NM2012S Lecture17 Polynomial Interpolation

8/12/2019 NM2012S Lecture17 Polynomial Interpolation

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Polynomial Interpolation

BerlinChenDepartmentofComputerScience&InformationEngineering

NationalTaiwan

Normal

University

Reference:

1.Applied Numerical Methods with MATLAB for Engineers, Chapter 17 & Teaching material


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Chapter Objectives (1/2)

Recognizing that evaluating polynomial coefficients with

simultaneous 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 Newtons

polynomial

Knowing how to perform an interpolation with a

Lagrange polynomial

NM Berlin Chen 2


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Chapter Objectives (2/2)

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

NM Berlin Chen 3


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Polynomial Interpolation

You will frequently have occasions to estimate

intermediate values between precise data points

The function you use to interpolate must pass throughthe actual data points - this makes interpolation more

restrictive than fitting

The most common method for this purpose is polynomialinterpolation, where an (n-1)th order polynomial is solved

that passes through n data points:

NM Berlin Chen 4

f(x) a1 a2x a3x2 anxn1

MATLAB version :

f(x) p1xn1 p2x

n2 pn1x pn


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Determining Coefficients

Since polynomial interpolation provides as many basis

functions as there are data points (n), the polynomial

coefficients can be found exactly using linear algebra For n data points, there is one and only one polynomial of order

(n-1) that passes through all the points

MATLABs built in polyfit and polyval commands canalso be used - all that is required is making sure the

order of the fit for n data points is n-1

NM Berlin Chen 5


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Polynomial Interpolation Problems

One problem that can occur with solving for the

coefficients of a polynomial is that the system to be

inverted is in the form:

Matrices such as that on the left are known as

Vandermonde 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 thedata

NM Berlin Chen 6

x1n1 x1

n2 x1 1

x2n1 x2

n2 x2 1

xn1n1 xn1

n2 xn1 1

xnn1 xn

n2 xn 1

p1p2

pn1pn

f x1 f x2

f xn1 f xn


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Newton Interpolating Polynomials

Another way to express a polynomial interpolation is to

use Newtons interpolating polynomial

The differences between a simple polynomial and

Newtons interpolating polynomial for first and second

order interpolations are:

NM Berlin Chen 7

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 )

12

122

11

)()(

)(

xx

xfxfb

xfb

13

12

12

23

23

3

)()()()(

xx

xx

xfxf

xx

xfxf

b


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Newton Interpolating Polynomials (1/3)

The first-order Newton

interpolating polynomial

may be obtained fromlinear interpolation and

similar triangles, as shown

The resulting formula

based on known pointsx1andx2 and the values of

the dependent function at

those points is:

NM Berlin Chen 8

112

1211

1211

xx

xx

xfxfxfxf

xxbbxf

12

12

1

11

xx

xfxf

xx

xfxf

designateafirstorderpolynomial


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Linear Interpolation: An Example

NM Berlin Chen 9

Thesmallertheintervalbetweenthe

data

points,

the

better

the

approximation.

Example 17.2


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The second-order Newton

interpolating polynomial

introduces some curvatureto the line connecting the

points, but still goes through

the first two points

The resulting formula based

on known pointsx1,x2, and

x3 and the values of the

dependent function at thosepoints is:

NM Berlin Chen 10

2113

12

12

23

23

1

12

1212

2131212

xxxxxx

xx

xfxf

xx

xfxf

xxxx

xfxfxfxf

xxxxbxxbbxf


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Quadratic Interpolation: An Example

NM Berlin Chen 11

Example 17.3


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In general, an (n-1)th Newton interpolating polynomialhas all the terms of the (n-2)th polynomial plus one extra

The general formula is:

where

and the f[] represent divided differences

NM Berlin Chen 12

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:

NM Berlin Chen 13

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

NM Berlin Chen 14


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Lagrange Interpolating Polynomials (1/3)

Another method that uses shifted values to express an

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 are weighting coefficients of thejth polynomial,which are functions ofx

NM Berlin Chen 15

3

332

321

312

23212

2221

211211

)()(2

)()(1

LagrangeSimpleOrder

xfLxfLxfLxfxaxaaxfnd

xfLxfLxfxaaxfst

niL

))((

))((

,))((

))((

,))((

))((

2313

2133

3212

3132

3121

3231

xxxx

xxxx

Lxxxx

xxxx

Lxxxx

xxxx

L

12

122

21

221 ,

xx

xxL

xx

xxL


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The first-order Lagrange

interpolating polynomial may

be obtained from a weighted

combination of two linearinterpolations, as shown

The resulting formula based

on known pointsx1 andx2

and the values of thedependent function at those

points is:

NM Berlin Chen 16

2121

121

21

12

122

21

221

2

2

21

2

11

)(

,

)(

xfxx

xx

xfxx

xx

xf

xx

xxL

xx

xxL

xfLxfLxf


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In general, the Lagrange polynomial interpolation for n

points is:

where is given by:

NM Berlin Chen 17

n

iL

n

ii

niin xfxLxf

11

n

ijj

ji

jni

xx

xxxL

1


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Lagrange Interpolating Polynomial: An Example

NM Berlin Chen 18


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MATLAB Implementation

NM Berlin Chen 19


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Inverse Interpolation (1/2)

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 isinverse interpolation

NM Berlin Chen 20

Someadjacent

points

of

f(x)

are

bunched

together

and

others

spread

out

widely.

Thiswouldleadtooscillationsintheresultingpolynomial.


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Inverse Interpolation (2/2)

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 and then solve thecorresponding roots problem:

f(x)-fdesired=0 forx

NM Berlin Chen 21


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Extrapolation

Extrapolation is the

process of estimating a

value of f(x) that liesoutside the range of the

known base pointsx1,x2,

,xn

Extrapolation represents a

step into the unknown,

and extreme care should

be exercised whenextrapolating!

NM Berlin Chen 22


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Extrapolation Hazards

The following shows the results of extrapolating a

seventh-order polynomial which is derived from the first

8 points (1920 to 1990) of the USA population data set:

NM Berlin Chen 23


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Oscillations

Higher-order polynomials can not only lead to round-off errors

due to ill-conditioning, but can also introduce oscillations to

an interpolation or fit where they should not be

In the figures below, the dashed line represents an function,

the circles represent samples of the function, and the solid

line represents the results of a polynomial interpolation:

NM Berlin Chen 24

function)s(Runge' 251

12

x

xf

NM2012S Lecture17 Polynomial Interpolation

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