libMesh::JacobiPolynomials Namespace Reference

Functions
Real	value (unsigned n, unsigned alpha, unsigned beta, Real x)
	The Jacobi polynomial value and derivative formulas are based on the corresponding Wikipedia article 0. More...
Real	deriv (unsigned n, unsigned alpha, unsigned beta, Real x)

Function Documentation

deriv()

Real libMesh::JacobiPolynomials::deriv ( unsigned  n, unsigned  alpha, unsigned  beta, Real  x  )

inline

Definition at line 74 of file jacobi_polynomials.h.

{
  // Call value() with elevated (alpha, beta) and decremented n.
  return n == 0 ? 0 : 0.5 * (1 + alpha + beta + n) * value(n-1, alpha+1, beta+1, x);
}

References value().

value()

Real libMesh::JacobiPolynomials::value ( unsigned  n, unsigned  alpha, unsigned  beta, Real  x  )

inline

The Jacobi polynomial value and derivative formulas are based on the corresponding Wikipedia article 0.

Note that we have shifted the indices used in the recurrence relation, otherwise this is identical to the recurrence relation given in the article. When alpha = beta = 0, the Jacobi polynomials reduce to the Legendre polynomials.

Definition at line 44 of file jacobi_polynomials.h.

{
  if (n == 0)
    return 1.;
 
  // Compute constants independent of loop index.
  unsigned int ab = alpha + beta;
  unsigned int a2 = alpha * alpha;
  unsigned int b2 = beta * beta;
 
  Real p0 = 1;
  Real p1 = (alpha + 1) + (ab + 2) * 0.5 * (x - 1);
 
  unsigned int i = 1;
  while (i < n)
    {
      // Note: we swap before updating p1, so p0 and p1 appear in
      // opposite positions than is usual in the update formula.
      std::swap(p0, p1);
      p1 = (((2*i + ab + 1) *
        ((2*i + ab + 2) * (2*i + ab) * x + a2 - b2)) * p0
        - 2 * (i + alpha) * (i + beta) * (2*i + ab + 2) * p1) /
       (2 * (i + 1) * (i + 1 + ab) * (2*i + ab));
 
      ++i;
    }
  return p1;
}

References libMesh::Real, and swap().

Referenced by deriv().