Hermite interpolation
Encyclopedia
In numerical analysis
, Hermite interpolation, named after Charles Hermite
, is a method of interpolating data points
as a polynomial function. The generated Hermite polynomial is closely related to the Newton polynomial
, in that both are derived from the calculation of divided differences
.
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.)
for all points.) Therefore, given 
data points 
, and values 
and 
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
. However, for some divided differences,
which is undefined!
In this case, we replace the divided difference by
. All others are calculated normally.
has k derivatives. Then the dataset 
contains k identical copies of 
. When creating the table, divided differences
of
identical values will be calculated as
.
For example,
etc.
. Evaluating the function and its first two derivatives at 
, we obtain the following data:
Since we have two derivatives to work with, we construct the set
. 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.
, the error function is
where c is an unknown within the range
, K is the total number of data-points plus one, and 
is the number of derivatives known at each 
plus one.
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
, Hermite interpolation, named after Charles Hermite
Charles Hermite
Charles Hermite was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra....
, is a method of interpolating data points
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
as a polynomial function. The generated Hermite polynomial is closely related to the Newton polynomial
Newton polynomial
In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form...
, in that both are derived from the calculation of divided differences
Divided differences
In mathematics divided differences is a recursive division process.The method can be used to calculate the coefficients in the interpolation polynomial in the Newton form.-Definition:Given n data points,\ldots,...
.
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.)
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 for all points.) Therefore, given data points , and values and for a function f that we want to interpolate, we create a new datasetsuch that
Now, we create a divided differences table
Divided differences
In mathematics divided differences is a recursive division process.The method can be used to calculate the coefficients in the interpolation polynomial in the Newton form.-Definition:Given n data points,\ldots,...
for the points
. However, for some divided differences,which is undefined!
In this case, we replace the divided difference by
. All others are calculated normally.General case
In the general case, suppose a given point has k derivatives. Then the dataset contains k identical copies of . When creating the table, divided differencesDivided differences
In mathematics divided differences is a recursive division process.The method can be used to calculate the coefficients in the interpolation polynomial in the Newton form.-Definition:Given n data points,\ldots,...
of
identical values will be calculated as.For example,
etc.
Example
Consider the function. Evaluating the function and its first two derivatives at , we obtain the following data:
| x | ƒ(x) | ƒ |
ƒ |
| −1 | 2 | −8 | 56 |
| 0 | 1 | 0 | 0 |
| 1 | 2 | 8 | 56 |
Since we have two derivatives to work with, we construct the set
. 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, the error function iswhere c is an unknown within the range
, K is the total number of data-points plus one, and is the number of derivatives known at each plus one.See also
- Cubic Hermite splineCubic Hermite splineIn the mathematical subfield of numerical analysis a cubic Hermite spline , named in honor of Charles Hermite, is a third-degree spline with each polynomial of the spline in Hermite form...
- Newton series, also known as finite differences
- Neville's schema
- Polynomial interpolationPolynomial interpolationIn numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial: given some points, find a polynomial which goes exactly through these points.- Applications :...
- Lagrange formLagrange polynomialIn numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points x_j and numbers y_j, the Lagrange polynomial is the polynomial of the least degree that at each point x_j assumes the corresponding value y_j...
of the interpolation polynomial - Bernstein formBernstein polynomialIn the mathematical field of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials....
of the interpolation polynomial - Chinese remainder theorem#Applications
External links
- Hermites Interpolating Polynomial at Mathworld