MAT 4725Numerical Analysis

Section 3.1

Interpolation and the Lagrange Polynomial

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HW 7b (d)

HW 7b (e)

Material Temperature.Temp

x

Material Temperature.Temp

x5

?

Interpolation

3.1 Goal

Find a polynomial P(x) that passes through all the data points (xi,yi), i=0,1,2,…,n

Use P(x) to estimate the function values

A Simple Situation

Suppose there are only 2 data points:

(x0,f(x0)), (x1,f(x1))

Let us find a degree one poly. P(x) that passes through them.

A Simple Situation

Suppose there are only 2 data points:

(x0,f(x0)), (x1,f(x1))

Let us find a degree one poly. P(x) that passes through them

Q: Why degree one?

A Simple Situation

Suppose there are only 2 data points:

(x0,f(x0)), (x1,f(x1))

Let us find a degree one poly. P(x) that passes through them

Q: We know easier way to find a straight line through two points. Why the trouble?

In General…

Suppose there are (n+1) data points:

(xi,f(xi)) i=0,1,2,…,n

Let us find a degree n poly. P(x) that passes through them

n-th Lagrange Interpolating Poly.

0

0 0 1 1

( ) ( ) ( )

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

n

k kk

n n

P x f x L x

f x L x f x L x f x L x

Example 1

Find the 2nd Lagrange Polynomial P(x)

0 1 2

1( ) ; 2, 2.5, 4f x x x x

x

Example 1 0 1 2

1( ) ; 2, 2.5, 4f x x x x

x

20

21

22

2

( ) 6.5 10

4( ) 6 8

31

( ) 4.5 53

( ) 0.05 0.425 1.15

L x x x

L x x x

L x x x

P x x x

Example 1 0 1 2

1( ) ; 2, 2.5, 4f x x x x

x

1y

x

20.05 0.425 1.15y x x

Example 1 0 1 2

1( ) ; 2, 2.5, 4f x x x x

x

1y

x

20.05 0.425 1.15y x x

Q: For what range will P(x) give good estimations?

Error Formula

We will skip the error analysis (similar to Taylor poly.)

We will see this again in section 4.1

( 1)

0 1

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

( 1)!

where [a,b], ( ) ( , )

n

n

i

f xf x P x x x x x x x

n

x x a b

Classwork 1, 2

Write a program to compute the 2nd Lagrange Polynomial

INPUT: (xi,f(xi)) i=0,1,2 OUTPUT: P(x)

Remark #1

(xi,f(xi)) are passed into the program as two arrays:xx=[x0,x1,x2], yy=[y0,y1,y2]

>xx:=array(0..2,[2, 2.5, 4]);

yy:=array(0..2,[0.5, 0.4, 0.25]);

Hints

Hints are provided in the handout.

Homework

Download Homework from the web. Read the first 4 pages of 3.5 for

Wednesday