/* lobatto_shape.c  zero at -1 and 1  page 25 Solin   */
/*                  and kernel functions  page 27     */
/*                  lobatto are integrated Legendre ? */
/*    normalized derivatives of Lobatto are Legendre  */

#include <math.h>

double L2(double x)
{
  const double c = 0.5*sqrt(1.5);
  return c * (x*x-1.0);
}

double L3(double x)
{
  const double c = 0.5*sqrt(2.5);
  return c * (x*x-1.0) * x;
}

double L4(double x)
{
  const double c = 0.125*sqrt(3.5);
  return c * (x*x-1.0) * (5.0*x*x-1.0);
}

double L5(double x)
{
  const double c = 0.125*sqrt(4.5);
  return c * (x*x-1.0) * (7.0*x*x-3.0) * x;
}

double L6(double x)
{
  const double c = 0.0625*sqrt(5.5);
  return c * (x*x-1.0) * (21.0*x*x*x*x-14.0*x*x+1.0);
}

double L7(double x)
{
  const double c = 0.0625*sqrt(6.5);
  return c * (x*x-1.0) * (33.0*x*x*x*x-30.0*x*x+5.0) * x;
}

double L8(double x)
{
  const double c = sqrt(7.5)/128.0;
  return c * (x*x-1.0) * (429.0*x*x*x*x*x*x-495.0*x*x*x*x+135.0*x*x-5.0);
}

/* kernel functions   -4.0*L_k-2(x)/(x*x-1.0)  */
double ph0(double x)
{
  const double c = -2.0*sqrt(1.5);
  return c;
}

double ph1(double x)
{
  const double c = -2.0*sqrt(2.5);
  return c * x;
}

double ph2(double x)
{
  const double c = -0.5*sqrt(3.5);
  return c * (5.0*x*x-1.0);
}

double ph3(double x)
{
  const double c = -0.5*sqrt(4.5);
  return c * (7.0*x*x-3.0) * x;
}

double ph4(double x)
{
  const double c = -0.25*sqrt(5.5);
  return c * (21.0*x*x*x*x-14.0*x*x+1.0);
}

double ph5(double x)
{
  const double c = -0.25*sqrt(6.5);
  return c * (33.0*x*x*x*x-30.0*x*x+5.0) * x;
}

double ph6(double x)
{
  const double c = -sqrt(7.5)/32.0;
  return c * (429.0*x*x*x*x*x*x-495.0*x*x*x*x+135.0*x*x-5.0);
}