//
// Spherical
//
/// <summary>
/// Returns the associated legendre polynomial with angle parameterization P{l,m}( cos(theta) )
/// </summary>
/// <param name="n"></param>
/// <param name="m"></param>
/// <param name="theta"></param>
/// <returns></returns>
static double LegendrePAngle(int n, int m, double theta)
{
if (n < 0)
{
n = -(n + 1); // P{n,m}(x) = P{-n-1,m}(x)
}
if (m == 0)
{
return(Math2.LegendreP(n, Math.Cos(theta)));
}
// if ( abs(m) < n ) return 0 by definition; handling int.MinValue too
if (m < -n || m > n)
{
return(0);
}
// P{n,-m} = (-1)^m * (n-m)!/(n+m)! * P{n,m}
if (m < 0)
{
double legP = LegendrePAngle(n, -m, theta);
if (legP == 0)
{
return(legP);
}
if (IsOdd(m))
{
legP = -legP;
}
int MaxFactorial = Math2.FactorialTable.Length - 1;
if (n - m <= MaxFactorial)
{
double ratio = Math2.FactorialTable[n + m] / Math2.FactorialTable[n - m];
return(ratio * legP);
}
if (n - m <= 300)
{
// break up the Tgamma ratio into two separate factors
// to prevent an underflow: a!/b! = (a!/170!) * (170!/b!)
// 170!/300! ~= 2.37e-308 > double.MinNormalizedValue
return((legP * (Math2.FactorialTable[n + m] / Math2.FactorialTable[MaxFactorial])) * Math2.TgammaRatio(MaxFactorial + 1, n + 1 - m));
}
#if false
// since we are setting Lmax = 128, so Mmax = 128, and Lmax+Mmax = 256
// so this should never be called, but left here for completeness
// 170!/300! ~= 2.37e-308 > double.MinNormalizedValue
double ratio = Math2.TgammaRatio(l + m + 1, l + 1 - m);
if (ratio >= DoubleLimits.MinNormalizedValue)
{
return(ratio * legP);
}
double value = Math.Exp(Math2.Lgamma(l + m + 1) - Math2.Lgamma(l + 1 - m) + Math.Log(Math.Abs(legP)));
return(Math.Sign(legP) * value);
#else
Policies.ReportNotImplementedError("LegendrePAngle(l: {0}, m: {1}, theta: {2}): Not implemented for |l| >= 128", n, m, theta);
return(double.NaN);
#endif
}
// at this point n >= 0 && 0 <= m <= n
Debug.Assert(n >= 0 && (m >= 0 && m <= n));
// use the recurrence relations
// P{n, n} = (-1)^n * (2*n - 1)!! * (1 - x^2)^(n/2)
// P{n+1, n} = (2*n + 1) * x * P{n, n}
// (n-m+1)*P{n+1,m} = (2*n + 1)* x * P{n,m} - (n+m)*P{n-1, m}
// see: http://en.wikipedia.org/wiki/Associated_Legendre_polynomials
double x = Math.Cos(theta);
double sin_theta = Math.Sin(theta);
// the following computes (2m-1)!! * sin_theta^m
// taking special care not to underflow too soon
// double pmm = Math.Pow(sin_theta, m) * Math2.Factorial2(2 * m - 1)
int exp = 0;
double mant = Math2.Frexp(sin_theta, out exp);
double pmmScaled = Math.Pow(mant, m) * Math2.Factorial2(2 * m - 1);
double pmm = Math2.Ldexp(pmmScaled, m * exp);
if (IsOdd(m))
{
pmm = -pmm;
}
if (n == m)
{
return(pmm);
}
// p0 - current
// p1 - next
double p0 = pmm;
double p1 = (2.0 * m + 1.0) * x * p0; //P{m+1,m}(x)
for (uint k = (uint)m + 1; k < (uint)n; k++)
{
double next = ((2 * k + 1) * x * p1 - (k + (uint)m) * p0) / (k - (uint)m + 1);
p0 = p1;
p1 = next;
}
return(p1);
}
/// <summary>
/// Returns the Associated Legendre Polynomial of the first kind
/// </summary>
/// <param name="n">Polynomial degree. Limited to |<paramref name="n"/>| ≤ 127</param>
/// <param name="m">Polynomial order</param>
/// <param name="x">Requires |<paramref name="x"/>| ≤ 1</param>
