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Chapter 4

Interpolation and

Approximation

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4.1 Lagrange Interpolation

The basic interpolation problem can be posedin one of two ways:

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exist

unique

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Example 4.1

e-1/2

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Discussion

The construction presented in this section iscalled Lagrange interpolation.

How good is interpolation at approximating a

function? (Sections 4.3, 4.11) Consider another example:

If we use a fourth-degree interpolating polynomialto approximate this function, the results are asshown in Figure 4.3 (a).

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Error for n=8n= 16

n= 8n= 4

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Discussion

There are circumstances in which polynomial interpolation asapproximation will work very well, and other circumstances in whichit will not.

The Lagrange form of the interpolating polynomial is not well suitedfor actual computations, and there is an alternative constructionthat is far superior to it.

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4.2 Newton Interpolation and

Divided Differences

The disadvantage of the Lagrange form

If we decide to add a point to the set of nodes,we have to completely re-compute all of the

functions.

Here we introduce an alternative form of the

polynomial: the Newton form It can allow us to easily write in terms of

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Newton Interpolation

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=0

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Example 4.2

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Discussion

The coefficients are called divided differences. We can use divided-difference table to find them.

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Example 4.3

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Example 4.3 (Con.)

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Table 4.5

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4.3 Interpolation Error

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4.5 Application: More

Approximations to the Derivative

dependson x

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4.5 Application: More

Approximations to the Derivative

The interpolating polynomial in Lagrange form is

The error is given as in (4.20), thus

We get

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We can use above equations to get:

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4.7 Piecewise Polynomial

Interpolation If we keep the order of the polynomial fixed and use

different polynomials over different intervals, with thelength of the intervals getting smaller and smaller, theninterpolation can be a very accurate and powerful

approximation tool.

For example:

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Example 4.6

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4.8 An Introduction to Splines

4.8.1 Definition of the problem

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Discussion

From the definition:

d: degree of approximation

Related to the number of unknown coefficients (the degrees offreedom)

N: degree of smoothness

Related to the number of constraints

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Discussion

We can make the first term vanish by setting

This establishes a relationship between the polynomialdegree of the spline and the smoothness degree.

For example: cubic splines

If we consider the common case of cubic splines, then

d=3 and N=2.

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4.8.2 Cubic B-Splines

B-Spline: assume an uniform grid

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Cubic B-Splines

How do we know that B(x) is a cubic spline

function?

Computer the one-sided derivatives at the knots:

and similarly for the second derivative.

If the one-sided values are equal to each other,

then the first and second derivatives are continuous,and hence B is a cubic spline.

Note that B is only locally defined, meaning that it

is nonzero on only a small interval.

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A Spline Approximation

We can use B to construct a spline approximation to anarbitrary function f.

Define the sequence of functions

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xi=0.4h=0.05

xi=0.75h=0.05

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n+1 equations inn+3 unknowns

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A Spline Approximation

Now, we need to come up with two additional constraintsin order to eliminate two of the unknowns.

Two common choices are The natural spline:

A simple construction

Leads to higher error near the end points

The complete spline: Better approximation properties

Do not actually require the derivative at the endpoints

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Natural SplineFrom

n-1

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Complete SplineFrom

n+1

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Example 4.7

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Example 4.8

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4 9 A li ti S l ti f

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4.9 Application: Solution of

Boundary Value Exercises Consider the two-point boundary value problem:

We construct the uniform grid of points:

We now look for our approximation in the form of a cubic splinedefine on this grid.

Consider the function:

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The advantage of this approach is we can get a continuoussmooth function.

Because we know the values of and its derivatives ateach of the nodes, we can easily reduce this to the systemof equations: (n+1 equations in n+3 unknown)

where

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We can eliminate the two extra unknowns by imposing

the boundary conditions on the approximation:

Substitute these into the first and last equations of therectangular system, we get

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We are then left with the square system:

where

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Example 4.9

h

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where

The solution we get

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4 10 Least Squares Concepts in

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4.10 Least Squares Concepts in

Approximation

4.10.1 An introduction to data fitting

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Least Square Data Fitting

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Example 4.10

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Example 4.11

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4 10 2 Least Squares Approximation

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4.10.2 Least Squares Approximation

and Orthogonal Polynomials

Let , we can seek such thatis minimized.

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Inner Productions

Inner productions of functions:

Inner product on real vector spaces:

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Inner Productions

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The definition of inner product will allow us to apply anumber of ideas from linear algebra to the construction

of approximations.

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The system can be organized along matrix-vector linesas

If our basis function satisfy the orthogonality condition

the special basis functions that satisfy this equation arecalled orthogonal polynomials.

Then the above matrix is a diagonal matrix, and we veryeasily have

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Orthogonal Polynomials Legendre polynomials:

Example 4 12

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Example 4.12

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Example4.13

4.11 Advanced Topics in

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4.11 Advanced Topics in

Interpolation Error

You can read it by yourselves.