# Hermite interpolation formula

A form of writing the polynomial $H _ {m}$ of degree $m$ that solves the problem of interpolating a function $f$ and its derivatives at points $x _ {0} \dots x _ {n}$, that is, satisfying the conditions

$$\tag{1 } \left . \begin{array}{c} {H _ {m} ( x _ {0} ) = f( x _ {0} ) \dots H _ {m} ^ {( \alpha _ {0} - 1) } ( x _ {0} ) = f ^ { ( \alpha _ {0} - 1) } ( x _ {0} ) , } \\ {\dots \dots \dots \dots \dots } \\ {H _ {m} ( x _ {n} ) = f ( x _ {n} ) \dots H _ {m} ^ {( \alpha _ {n} - 1 ) } ( x _ {n} ) = f ^ { ( \alpha _ {n} - 1 ) } ( x _ {n} ), } \\ {m = \sum _ { i= } 0 ^ { n } \alpha _ {i} - 1 . } \end{array} \right \}$$

The Hermite interpolation formula can be written in the form

$$H _ {m} ( x) = \sum _ { i= } 0 ^ { n } \sum _ { j= } 0 ^ { {\alpha _ i} - 1 } \ \sum _ { k= } 0 ^ { {\alpha _ i} - j - 1 } f ^ { ( j) } ( x _ {i} ) \frac{1}{k!} \frac{1}{j!} \left [ \frac{( x - x _ {i} ) ^ {\alpha _ {i} } }{\Omega ( x) } \right ] _ {x = x _ {i} } ^ {(} k) \times$$

$$\times \frac{\Omega ( x) }{( x - x _ {i} ) ^ {\alpha _ {i} - j - k } } ,$$

where $\Omega ( x) = ( x - x _ {0} ) ^ {\alpha _ {0} } \dots ( x - x _ {n} ) ^ {\alpha _ {n} }$.

How to Cite This Entry:
Hermite interpolation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_interpolation_formula&oldid=47214
This article was adapted from an original article by M.K. Samarin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article