SpherepackIterator Class Reference

Iterates over spectral coefficients stored in SPHEREPACK format. More...

#include <SpherepackIterator.hpp>

Public Types
enum class	CoefficientArray { a , b }
	SPHEREPACK has two coefficient variables, 'a' and 'b', that hold the cos(m*phi) and sin(m*phi) parts of the spectral coefficients.

Public Member Functions
	SpherepackIterator (size_t l_max_input, size_t m_max_input, size_t stride=1)
size_t	l_max () const noexcept
size_t	m_max () const noexcept
size_t	n_th () const noexcept
size_t	n_ph () const noexcept
size_t	stride () const noexcept
size_t	spherepack_array_size () const noexcept
	Size of a SPHEREPACK coefficient array (a and b combined), not counting stride. For non-unit stride, the size of the array should be spherepack_array_size()*stride.
SpherepackIterator &	operator++ () noexcept
	operator bool () const noexcept
size_t	operator() () const noexcept
	Current index into a SPHEREPACK coefficient array.
size_t	l () const noexcept
	Current values of l and m.
size_t	m () const noexcept
CoefficientArray	coefficient_array () const noexcept
	Whether the iterator points to an element of 'a' or 'b', i.e. points to the cos(m*phi) or sin(m*phi) part of a spectral coefficient.
SpherepackIterator &	reset () noexcept
	Reset iterator back to beginning value. Returns *this.
SpherepackIterator &	set (size_t l_input, size_t m_input, CoefficientArray coefficient_array_input) noexcept
	Set iterator to specific value of l, m, array. Returns *this.
SpherepackIterator &	set (size_t l_input, int m_input) noexcept
	Same as 'set' above, but assumes CoefficientArray is 'a' for m>=0 and 'b' for m<0. This is useful when converting between true spherical harmonics (which allow negative values of m) and SPHEREPACK coefficients (which have only positive values of m, but two arrays for sin(m*phi) and cos(m*phi) parts).

Detailed Description

Iterates over spectral coefficients stored in SPHEREPACK format.

Details

The internal SPHEREPACK ordering is not intuitive, so SpherepackIterator exists for the purpose of iterating over an array containing SPHEREPACK coefficients and determining the (l,m) of each entry in the array.

SPHEREPACK expands \(f(\theta,\phi)\) as

\[ f(\theta,\phi) = \frac{1}{2} \sum_{l=0}^{l_{max}} \bar{P}_l^0 a(0,l) + \sum_{m=1}^{m_{max}} \sum_{l=m}^{l_{max}} \bar{P}_l^m \left( a(m,l) \cos(m \phi) - b(m,l) \sin(m \phi)\right) \]

where \(a(m,l)\) and \(b(m,l)\) are the SPHEREPACK spectral coefficients, and \(\bar{P}_l^m\) are unit-orthonormal associated Legendre polynomials:

\[ \bar{P}_l^m = (-1)^m \sqrt{\frac{(2l+1)(l-m)!}{2(l+m)!}} P_l^m, \]

where \(P_l^m\) are the associated Legendre polynomials as defined for example in Jackson "Classical Electrodynamics".

Example

  const size_t l_max = 4;
  const size_t m_max = 2;
  const size_t stride = 5;
  SpherepackIterator iter(l_max, m_max, stride);
  // Allocate space for a SPHEREPACK array
  std::vector<double> array(iter.spherepack_array_size() * stride);
  // Set each array element equal to l+m for real part
  // and l-m for imaginary part.
  size_t i = 0;
  for (iter.reset(); iter; ++iter, ++i) {
    if (iter.coefficient_array() == SpherepackIterator::CoefficientArray::a) {
      array[iter()] = iter.l() + iter.m();
    } else {
      array[iter()] = iter.l() - iter.m();
    }
    CHECK(iter.l() == test_l[i]);
    CHECK(iter.m() == test_m[i]);
    CHECK(iter() == test_index[i]);
  }