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