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145 lines
5.6 KiB
Plaintext
145 lines
5.6 KiB
Plaintext
[section:sph_harm Spherical Harmonics]
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[h4 Synopsis]
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``
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#include <boost/math/special_functions/spherical_harmonic.hpp>
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``
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namespace boost{ namespace math{
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template <class T1, class T2>
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std::complex<``__sf_result``> spherical_harmonic(unsigned n, int m, T1 theta, T2 phi);
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template <class T1, class T2, class ``__Policy``>
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std::complex<``__sf_result``> spherical_harmonic(unsigned n, int m, T1 theta, T2 phi, const ``__Policy``&);
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template <class T1, class T2>
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``__sf_result`` spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi, const ``__Policy``&);
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template <class T1, class T2>
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``__sf_result`` spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi, const ``__Policy``&);
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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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when T1 and T2 are different types.
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[optional_policy]
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template <class T1, class T2>
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std::complex<``__sf_result``> spherical_harmonic(unsigned n, int m, T1 theta, T2 phi);
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template <class T1, class T2, class ``__Policy``>
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std::complex<``__sf_result``> spherical_harmonic(unsigned n, int m, T1 theta, T2 phi, const ``__Policy``&);
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Returns the value of the Spherical Harmonic Y[sub n][super m](theta, phi):
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[equation spherical_0]
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The spherical harmonics Y[sub n][super m](theta, phi) are the angular
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portion of the solution to Laplace's equation in spherical coordinates
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where azimuthal symmetry is not present.
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[caution Care must be taken in correctly identifying the arguments to this
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function: [theta][space] is taken as the polar (colatitudinal) coordinate
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with [theta][space] in \[0, [pi]\], and [phi][space] as the azimuthal (longitudinal)
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coordinate with [phi][space] in \[0,2[pi]). This is the convention used in Physics,
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and matches the definition used by
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[@http://documents.wolfram.com/mathematica/functions/SphericalHarmonicY
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Mathematica in the function SpericalHarmonicY],
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but is opposite to the usual mathematical conventions.
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Some other sources include an additional Condon-Shortley phase term of
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(-1)[super m] in the definition of this function: note however that our
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definition of the associated Legendre polynomial already includes this term.
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This implementation returns zero for m > n
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For [theta][space] outside \[0, [pi]\] and [phi][space] outside \[0, 2[pi]\] this
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implementation follows the convention used by Mathematica:
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the function is periodic with period [pi][space] in [theta][space] and 2[pi][space] in
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[phi]. Please note that this is not the behaviour one would get
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from a casual application of the function's definition. Cautious users
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should keep [theta][space] and [phi][space] to the range \[0, [pi]\] and
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\[0, 2[pi]\] respectively.
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See: [@http://mathworld.wolfram.com/SphericalHarmonic.html
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Weisstein, Eric W. "Spherical Harmonic."
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From MathWorld--A Wolfram Web Resource]. ]
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template <class T1, class T2>
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``__sf_result`` spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` spherical_harmonic_r(unsigned n, int m, T1 theta, T2 phi, const ``__Policy``&);
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Returns the real part of Y[sub n][super m](theta, phi):
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[equation spherical_1]
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template <class T1, class T2>
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``__sf_result`` spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` spherical_harmonic_i(unsigned n, int m, T1 theta, T2 phi, const ``__Policy``&);
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Returns the imaginary part of Y[sub n][super m](theta, phi):
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[equation spherical_2]
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[h4 Accuracy]
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The following table shows peak errors 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. Peak errors are the same
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for both the real and imaginary parts, as the error is dominated by
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calculation of the associated Legendre polynomials: especially near the
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roots of the associated Legendre function.
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All values are in units of epsilon.
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[table_spherical_harmonic_r]
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[table_spherical_harmonic_i]
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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
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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 fairly naively using the formulae
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given above. Some extra care is taken to prevent roundoff error
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when converting from polar coordinates (so for example the
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['1-x[super 2]] term used by the associated Legendre functions is calculated
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without roundoff error using ['x = cos(theta)], and
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['1-x[super 2] = sin[super 2](theta)]). The limiting factor in the error
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rates for these functions is the need to calculate values near the roots
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of the associated Legendre functions.
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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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