...one of the most highly
regarded and expertly designed C++ library projects in the
world.

— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards

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

namespace boost{ namespace math{ template <class T>calculated-result-typelegendre_p(int n, T x); template <class T, class Policy>calculated-result-typelegendre_p(int n, T x, const Policy&); template <class T>calculated-result-typelegendre_p(int n, int m, T x); template <class T, class Policy>calculated-result-typelegendre_p(int n, int m, T x, const Policy&); template <class T>calculated-result-typelegendre_q(unsigned n, T x); template <class T, class Policy>calculated-result-typelegendre_q(unsigned n, T x, const Policy&); template <class T1, class T2, class T3>calculated-result-typelegendre_next(unsigned l, T1 x, T2 Pl, T3 Plm1); template <class T1, class T2, class T3>calculated-result-typelegendre_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.

template <class T>calculated-result-typelegendre_p(int l, T x); template <class T, class Policy>calculated-result-typelegendre_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) = P_{l}(x)

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

template <class T>calculated-result-typelegendre_p(int l, int m, T x); template <class T, class Policy>calculated-result-typelegendre_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 | |
---|---|

The definition of the associated Legendre polynomial used here includes
a leading Condon-Shortley phase term of (-1) 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-typelegendre_q(unsigned n, T x); template <class T, class Policy>calculated-result-typelegendre_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-typelegendre_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-typelegendre_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*.

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 3.14. 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 3.15. 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 3.16. 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.

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.

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.