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Hermite interpolation passes through the function and its first p derivatives at N + 1 data points. This results in a polynomial function of degree p + 1 N + 1 – 1 .
Hermite Interpolation Example (Lagrange interpolation) We search for a polynomial p(x) of degree m such that p(x 0) = f(x 0); p(x 1) = f(x 1); ::: p(x m) = f(x m) where x 0;x 1;:::;x m 2R are m + 1 pairwise distinct points. Here, n = m; k 0 = k 1 = = k n = 0; Example (Taylor interpolation) We search for a polynomial p of degree m such that p(x ...
The Hermite polynomials H(x) agree with f(x) and the derivatives of the Hermite polynomials H ′ (x) agree with f ′ (x). The degree of the Hermite polynomial is 2n +1 since 2n +2
Example 3 Find the Hermite interpolation polynomial for a function f for which we know f(0) = 1;f0(0) = 2 and f(1) = 3 (equivalent with x0 = 0 multiple node of order 2, x1 = 1 simple node).
The following function computes the Hermite interpolating polynomial, using the divided differences obtained from hdiff, and then evaluates the polynomial at \(w\).
• Idea: Use piecewise polynomial interpolation, i.e, divide the interval into smaller sub-intervals, and construct different low degree polynomial approximations (with small oscillations) on the sub-intervals.
Hermite interpolation over n data points — Hermite basis Proposition 4.1 (Hermite basis) The set of polynomials h i(x); h i(x); i = 0;1;:::;n form a basis of the vector space R 2n+1[x]. This basis is called the polynomial Hermite interpolation basis relative to data points x i. Polynomials h i(x) and h i(x) are named Hermite interpolation ...
