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.

References value().

{
  // 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);
}

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.

References libMesh::Real.

Referenced by deriv().

{
  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;
}