UIN Malang · Analisis Numerik (P6) 2 Introduction Interpolation was used for long time to provide...

Analisis Numerik (P6) 1

Analisis NumerikPertemuan 6:

Interpolation

UIN Malang

Read Chapter 18, Sections 1-5


Introduction

Interpolation was used for long time to provide an estimate of a tabulated function at values that are not available in the table.

What is sin (0.15)?

x sin(x)

0 0.0000

0.1 0.0998

0.2 0.1987

0.3 0.2955

0.4 0.3894

Using Linear Interpolation sin (0.15) ≈ 0.1493

True value (4 decimal digits) sin (0.15) = 0.1494


The Interpolation Problem

Given a set of n+1 points,

Find an nth order polynomial

that passes through all points, such that:

)(,....,,)(,,)(, 1100 nn xfxxfxxfx

)(xfn

niforxfxf iin ,...,2,1,0)()(


Example

An experiment is used to determine the viscosity of water as a function of temperature. The following table is generated:

Problem: Estimate the viscosity when the temperature is 8 degrees.

Temperature

(degree)

Viscosity

0 1.792

5 1.519

10 1.308

15 1.140


Interpolation Problem

Find a polynomial that fits the data points exactly.

)V(TV

TaV(T)

ii

n

k

k

k

0

tscoefficien

Polynomial:

eTemperatur:

Viscosity:

ka

T

V

Linear Interpolation: V(T)= 1.73 − 0.0422 T

V(8)= 1.3924


Existence and Uniqueness

Given a set of n+1 points:

Assumption: are distinct

Theorem:

There is a unique polynomial fn(x) of order ≤ nsuch that:

,...,n,iforxfxf iin 10)()(

nxxx ,...,, 10

)(,....,,)(,,)(, 1100 nn xfxxfxxfx


Examples of Polynomial Interpolation

Linear Interpolation

Given any two points, there is one polynomial of order ≤ 1 that passes through the two points.

Quadratic Interpolation

Given any three points there is one polynomial of order ≤ 2 that passes through the three points.


Linear InterpolationGiven any two points,

The line that interpolates the two points is:

Example :

Find a polynomial that interpolates (1,2) and (2,4).

)(,,)(, 1100 xfxxfx

001

0101

)()()()( xx

xx

xfxfxfxf

xxxf 2112

242)(1


Quadratic Interpolation Given any three points:

The polynomial that interpolates the three points is:

)(, ,)(,,)(, 221100 xfxandxfxxfx

02

01

01

12

12

2102

01

01101

00

1020102

)()()()(

],,[

)()(],[

)(

:

)(

xx

xx

xfxf

xx

xfxf

xxxfb

xx

xfxfxxfb

xfb

where

xxxxbxxbbxf


General nth Order Interpolation

Given any n+1 points:

The polynomial that interpolates all points is:

)(,..., ,)(,,)(, 1100 nn xfxxfxxfx

],...,,[

....

],[

)(

......)(

10

101

00

10102010

nn

nnn

xxxfb

xxfb

xfb

xxxxbxxxxbxxbbxf


Divided Differences

0

1102110

02

1021210

01

0110

],...,,[],...,,[],...,,[

............

DDorder Second],[],[

],,[

DDorder First ][][

],[

DDorder Zeroth )(][

xx

xxxfxxxfxxxf

xx

xxfxxfxxxf

xx

xfxfxxf

xfxf

k

kkk

kk


Divided Difference Table

x F[ ] F[ , ] F[ , , ] F[ , , ,]

x0 F[x0] F[x0,x1] F[x0,x1,x2] F[x0,x1,x2,x3]

x1 F[x1] F[x1,x2] F[x1,x2,x3]

x2 F[x2] F[x2,x3]

x3 F[x3]

n

i

i

j

jin xxxxxFxf0

1

0

10 ],...,,[)(



f(xi)

0 -5

1 -3

-1 -15

ixx F[ ] F[ , ] F[ , , ]

0 -5 2 -4

1 -3 6

-1 -15

Entries of the divided difference table are obtained from the data table using simple operations.



f(xi)

0 -5

1 -3

-1 -15

ixx F[ ] F[ , ] F[ , , ]

0 -5 2 -4

1 -3 6

-1 -15

The first two column of the

table are the data columns.

