gregory_weights()

template<typename T = double>

T integration::Integrator< T >::gregory_weights ( int  n, int  j  )

inline

Returns the Gregory weights for integration.

Purpose

The Gregory integration formula of order \(k\) is defined by

\begin{align*} \int^{x_m}_0 dx \, f(x) = \sum^{\mathrm{max}(k,m)}_{l=0} w_{m,l}\, f(x_l) \end{align*}

For \(m > 2 k\), the weights have a simpler form:

\begin{align*} w_{m,j} = \begin{cases} \omega_j & : j \le k \\ 1 & : k < j < m-k \\ \omega_{m-j} & : j \ge m-k \end{cases} \end{align*}

Here, \(\omega_j\) are tabulated weights for \(0 \le j \le k\).

gregory_weights returns \(w_{n,j}\) for given \(n, j\).

Parameters

n	Integration runs up to \(x_n\).
j	Index of the Gregory weight.

Definition at line 382 of file integration.hpp.

                                {
  int k1=k_+1,k2=2*k1;
  assert( j>=0 && ( j<=k_ || j<=n) );
  if( n>=k2 ){
  if( j > k_ && j < n-k_ ) { return 1;}
  else if ( j<k1 ) { return gregory_omega_[j];}
  else { return gregory_omega_[n-j]; }
}else{ return gregory_weights_[n*k2 + j];}
}