Hermite Interpolation of High Order
by admin in Curve Fitting , Interpolation , Math, Statistics, and Optimization , MATLAB Family on June 18, 2019Hermite interpolation, named after Charles Hermite, is a method of interpolating data points as a polynomial function. The generated Hermite interpolating polynomial is closely related to the Newton polynomial, in that both are derived from the calculation of divided differences. However, the Hermite interpolating polynomial may also be computed without using divided differences, see Chinese remainder theorem § Hermite interpolation.
Unlike Newton interpolation, Hermite interpolation matches an unknown function both in observed value, and the observed value of its first m derivatives. This means that n(m + 1) values
must be known, rather than just the first n values required for Newton interpolation. The resulting polynomial may have degree at most n(m + 1) − 1, whereas the Newton polynomial has maximum degree n − 1. (In the general case, there is no need for m to be a fixed value; that is, some points may have more known derivatives than others. In this case the resulting polynomial may have degree N − 1, with N the number of data points.)
Example On Using This Code
Input
n = 4; % n for (n+1) nodes xf = [0 1 2 3 5]'; % X values f =@(x) exp(x); % f(x) function fp =@(x) exp(x); % f'(x) function xx = 2.7; % finding interpolation at xx
Output
Coefficients of the Hermite polynomial are: [1.00000000 1.00000000 0.71828183 0.28171817 0.09726402 0.02375378 0.00537312 0.00095299 0.00018537 0.00002859] The Interpolated value at 2.7 is equal to = 14.8797 Hermite Interpolated Function is H(x) = 0.000028*x^9  0.0003006*x^8 + 0.0021023*x^7  0.004861*x^6 + 0.0204684*x^5 + 0.0279085*x^4 + 0.1750319*x^3 + 0.497903584*x^2 + x + 1.0
Contents
 Usage
 Simple case
 General case
 Example
 Error
 References
 External links
Usage
Simple Case
When using divided differences to calculate the Hermite polynomial of a function f, the first step is to copy each point m times. (Here we will consider the simplest case m=1 for all points.) Therefore, given n+1 data points x_(0), x_(1), x_(2),… ,x_(n), and values f(x_(0)), f(x_(1)), f(x_(2)),… ,f(x_(n)) and f'(x_(0)), f'(x_(1)), f'(x_(2)),… ,f'(x_(n)) for a function f that we want to interpolate, we create a new dataset
such that
Now, we create a divided differences table for the points z_(0), z_(1),… ,z_(2n+1). However, for some divided differences,
which is undefined. In this case, the divided difference is replaced by f'(z_(i)). All others are calculated normally.
General Case
In the general case, suppose a given point x_(i) has k derivatives. Then the dataset z_(0), z_(1),… ,z_(N) contains k identical copies of x_(i). When creating the table, divided differencesof j = 2, 3 ,… ,k identical values will be calculated as
For example,
etc.
Example
Consider the function f(x) = x^(8) + 1. Evaluating the function and its first two derivatives at x in {1,0,1}, we obtain the following data:

x ƒ(x) ƒ‘(x) ƒ”(x) −1 2 −8 56 0 1 0 0 1 2 8 56
Since we have two derivatives to work with, we construct the set z_(i) = {1,1,1,0,0,0,1,1,1}. Our divided difference table is then:
and the generated polynomial is
by taking the coefficients from the diagonal of the divided difference table, and multiplying the kth coefficient by
, as we would when generating a Newton polynomial.
Error
Call the calculated polynomial H and original function f. Evaluating a point x in [x_(0), x_(n)], the error function is
where c is an unknown within the range [x_(0), x_(N)], K is the total number of datapoints, and k_(i) is the number of derivatives known at each x_(i) plus one.
References
 Burden, Richard L.; Faires, J. Douglas (2004). Numerical Analysis. Belmont: Brooks/Cole.
 Spitzbart, A. (January 1960), “A Generalization of Hermite’s Interpolation Formula”, American Mathematical Monthly, 67 (1): 42–46, doi:10.2307/2308924, JSTOR 2308924
External Links
 Hermites Interpolating Polynomial at Mathworld
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