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Radau Quadrature

A Gaussian Quadrature-like formula for numerical estimation of integrals. It requires $m+1$ points and fits all Polynomials to degree $2m$, so it effectively fits exactly all Polynomials of degree $2m-1$. It uses a Weighting Function $W(x)=1$ in which the endpoint $-1$ in the interval $[-1,1]$ is included in a total of $n$ Abscissas, giving $r=n-1$ free abscissas. The general formula is

\begin{displaymath}
\int_{-1}^1 f(x)\,dx=w_1 f(-1)+\sum_{i=2}^{n} w_if(x_i).
\end{displaymath} (1)

The free abscissas $x_i$ for $i=2$, ..., $n$ are the roots of the Polynomial
\begin{displaymath}
{P_{n-1}(x)+P_n(x)\over 1+x},
\end{displaymath} (2)

where $P(x)$ is a Legendre Polynomial. The weights of the free abscissas are
\begin{displaymath}
w_i={1-x_i\over n^2[P_{n-1}(x_i)]^2}={1\over(1-x_i)[P_{n-1}'(x_i)]^2},
\end{displaymath} (3)

and of the endpoint
\begin{displaymath}
w_1={2\over n^2}.
\end{displaymath} (4)

The error term is given by
\begin{displaymath}
E={2^{2n-1}n[(n-1)!]^4\over[(2n-1)!]^3} f^{(2n-1)}(\xi),
\end{displaymath} (5)

for $\xi\in(-1,1)$.

$n$ $x_i$ $w_i$
2 $-1$ 0.5
  0.333333 1.5
3 $-1$ 0.222222
  $-0.289898$ 1.02497
  0.689898 0.752806
4 $-1$ 0.125
  $-0.575319$ 0.657689
  0.181066 0.776387
  0.822824 0.440924
5 $-1$ 0.08
  $-0.72048$ 0.446208
  $-0.167181$ 0.623653
  0.446314 0.562712
  0.885792 0.287427

The Abscissas and weights can be computed analytically for small $n$.

$n$ $x_i$ $w_i$
2 $-1$ ${\textstyle{1\over 2}}$
  ${\textstyle{1\over 3}}$ ${\textstyle{3\over 2}}$
3 $-1$ ${\textstyle{2\over 9}}$
  ${\textstyle{1\over 5}}(1-\sqrt{6}\,)$ ${\textstyle{1\over 18}}(16+\sqrt{6}\,)$
  ${\textstyle{1\over 5}}(1+\sqrt{6}\,)$ ${\textstyle{1\over 18}}(16-\sqrt{6}\,)$

See also Chebyshev Quadrature, Lobatto Quadrature


References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 888, 1972.

Chandrasekhar, S. Radiative Transfer. New York: Dover, p. 61, 1960.

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 338-343, 1956.



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© 1996-9 Eric W. Weisstein
1999-05-25