Hermite's Interpolating Polynomial (original) (raw)

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Let l(x) be an th degree polynomial with zeros at x_1 , ..., x_n . Then the fundamental Hermite interpolating polynomials of the first and second kinds are defined by

	(1)

and

	(2)

for nu=1 , 2, ..., where the fundamental polynomials of Lagrange interpolation are defined by

	(3)

They are denoted h_nu(x) and h_nu(x) , respectively, by Szegö (1975, p. 330).

These polynomials have the properties

for mu,nu=1 , 2, ..., . Now let f_1 , ..., f_n and f_1^' , ..., f_n^' be values. Then the expansion

	(8)

gives the unique Hermite interpolating fundamental polynomial for which

If f_nu^'=0 , these are called Hermite's interpolating polynomials.

The fundamental polynomials satisfy

	(11)

and

	(12)

Also, if dalpha(x) is an arbitrary distribution on the interval [a,b] , then

where lambda_nu are Christoffel numbers.

See also

Christoffel Number, Lagrange Interpolating Polynomial

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References

Bartels, R. H.; Beatty, J. C.; and Barsky, B. A. "Hermite and Cubic Spline Interpolation." Ch. 3 in An Introduction to Splines for Use in Computer Graphics and Geometric Modelling. San Francisco, CA: Morgan Kaufmann, pp. 9-17, 1998.Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 314-319, 1956.Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 330-332, 1975.

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Hermite's Interpolating Polynomial

Cite this as:

Weisstein, Eric W. "Hermite's Interpolating Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HermitesInterpolatingPolynomial.html

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