are:

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.

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 effectively zero error.

Table 6.32. Error rates for legendre_p

Microsoft Visual C++ version 12.0
Win32
double

GNU C++ version 5.1.0
linux
double

GNU C++ version 5.1.0
linux
long double

Sun compiler version 0x5130
Sun Solaris
long double

Legendre Polynomials: Small Values

Max = 211ε (Mean = 20.4ε)

Max = 0.732ε (Mean = 0.0619ε)

(GSL 1.16: Max = 211ε (Mean = 20.4ε))

Max = 69.2ε (Mean = 9.58ε)

(<tr1/cmath>: Max = 124ε (Mean = 13.2ε))

Max = 69.2ε (Mean = 9.58ε)

Legendre Polynomials: Large Values

Max = 300ε (Mean = 33.2ε)

Max = 0.632ε (Mean = 0.0693ε)

(GSL 1.16: Max = 300ε (Mean = 33.2ε))

Max = 699ε (Mean = 59.6ε)

(<tr1/cmath>: Max = 343ε (Mean = 32.1ε))

Max = 699ε (Mean = 59.6ε)


Table 6.33. Error rates for legendre_q

Microsoft Visual C++ version 12.0
Win32
double

GNU C++ version 5.1.0
linux
double

GNU C++ version 5.1.0
linux
long double

Sun compiler version 0x5130
Sun Solaris
long double

Legendre Polynomials: Small Values

Max = 46.4ε (Mean = 7.32ε)

Max = 0.612ε (Mean = 0.0517ε)

(GSL 1.16: Max = 46.4ε (Mean = 7.46ε))

Max = 50.9ε (Mean = 9ε)

Max = 50.9ε (Mean = 8.98ε)

Legendre Polynomials: Large Values

Max = 4.6e+003ε (Mean = 366ε)

Max = 2.49ε (Mean = 0.202ε)

(GSL 1.16: Max = 4.6e+03ε (Mean = 366ε))

Max = 5.98e+03ε (Mean = 478ε)

Max = 5.98e+03ε (Mean = 478ε)


Table 6.34. Error rates for legendre_p (associated)

Microsoft Visual C++ version 12.0
Win32
double

GNU C++ version 5.1.0
linux
double

GNU C++ version 5.1.0
linux
long double

Sun compiler version 0x5130
Sun Solaris
long double

Associated Legendre Polynomials: Small Values

Max = 121ε (Mean = 7.14ε)

Max = 0.999ε (Mean = 0.05ε)

(GSL 1.16: Max = 121ε (Mean = 6.75ε) And other failures.)

Max = 175ε (Mean = 9.88ε)

(<tr1/cmath>: Max = 175ε (Mean = 9.36ε) And other failures.)

Max = 77.7ε (Mean = 5.59ε)


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.

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 NTL::RR at 1000-bit precision.

Implementation

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


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