info prev up next book cdrom email home

Hermite's Interpolating Polynomial

Let $l(x)$ be an $n$th degree Polynomial with zeros at $x_1$, ..., $x_n$. Then the fundamental Polynomials are

h^{(1)}_\nu(x)=\left[{1-{l''(x_\nu)\over l'(x_\nu)}}\right][l_\nu(x)]^2
\end{displaymath} (1)

\end{displaymath} (2)

for $\nu=1$, 2, ...$n$. These polynomials have the properties
$\displaystyle h^{(1)}_\nu(x_\mu)$ $\textstyle =$ $\displaystyle \delta_{\nu\mu}$ (3)
$\displaystyle {h^{(1)}_\nu}'(x_\mu)$ $\textstyle =$ $\displaystyle 0$ (4)
$\displaystyle h^{(2)}_\nu(x_\mu)$ $\textstyle =$ $\displaystyle 0$ (5)
$\displaystyle {h^{(2)}_\nu}'(x_\mu)$ $\textstyle =$ $\displaystyle \delta_{\nu\mu}.$ (6)

for $\mu,\nu=1$, 2, ..., $n$. Now let $f_1$, ..., $f_n$ and $f_1'$, ..., $f_n'$ be values. Then the expansion
W_n(x)=\sum_{\nu=1}^n f_\nu h^{(1)}_\nu(x)+\sum_{\nu=1}^n f_\nu'h^{(2)}_\nu(x)
\end{displaymath} (7)

gives the unique Hermite interpolating fundamental polynomial for which
$\displaystyle W_n(x_\nu)$ $\textstyle =$ $\displaystyle f_\nu$ (8)
$\displaystyle W_n'(x_\nu)$ $\textstyle =$ $\displaystyle f_\nu'.$ (9)

If $f_\nu'=0$, these are called Step Polynomials. The fundamental polynomials satisfy
\end{displaymath} (10)

\sum_{\nu=1}^n x_\nu h^{(1)}_\nu(x)+\sum_{\nu=1}^n h^{(2)}_\nu(x)=x.
\end{displaymath} (11)

Also, if $d\alpha(x)$ is an arbitrary distribution on the interval $[a,b]$, then
$\displaystyle \int_a^b h^{(1)}_\nu(x)\,d\alpha(x)$ $\textstyle =$ $\displaystyle \lambda_\nu$ (12)
$\displaystyle \int_a^b {h^{(1)}_\nu}'(x)\,d\alpha(x)$ $\textstyle =$ $\displaystyle 0$ (13)
$\displaystyle \int_a^b x{h^{(1)}_\nu}'(x)\,d\alpha(x)$ $\textstyle =$ $\displaystyle 0$ (14)
$\displaystyle \int_a^b h^{(2)}_\nu(x)\,d\alpha(x)$ $\textstyle =$ $\displaystyle 0$ (15)
$\displaystyle \int_a^b {h^{(2)}_\nu}'(x)\,d\alpha(x)$ $\textstyle =$ $\displaystyle \lambda_\nu$ (16)
$\displaystyle \int_a^b x{h^{(2)}_\nu}'(x)\,d\alpha(x)$ $\textstyle =$ $\displaystyle \lambda_\nu x_\nu,$ (17)

where $\lambda_\nu$ are Christoffel Numbers.


Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 314-319, 1956.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 330-332, 1975.

info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein