Hermite interpolation formula
A form of writing the polynomial 
of degree 
that solves the problem of interpolating a function 
and its derivatives at points 
, that is, satisfying the conditions
 | (1) |
The Hermite interpolation formula can be written in the form
 |
 |
where 
.
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 
and its derivatives are known at given points 
(whereas there is complete information in the case of Hermite interpolation). Data such as (1) naturally give rise to a matrix 
, a so-called interpolation matrix, constructed as follows. Write 
for 
and 
. Put 
if the constant 
is known (given) and 
if it is not (for Hermite interpolation all 
). Now 
.
Such a matrix 
is called order regular if it is associated to a solvable problem (i.e. (1) is solvable for all choices of 
for which 
). (Similarly, if the set 
of interpolation points may vary over a given class, a pair 
is called regular if (1) is solvable for all 
in this class and all choices of 
for which 
.) A basic theme in Birkhoff interpolation is to find the regular pairs 
. 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 |
Hermite interpolation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_interpolation_formula&oldid=47214