# Hermite interpolation formula

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

#### References

 [1] I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)

Hermite interpolation can be regarded as a special case of Birkhoff interpolation (also called lacunary interpolation). In the latter, not all values of a function $f$ and its derivatives are known at given points $x _ {0} < \dots < x _ {n}$( whereas there is complete information in the case of Hermite interpolation). Data such as (1) naturally give rise to a matrix $E$, a so-called interpolation matrix, constructed as follows. Write $f ^ { ( k) } ( x _ {i} ) = c _ {i,k}$ for $k = k ( i) = 0 \dots \alpha _ {i} - 1$ and $i = 0 \dots n$. Put $e _ {i,k} = 1$ if the constant $c _ {i,k}$ is known (given) and $e _ {i,k} = 0$ if it is not (for Hermite interpolation all $e _ {i,k} = 1$). Now $E = ( e _ {i,k} ) _ {i,k}$.
Such a matrix $E$ is called order regular if it is associated to a solvable problem (i.e. (1) is solvable for all choices of $c _ {i,k}$ for which $e _ {i,k} = 1$). (Similarly, if the set $X$ of interpolation points may vary over a given class, a pair $E , X$ is called regular if (1) is solvable for all $X$ in this class and all choices of $c _ {i,k}$ for which $e _ {i,k} = 1$.) A basic theme in Birkhoff interpolation is to find the regular pairs $E , X$. More information can be found in [a1].