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

\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)

Comments

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].

References

[a1]	G.G. Lorentz, K. Jetter, S.D. Riemenschneider, "Birkhoff interpolation" , Addison-Wesley (1983)
[a2]	I.P. Mysovskih, "Lectures on numerical methods" , Wolters-Noordhoff (1969) pp. Chapt. 2, Sect. 10
[a3]	B. Wendroff, "Theoretical numerical analysis" , Acad. Press (1966) pp. Chapt. 1