Interpolation and Extrapolation


This course belongs to the track Numerical Algorithms in the Department of Scientific Computing in the School of Computer Science.

In this course, students will learn how to approximate discrete data. Different representations and techniques for the approximation are discussed.

Discrete DataEdit

Discrete data arises from measurements of   or when   is difficult or expensive to evaluate.

In general, discrete data is given by a set of nodes  ,   and the function   evaluated at these points:  ,  .

In the following, we will refer to the individual nodes and function values as   and   respectively. The vectors of the   nodes or function values are written as   and   respectively.

We will also always assume that the   are distinct.

Polynomial InterpolationEdit

Although there are many ways of representing a function  , polynomials are the tool of choice for interpolation. As the Taylor series expansion of a function   around   shows, every continuous, differentiable function has a polynomial representation

     
   
   

in which the   are the exact weights.

If a function   is assumed to be continuous and differentiable and we assume that it's derivatives   are zero or of negligible magnitude for increasing  , it is a good idea to approximate the function with a polynomial.

Given   distinct nodes  ,   at which the function values   are known, we can construct an interpolating polynomial   such that   for all  . Such an interpolating polynomial with degree   is unique. However infinitely many such polynomials can be found with degree  .

Any nice way of deriving the interpolation error?

The Vandermonde MatrixEdit

By definition as given above, our polynomial

 

must interpolate the function   at the   nodes  . This leads to the following system of   linear equations in the coefficients  :

     
     
 
     

We can re-write this system of linear equation in matrix notation as

 

where   is the vector of the  ,   coefficients and   is the so-called moment matrix or Vandermonde matrix with

 

or

 

This linear system of equations can be solved for   using Gauss elimination or other methods.

The following shows how this can be done in Matlab or GNU Octave.

% declare our function f
f = inline('sin(pi*2*x)','x')

% order of the interpolation?
n = 5

% declare the nodes x_i as random in [0,1]
x = rand(n,1)

% create the moment matrix
M = zeros(n)
for i=1:n
    M(:,i) = x.^(i-1);
end

% solve for the coefficients a
a = M \ f(x)

% compute g(x)
g = ones(n,1) * a(1)
for i=2:n
    g = g + x.^(i-1)*a(i);
end;

% how good was the approximation?
g - f(x)

However practical it may seem, the Vandermonde matrix is not especially suited for more than just a few nodes. Due to it's special structure, the Condition Number of the Vandermode matrix increases with  .

If in the above example we set   and compute   with

% solve for the coefficients a
a = inv(M) * f(x)

the approximation is only good up to 10 digits. For   it is accurate only to one digit.

We must therefore consider other means of computing and/or representing  .

Lagrange InterpolationEdit

Given a set of   nodes  ,  , we define the  th Lagrange Polynomial as the polynomial of degree   that satisifies

 

Maybe add a nice plot of a Lagrange polynomial over a few nodes?

The first condition, namely that   for   can be satisfied by construction. Since every polynomial of degree   can be represented as the product

 

where the   are the roots of   and   is some constant, we can construct   as

 .

The polynomial   is of order   and satisifies the first condition, namely that   for  .

Since, by construction, the   roots of   are at the nodes  ,  , we know that the value of   cannot be zero. This value, however, should be   by definition. We can enforce this by scaling   by  :

     
   

If we insert any  ,   into the above equation, the nominator becomes   and the first condition is satisified. If we insert  , the nominator and the denominator are the same and we get  , which means that the second condition is also satisfied.

We can now use the Lagrange polynomials   to construct a polynomial   of degree   which interpolates the   function values   at the points  .

Consider that, given the definition of  ,

 .

Therefore, if we define   as

 

then   for all  . Since the   are all of degree  , the weighted sum of these polynomials is also of degree  .

Therefore,   is the polynomial of degree   which interpolates the   values   at the nodes  .

The representation with Lagrange polynomials has some advantages over the monomial representation   described in the previous section.

First of all, once the Lagrange polynomials have been computed over a set of nodes  , computing the interpolating polynomial for a new set of function values   is trivial, since these only need to be multiplied with the Lagrange polynomials, which are independent of the  .

Another advantage is that the representation can easily be extended if additional points are added. The Lagrange polynomial  , obtained by adding a point   to the interpolation, is

 .

Only the new polynomial   would need to be re-computed from scratch.

In the monomial representation, adding a point would involve adding an extra line and column to the Vandermonde matrix and re-computing the solution.

  • Add a word on Barycentric interpolation to facilitate adding points.

Newton InterpolationEdit

Given a set of   function values   evaluated at the nodes  , the Newton polynomial is defined as the sum

 

where the   are the   Newton basis polynomials defined by

     
     .

An interpolation polynomial of degree n+1 can be easily obtained from that of degree n by just adding one more node point   and adding a polynomial of degree n+1 to  .

The Newton form of the interpolating polynomial is particularly suited to computations by hand, and underlies Neville's algorithm for polynomial interpolation. It is also directly related to the exact formula for the error term  . Any good text in numerical analysis will prove this formula and fully specify Neville's algorithm, usually doing both via divided differences.

(Should link to error formula, divided differences and Neville's algorithm, adding articles where they don't exist)

Interpolation errorEdit

When interpolating a given function f by a polynomial of degree n at the nodes x0,...,xn we get the error

 

where

 

is the notation for divided differences.

If f is n + 1 times continuously differentiable on a closed interval I and   be a polynomial of degree at most n that interpolates f at n + 1 distinct points {xi} (i=0,1,...,n) in that interval. Then for each x in the interval there exists   in that interval such that

 

The above error bound suggests choosing the interpolation points xi such that the product | Π (xxi) | is as small as possible.

ProofEdit

Let's set the error term is

 

and set up a auxiliary function   and the function is

 

where

 

and

 

Since   are roots of function f and  , so we will have

 

and

 

Then   has n+2 roots. From Rolle's theorem,   has n+1 roots, then   has one root  , where   is in the interval I.

So we can get

 

Since   is a polynomial of degree at most n, then

 

Thus

 

Since   is the root of  , so

 

Therefore

 .

Switching Between RepresentationsEdit

Arbitrary Basis FunctionsEdit

Piecewise Interpolation / SplinesEdit

B Splines

Radial Basis FunctionsEdit

ExtrapolationEdit

Interpolation is the technique of estimating the value of a function for any intermediate value of the independent variable while the process of computing the value of the function outside the given range is called extrapolation.

Caution must be used when extrapolating, as assumptions outside the data region (linearization, general shape and slope) may breakdown.

ExercisesEdit