Chapter17rev1(Polinomial Interpolation)

8/13/2019 Chapter17rev1(Polinomial Interpolation)

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Part 4Chapter 17

Polynomial Interpolation

PowerPoints organized by Dr. Michael R. Gustafson II, Duke UniversityAll images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.


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Chapter Objectives

Recognizing that evaluating polynomial coefficients withsimultaneous equations is an ill-conditioned problem.

Knowing how to evaluate polynomial coefficients andinterpolate with MATLABs polyfit and polyval functions.

Knowing how to perform an interpolation with Newtonspolynomial.

Knowing how to perform an interpolation with a Lagrangepolynomial.

Knowing how to solve an inverse interpolation problem by

recasting it as a roots problem. Appreciating the dangers of extrapolation.

Recognizing that higher-order polynomials can manifest

large oscillations.


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Polynomial Interpolation

You will frequently have occasions to estimateintermediate values between precise data points.

The function you use to interpolate must pass

through the actual data points - this makes

interpolation more restrictive than fitting.

The most common method for this purpose is

polynomial interpolation, where an (n-1)thorder

polynomial is solved that passes through ndatapoints: f(x) a1 a2x a3x

2 anxn1

MATLAB version :

f(x) p1xn1

p2xn2

pn1x pn


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Determining Coefficients

Since polynomial interpolation provides asmany basis functions as there are data

points (n), the polynomial coefficients can be

found exactly using linear algebra. MATLABs built in polyfit and polyval

commands can also be used - all that is

required is making sure the order of the fit forndata points is n-1.


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Polynomial Interpolation Problems

One problem that can occur with solving for the coefficientsof a polynomial is that the system to be inverted is in theform:

Matrices such as that on the left are known asVandermonde matrices, and they are very ill-conditioned -meaning their solutions are very sensitive to round-offerrors.

The issue can be minimized by scaling and shifting the data.

x1n1 x1

n2 x1 1

x2n1 x2

n2 x2 1

xn1n1

xn1n2

xn1 1xn

n1 xnn2 xn 1

p1p2

pn1pn

f x1 f x2

f xn1 f xn


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Newton Interpolating Polynomials

Another way to express a polynomialinterpolation is to use Newtons interpolating

polynomial.

The differences between a simple polynomialand Newtons interpolating polynomial for

first and second order interpolations are:

Order Simple Newton1st f1(x) a1 a2x f1(x) b1 b2(x x1)2nd f2(x) a1 a2x a3x

2 f2(x) b1 b2(x x1) b3(x x1)(x x2 )


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(cont) The first-order Newton

interpolating polynomial

may be obtained from

linear interpolation and

similar triangles, as shown. The resulting formula

based on known pointsx1

andx2and the values of

the dependent function at

those points is:

f1 x f x1 f x2 f x1

x2x1x x1


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(cont) The second-order Newton

interpolating polynomial

introduces some curvature to

the line connecting the points,

but still goes through the firsttwo points.

The resulting formula based

on known pointsx1,x2, andx3

and the values of the

dependent function at those

points is:

f2 x f x1 f x2 f x1

x2x1x x1

f x3 f x2 x3x2

f x2 f x1

x2x1x3x1

x x1 x x2


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(cont)

In general, an (n-1)thNewton interpolatingpolynomial has all the terms of the (n-2)thpolynomial plus one extra.

The general formula is:

where

and the f[] represent divided differences.

fn1 x b1 b2 x x1 bn x x1 x x2 x xn1

b1 f x1 b2 f x2,x1

b3 f x3,x2,x1

bn f xn,xn1, ,x2 ,x1


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Divided Differences

Divided difference are calculated as follows:

Divided differences are calculated using divided

difference of a smaller number of terms:

f xi,xj f xi f xj

xixj

f xi,xj ,xk f xi ,xj f xj ,xk

xixk

f xn,xn1, ,x2 ,x1 f xn,xn1, ,x2 f xn1,xn2, ,x1

xnx1


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MATLAB Implementation


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Lagrange Interpolating Polynomials

Another method that uses shifted value to expressan interpolating polynomial is the Lagrange

interpolating polynomial.

The differences between a simply polynomial and

Lagrange interpolating polynomials for first and

second order polynomials is:

where the Liare weighting coefficients that are

functions ofx.

Order Simple Lagrange

1st f1

(x) a1

a2

x f1

(x) L1

f x1

L

2

f x2

2nd f2(x) a1 a2x a3x2 f2(x) L1f x1 L2f x2 L3f x3


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(cont) The first-order Lagrange

interpolating polynomial may

be obtained from a weighted

combination of two linear

interpolations, as shown.

The resulting formula basedon known pointsx1andx2

and the values of the

dependent function at those

points is:

f1(x) L1f x1 L2f x2

L1 x x2x1x2

,L2 x x1x2x1

f1(x)

x x2x1x2 f x1

x x1x2x1 f x2


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(cont)

In general, the Lagrange polynomialinterpolation for npoints is:

where Liis given by:

fn1 xi Li x f xi i1

n

Li x x xj

xixjj1ji

n


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MATLAB Implementation


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Inverse Interpolation

Interpolation general means finding some value f(x)for somexthat is between given independent data

points.

Sometimes, it will be useful to find thexfor which

f(x) is a certain value - this is inverse interpolation.

Rather than finding an interpolation ofxas a

function of f(x), it may be useful to find an equation

for f(x) as a function ofxusing interpolation andthen solve the corresponding roots problem:

f(x)-fdesired=0 forx.


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Extrapolation

Extrapolationis theprocess of estimating a

value of f(x) that lies

outside the range of the

known base pointsx1,x2,,xn.

Extrapolation represents

a step into the unknown,

and extreme care should

be exercised when

extrapolating!


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Extrapolation Hazards

The following shows the results ofextrapolating a seventh-order population

data set:


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Chapter17rev1(Polinomial Interpolation)