Third column: First order differences.

Fourth column: Second order differences.



0 -5

1 -3

-1 -15

iyixx F[ ] F[ , ] F[ , , ]

0 -5 2 -4

1 -3 6

-1 -15

201

)5(3

01

0110

][][],[

xx

xfxfxxf



0 -5

1 -3

-1 -15

iyixx F[ ] F[ , ] F[ , , ]

0 -5 2 -4

1 -3 6

-1 -15

611

)3(15

12

1221

][][],[

xx

xfxfxxf



0 -5

1 -3

-1 -15

iyixx F[ ] F[ , ] F[ , , ]

0 -5 2 -4

1 -3 6

-1 -15

4)0(1

)2(6

02

1021210

],[],[],,[

xx

xxfxxfxxxf



0 -5

1 -3

-1 -15

iyixx F[ ] F[ , ] F[ , , ]

0 -5 2 -4

1 -3 6

-1 -15

)1)(0(4)0(25)(2 xxxxf

f2(x)= F[x0]+F[x0,x1] (x-x0)+F[x0,x1,x2] (x-x0)(x-x1)


Two Examples

x y

1 0

2 3

3 8

Obtain the interpolating polynomials for the two examples:

x y

2 3

1 0

3 8

What do you observe?


Two Examples

1

)2)(1(1)1(30)(

2

2

x

xxxxP

x Y

1 0 3 1

2 3 5

3 8

x Y

2 3 3 1

1 0 4

3 8

1

)1)(2(1)2(33)(

2

2

x

xxxxP

Ordering the points should not affect the interpolating polynomial.


Properties of Divided Difference

],,[],,[],,[ 012021210 xxxfxxxfxxxf

Ordering the points should not affect the divided difference:


Example

Find a polynomial to interpolate the data.

x f(x)

2 3

4 5

5 1

6 6

7 9


Example

x f(x) f[ , ] f[ , , ] f[ , , , ] f[ , , , , ]

2 3 1 -1.6667 1.5417 -0.6750

4 5 -4 4.5 -1.8333

5 1 5 -1

6 6 3

7 9

)6)(5)(4)(2(6750.0

)5)(4)(2(5417.1)4)(2(6667.1)2(134

xxxx

xxxxxxf


Summary

.polynomial inginterpolat affect thenot should points theOrdering

methodsOther -

2] 18.[Section ion Interpolat Lagrange -

] 18.1[Section Difference DividedNewton -

itobtain toused becan methodsDifferent *

unique. is Polynomial inginterpolat The *

..., ,2 ,1 ,0)()(:Condition ingInterpolat niforxfxf ini


Lecture 21

Lagrange Interpolation


The Interpolation Problem

Given a set of n+1 points:

Find an nth order polynomial:

that passes through all points, such that:

)(,....,,)(,,)(, 1100 nn xfxxfxxfx

)(xfn

niforxfxf iin ,...,2,1,0)()(



Problem:

Given

Find the polynomial of least order such that:

Lagrange Interpolation Formula:

niforxfxf iin ,...,1,0)()(

)(xfn

….

….

1x nx

0y 1y ny

ix

iy

n

ijj ji

j

i

n

i

iin

xx

xxx

xxfxf

,0

0

)(

)()(

0x



ji

jix

x

ji

th

i

1

0)(

:spolynomialorder n are cardinals The

cardinals. thecalled are)(


Lagrange Interpolation Example

x 1/3 1/4 1

y 2 -1 7

)4/1)(3/1(27

)1)(3/1(161)1)(4/1(182)(

4/11

4/1

3/11

3/1)(

14/1

1

3/14/1

3/1)(

13/1

1

4/13/1

4/1)(

)()()()()()()(

2

12

1

02

02

21

2

01

01

20

2

10

10

2211002

xx

xxxxxP

xx

xx

xx

xx

xxx

xx

xx

xx

xx

xxx

xx

xx

xx

xx

xxx

xxfxxfxxfxP


Example

Find a polynomial to interpolate:

