Hermite Interpolation of High Order

by admin in , , , on June 18, 2019

Hermite 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  [email protected](x) exp(x);           % f(x) function
fp [email protected](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 functionthat 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 data-points, 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 Monthly67 (1): 42–46, doi:10.2307/2308924, JSTOR 2308924

External Links

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    June 18, 2019

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    June 19, 2019

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