public static double LegendreP(int n, int m, double x)
{
if (!(x >= -1 && x <= 1))
{
Policies.ReportDomainError("LegendreP(n: {0}, m: {1}, x: {2}): Requires |x| <= 1", n, m, x);
return(double.NaN);
}
if (n <= -128 || n >= 128)
{
Policies.ReportNotImplementedError("LegendreP(n: {0}, m: {1}, x: {2}): Not implemented for |n| >= 128", n, m, x);
return(double.NaN);
}
// P{l,m}(x) == P{-l-1,m}(x); watch overflow
if (n < 0)
{
n = -(n + 1);
}
if (m == 0)
{
return(Math2.LegendreP(n, x));
}
// by definition P{n,m}(x) == 0 when |m| > l
if (m < -n || m > n)
{
return(0);
}
// P{n,-m} = (-1)^m * (n-m)!/(n+m)! * P{n,m}
if (m < 0)
{
double legP = LegendreP(n, -m, x);
if (legP == 0)
{
return(legP);
}
if (IsOdd(m))
{
legP = -legP;
}
int MaxFactorial = Math2.FactorialTable.Length - 1;
if (n - m <= MaxFactorial)
{
double ratio = Math2.FactorialTable[n + m] / Math2.FactorialTable[n - m];
return(ratio * legP);
}
if (n - m <= 300)
{
// break up the Tgamma ratio into two separate factors
// to prevent an underflow: a!/b! = (a!/170!) * (170!/b!)
// 170!/300! ~= 2.37e-308 > double.MinNormalizedValue
return((legP * (Math2.FactorialTable[n + m] / Math2.FactorialTable[MaxFactorial])) * Math2.TgammaRatio(MaxFactorial + 1, n + 1 - m));
}
#if false
// since we are setting Nmax = 128, so Mmax = 128, and Nmax+Mmax = 256
// so this should never be called, but left here for completeness
// 170!/300! ~= 2.37e-308 > double.MinNormalizedValue
double ratioL = Math2.TgammaRatio(n + m + 1, n + 1 - m);
if (ratioL >= DoubleLimits.MinNormalizedValue)
{
return(ratioL * legP);
}
double value = Math.Exp(Math2.Lgamma(n + m + 1) - Math2.Lgamma(n + 1 - m) + Math.Log(Math.Abs(legP)));
return(Math.Sign(legP) * value);
#else
Policies.ReportNotImplementedError("LegendreP(n: {0}, m: {1}, x: {2}): Not implemented for |n| >= 128", n, m, x);
return(double.NaN);
#endif
}
// at this point n >= 0 && 0 <= m <= n
Debug.Assert(n >= 0 && (m >= 0 && m <= n));
// use the following recurrence relations:
// P{n, n} = (-1)^n * (2*n - 1)!! * (1 - x^2)^(n/2)
// P{n+1, n} = (2*n + 1) * x * P{n, n}
// (n-m+1)*P{n+1,m} = (2*n + 1)* x * P{n,m} - (n+m)*P{n-1, m}
// see: http://en.wikipedia.org/wiki/Associated_Legendre_polynomials
// the following is:
// pmm = Math2.Factorial2(2 * m - 1) * Math.Pow(1.0 - x*x, m/2.0);
// being careful not to underflow too soon
double pmm = 0;
if (Math.Abs(x) > Constants.RecipSqrt2)
{
double inner = (1 - x) * (1 + x);
int exp = 0;
double mant = Math2.Frexp(inner, out exp);
// careful: use integer division
double pmm_scaled = Math.Pow(mant, m / 2) * Math2.Factorial2(2 * m - 1);
pmm = Math2.Ldexp(pmm_scaled, (m / 2) * exp);
if (IsOdd(m))
{
pmm *= Math.Sqrt(inner);
}
}
else
{
double p_sin_theta = Math.Exp(m / 2.0 * Math2.Log1p(-x * x)); //Math.Pow(1.0 - x*x, m/2.0);
pmm = Math2.Factorial2(2 * m - 1) * p_sin_theta;
}
if (IsOdd(m))
{
pmm = -pmm;
}
if (n == m)
{
return(pmm);
}
// p0 - current
// p1 - next
double p0 = pmm;
double p1 = (2.0 * m + 1.0) * x * p0; //P{m+1,m}(x)
for (uint k = (uint)m + 1; k < (uint)n; k++)
{
double next = ((2 * k + 1) * x * p1 - (k + (uint)m) * p0) / (k - (uint)m + 1);
p0 = p1;
p1 = next;
}
return(p1);
}