Both Newton’s interpolation

method and Lagrange interpolation method must give the same answer.

x y

0 1

1 3

2 2

3 5

4 4


Newton’s Interpolation Method

0 1 2 -3/2 7/6 -5/8

1 3 -1 2 -4/3

2 2 3 -2

3 5 -1

4 4


Interpolating Polynomial

4324

4

8

5

12

59

8

95

12

1151)(

)3)(2)(1(8

5

)2)(1(6

7)1(

2

3)(21)(

xxxxxf

xxxx

xxxxxxxf


Interpolating Polynomial Using

Lagrange Interpolation Method

24

)3)(2)(1(

)34(

)3(

)24(

)2(

)14(

)1(

)04(

)0(

6

)4)(2)(1(

)43(

)4(

)23(

)2(

)13(

)1(

)03(

)0(

4

)4)(3)(1(

)42(

)4(

)32(

)3(

)12(

)1(

)02(

)0(

6

)4)(3)(2(

)41(

)4(

)31(

)3(

)21(

)2(

)01(

)0(

24

)4)(3)(2)(1(

)40(

)4(

)30(

)3(

)20(

)2(

)10(

)1(

4523)()(

4

3

2

1

0

43210

4

0

4

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxxx

xfxf

i

ii


Inverse Interpolation

given is where,)( : thatsuch Find

valuesof a table Given :Problem

kk yyxfx

….

….

1xnx

0y 1y ny

ix

iy

One approach:

Use polynomial interpolation to obtain fn(x) to interpolate the

data then use Newton’s method to find a solution to x

kn yxf )(

0x



….

….

1x nx

0y 1y ny

ix

iy

Inverse interpolation:

1. Exchange the roles

of x and y.

2. Perform polynomial

Interpolation on the

new table.

3. Evaluate

)( kn yfx

….

….

iy 0y 1y ny

ix 0x 1x nx

0x



x

y

y

x



Question:

What is the limitation of inverse interpolation?

• The original function has an inverse.

• y1, y2, …, yn must be distinct.


Inverse Interpolation Example

5.2)( that such Find table. theGiven

:Problem

xfx

x 1 2 3

y 3.2 2.0 1.6

3.2 1 -.8333 1.0417

2.0 2 -2.5

1.6 3

2187.1)5.0)(7.0(0417.1)7.0(8333.01)5.2(

)2)(2.3(0417.1)2.3(8333.01)(

2

2

fx

yyyyfx


Errors in polynomial Interpolation

Polynomial interpolation may lead to large errors (especially for high order polynomials).

BE CAREFUL

When an nth order interpolating polynomial is used, the error is related to the (n+1)th order derivative.


-5 -4 -3 -2 -1 0 1 2 3 4 5-0.5

0

0.5

1

1.5

2

true function

10 th order interpolating polynomial

10th Order Polynomial Interpolation


1

)1()1(

)1(4)()(

:Then points). end the(including b][a, in

points spacedequally 1at esinterpolatthat

n degree of polynomialany beLet

.)( and b],[a, on continuous is )(

: thatsuch functiona be)(Let

n

nn

n

ab

n

Mx-Pxf

nf

P(x)

Mxfxf

xf

Errors in polynomial InterpolationTheorem


Example

9

10

1

)1(

th

1034.19

6875.1

)10(4

1

)1(4

9 ,1

01

.[0,1.6875] interval in the points) spacedequally 01 (using

f(x) einterpolat topolynomialorder 9 use want toWe

sin

f(x)-P(x)

n

ab

n

Mf(x)-P(x)

nM

nforf

(x) f(x)

n

n


Summary

The interpolating polynomial is unique.

Different methods can be used to obtain it.

Newton’s divided difference

Lagrange interpolation

Others

Polynomial interpolation can be sensitive to data.

BE CAREFUL when high order polynomials are used.

UIN Malang · Analisis Numerik (P6) 2 Introduction Interpolation was used for long time to provide...

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