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/***********************************************/
/**
* @file legendreFunction.cpp
*
* @brief Associated Legendre functions.
* (fully normalized).
*
* @author Torsten Mayer-Guerr
* @author Annette Eicker
* @date 2001-05-31
*
*/
/***********************************************/
#include "base/importStd.h"
#include "base/matrix.h"
#include "legendreFunction.h"
/***** VARIABLES *******************************/
Matrix LegendreFunction::factor1;
Matrix LegendreFunction::factor2;
Matrix LegendreFunction::factor1Integral;
Matrix LegendreFunction::factor2Integral;
Vector LegendreFunction::factorSmall; //integration for small thetas
/***********************************************/
void LegendreFunction::computeFactors(UInt degree)
{
// Enough factors already computed?
if(factor1.rows()>degree)
return;
factor1 = Matrix(degree+1, Matrix::TRIANGULAR, Matrix::LOWER);
factor2 = Matrix(degree+1, Matrix::TRIANGULAR, Matrix::LOWER);
// factors for recursion P[n-1][n-1] -> P[n][n]
if(degree>0) factor1(1,1) = std::sqrt(3.0);
for(UInt n=2; n<=degree; n++)
factor1(n,n) = std::sqrt((2.*n+1)/(2.*n));
// factors for recursion P[m][n-1] and P[m][n-2] -> P[m][n]
for(UInt m=0; m<degree; m++)
for(UInt n=m+1; n<=degree; n++)
{
Double f = (2.*n+1)/((n+m)*(n-m));
factor1(n,m) = std::sqrt(f*(2.*n-1));
factor2(n,m) = -std::sqrt(f*(n-m-1.)*(n+m-1.)/(2.*n-3));
}
}
/***********************************************/
void LegendreFunction::computeFactorsIntegral(UInt degree)
{
// Enough factors already computed?
if(factor1Integral.rows()>degree)
return;
factor1Integral = Matrix(degree+1, Matrix::TRIANGULAR, Matrix::LOWER);
factor2Integral = Matrix(degree+1, Matrix::TRIANGULAR, Matrix::LOWER);
// factors for recursion P[n-1][n-2] and int_P[n-2][n-2] -> P[n][n]
for(UInt n=2; n<=degree; n++)
{
factor1Integral(n,n) = 1./(2.*n+2.)*std::sqrt((2.*n+1.)/(n*(n-1.)));
factor2Integral(n,n) = 1./(2.*n+2.)*std::sqrt(n*(2.*n+1.)*(2.*n-1.)/(n-1.));
}
// factors for recursion P[n-1][m] and int_P[n-2][m] -> P[n][m]
for(UInt m=0; m<degree; m++)
for(UInt n=m+1; n<=degree; n++)
{
factor1Integral(n,m) = -1.0/(n+1.)*std::sqrt((2.*n+1.)*(2.*n-1.)/((n-m)*(n+m)));
factor2Integral(n,m) = (n-2.)/(n+1.)*std::sqrt((2.*n+1.)*(n+m-1.)*(n-m-1.)/((2.*n-3.)*(n+m)*(n-m)));
}
// factors for Hmain diagonal for small thetas
factorSmall = Vector(degree+1);
for(UInt n=3; n<=degree; n++)
{
factorSmall(n)=1.0;
for(UInt k=5; k<=(2*n-1); k+=2)
factorSmall(n) *= k/(k+1.);
factorSmall(n) = std::sqrt(factorSmall(n)*(2*n+1));
}
}
/***********************************************/
const Matrix LegendreFunction::compute(Double t, UInt degree)
{
computeFactors(degree);
Matrix Fkt(degree+1, Matrix::TRIANGULAR, Matrix::LOWER);
Fkt(0,0) = 1e280; // dirty trick: to account for small numbers in very high degrees.
// recursion P[n-1][n-1] -> P[n][n] (main diagonals)
for(UInt n=1; n<=degree; n++)
Fkt(n,n) = factor1(n,n) *std::sqrt(1-t*t)* Fkt(n-1, n-1);
// recursion P[m][n-1] and P[m][n-2] -> P[m][n]
// secondary diagonal m=n-1
for(UInt n=1; n<=degree; n++)
Fkt(n,n-1) = factor1(n,n-1) * t * Fkt(n-1,n-1);
// all other functions
for(UInt m=0; m<(degree-1); m++)
for(UInt n=m+2; n<=degree; n++)
Fkt(n,m) = factor1(n,m)*t*Fkt(n-1,m) + factor2(n,m)*Fkt(n-2,m);
return 1e-280 * Fkt; // dirty trick: to account for small numbers in very high degrees.
}
/***********************************************/
const Matrix LegendreFunction::integral(Double t1, Double t2, UInt degree)
{
computeFactorsIntegral(degree);
Matrix intP(degree+1, Matrix::TRIANGULAR, Matrix::LOWER);
Double qt1 = t1*t1;
Double qt2 = t2*t2;
Double y1 = std::sqrt(1-qt1);
Double y2 = std::sqrt(1-qt2);
Double qy1 = 1-qt1;
Double qy2 = 1-qt2;
Double theta1 = acos(t1);
Double theta2 = acos(t2);
Matrix P1 = LegendreFunction::compute(t1, degree);
Matrix P2 = LegendreFunction::compute(t2, degree);
intP(0,0) = t2-t1;
intP(1,1) = std::sqrt(3.0)/2.0 * (t2*y2-theta2-t1*y1+theta1);
intP(2,2) = std::sqrt(5.0/12.0) * (3*t2-std::pow(t2, 3)-3*t1+std::pow(t1, 3));
// recursion P[n-1][n-2] and int_P[n-2][n-2] -> int_P[n][n]
// recursions are instable for small theta
if(theta1<=10.0*DEG2RAD && theta2<=10.0*DEG2RAD)
{
for(UInt n=3; n<=degree; n++)
{
Double x1 = 0.0;
Double x2 = 0.0;
Double fak = 1.0;
for(UInt k=0; k<=20; k++)
{
for(UInt i=1; i<=k; i++)
fak *= (2.*i-1.)/(2.*i);
x1 += fak*std::pow(y1, 2*k)/(n+2+2*k);
x2 += fak*std::pow(y2, 2*k)/(n+2+2*k);
}
intP(n,n) = -factorSmall(n)*(std::pow(y2, n+2)*x2 - std::pow(y1, n+2)*x1);
}
}
else // for large thetas
{
for(UInt n=3; n<=degree; n++)
intP(n,n) = factor1Integral(n,n) * (qy2*P2(n-1,n-2)-qy1*P1(n-1,n-2)) + factor2Integral(n,n) * intP(n-2,n-2);
}
// recursion P[n-1][m] and int_P[n-2][m] -> int_P[n][m]
for(UInt m=0; m<degree; m++)
{
intP(m+1,m) = factor1Integral(m+1,m) * (qy2*P2(m,m)-qy1*P1(m,m));
for(UInt n=m+2; n<=degree; n++)
intP(n,m) = factor1Integral(n,m) * (qy2*P2(n-1,m)-qy1*P1(n-1,m))
+ factor2Integral(n,m) * intP(n-2,m);
}
return intP;
}
/***********************************************/
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