Chapter 4 Chapter 4 Interpolation and Interpolation and ApproximationApproximation

4.1 Lagrange Interpolation4.1 Lagrange Interpolation

The basic interpolation problem can be The basic interpolation problem can be posed in one of two ways:posed in one of two ways:

Example 4.1Example 4.1

e-1/2

Discussion Discussion

The construction presented in this The construction presented in this section is called Lagrange interpolation.section is called Lagrange interpolation.

How good is interpolation at How good is interpolation at approximating a function? (Sections 4.3, approximating a function? (Sections 4.3, 4.11)4.11)

Consider another example:Consider another example: If we use a fourth-degree interpolating If we use a fourth-degree interpolating

polynomial to approximate this function, the polynomial to approximate this function, the results are as shown in Figure 4.3 (a).results are as shown in Figure 4.3 (a).

Error for n=8

DiscussionDiscussion

There are circumstances in which There are circumstances in which polynomial interpolation as approximation polynomial interpolation as approximation will work very well, and other will work very well, and other circumstances in which it will not.circumstances in which it will not.

The Lagrange form of the interpolating The Lagrange form of the interpolating polynomial is not well suited for actual polynomial is not well suited for actual computations, and there is an alternative computations, and there is an alternative construction that is far superior to it.construction that is far superior to it.

4.2 Newton Interpolation and 4.2 Newton Interpolation and Divided DifferencesDivided Differences

The disadvantage of the Lagrange formThe disadvantage of the Lagrange form If we decide to add a point to the set of nodes, If we decide to add a point to the set of nodes,

we have to completely recompute all of the fuwe have to completely recompute all of the functions.nctions.

Here we introduce an alternative form of thHere we introduce an alternative form of the polynomial: the Newton form.e polynomial: the Newton form. It can allow us to easily write in terms of It can allow us to easily write in terms of

=0

Example 4.2Example 4.2

)2

(

x

DiscussionDiscussion The coefficients are called divided The coefficients are called divided

differences.differences. We can use divided-difference table to find We can use divided-difference table to find

them.them.

Example 4.3Example 4.3

Example 4.3 (Con.)Example 4.3 (Con.)

Table 4.5

4.3 Interpolation Error4.3 Interpolation Error

NormsNorms

The infinity norm of pointwise norm:The infinity norm of pointwise norm:

The 2-norm:The 2-norm:

The interpolation error of linear The interpolation error of linear interpolationinterpolation

Example 4.5Example 4.5

Example 4.5 (Con.)Example 4.5 (Con.)

4.4 Application: Muller’s Method 4.4 Application: Muller’s Method and Inverse Quadratic Interpolationand Inverse Quadratic Interpolation

We can use the idea of interpolation to develop We can use the idea of interpolation to develop more sophisticated root-finding methods.more sophisticated root-finding methods.

For example: Muller’s Method For example: Muller’s Method Given three points we find the Given three points we find the

quadratic polynomial such that quadratic polynomial such that 0,1,2; and then define as the root of that is 0,1,2; and then define as the root of that is closest to .closest to .

compare

An Alternative to Muller’s MethodAn Alternative to Muller’s Method

Inverse quadratic interpolation:Inverse quadratic interpolation:

DiscussionDiscussion One great utility of Muller’s method is that it is able tOne great utility of Muller’s method is that it is able t

o find complex roots of real-valued functions, because o find complex roots of real-valued functions, because of the square root in the computation.of the square root in the computation.

Inverse quadratic interpolation is used as part of BrenInverse quadratic interpolation is used as part of Brent-Dekker root-finding algorithm, which is a commonly t-Dekker root-finding algorithm, which is a commonly implemented automatic root-finding program.implemented automatic root-finding program.

4.5 Application: More 4.5 Application: More Approximations to the DerivativeApproximations to the Derivative

depends

on x

4.5 Application: More 4.5 Application: More Approximations to the DerivativeApproximations to the Derivative

The interpolating polynomial in Lagrange The interpolating polynomial in Lagrange form isform is

The error is given as in (4.20), thusThe error is given as in (4.20), thus

We getWe get

We can use above equations to get:

4.6 Hermite Interpolation4.6 Hermite Interpolation

Hermite interpolation problem:Hermite interpolation problem:

Can we do this? Yes.Can we do this? Yes.

Divided-Difference TableDivided-Difference Table

An exampleAn example

error

Hermite Interpolation Error TheoreHermite Interpolation Error Theoremm

4.7 Piecewise Polynomial 4.7 Piecewise Polynomial InterpolationInterpolation

If we keep the order of the polynomial fixed If we keep the order of the polynomial fixed and use different polynomials over different and use different polynomials over different intervals, with the length of the intervals intervals, with the length of the intervals getting smaller and smaller, then getting smaller and smaller, then interpolation can be a very accurate and interpolation can be a very accurate and powerful approximation tool.powerful approximation tool.

For example:For example:

Example 4.6Example 4.6