Loading in 5 sec....

Chapter 8 Interpolation (1)PowerPoint Presentation

Chapter 8 Interpolation (1)

- 529 Views
- Updated On :

Chapter 8 Interpolation (1). Table of Contents. 8.1 Polynomial Interpolation 8.1.1 Lagrange Interpolation 8.1.2 Newton Interpolation 8.1.3 Difficulties with Polynomial Interpolation 8.2 Hermite Interpolation 8.3 Rational-Function Interpolation. Lagrange Interpolation Polynomials.

Related searches for chapter 8 interpolation 1

Download Presentation
## PowerPoint Slideshow about 'chapter 8 interpolation 1' - bernad

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

Chapter 8Interpolation (1)

Copyleft 2005 by MediaLab

Table of Contents

- 8.1 Polynomial Interpolation
- 8.1.1 Lagrange Interpolation
- 8.1.2 Newton Interpolation
- 8.1.3 Difficulties with Polynomial Interpolation

- 8.2 Hermite Interpolation
- 8.3 Rational-Function Interpolation

Copyleft 2005 by MediaLab

Lagrange Interpolation Polynomials

- Basic concept
- The Lagrange interpolating polynomial is the polynomial of degree n-1 that passes through the n points.
- Using given several point, we can find Lagrange interpolation polynomial.

Copyleft 2005 by MediaLab

General Form of Lagrange

- The general form of the polynomial is
p(x) = L1y1 + L2y2 + … + Lnyn

where the given points are (x1,y1), ….. , (xn,yn).

- The equation of the line passing through two points (x1,y1) and (x2,y2) is
- The equation of the parabola passing through three points (x1,y1), (x2,y2), and (x3,y3) is
- http://math.fullerton.edu/mathews/n2003/lagrangepoly/LagrangePolyProof.pdf

Copyleft 2005 by MediaLab

Example of Lagrange Interpolation

- Example
- Given points (x1,y1)=(-2,4), (x2,y2)=(0,2), (x3,y3)=(2,8)

Copyleft 2005 by MediaLab

MATLAB Function for Lagrange Interpolation

- We can represent the Lagrange polynomial with coefficient ck.
- p(x)=c1N1+c2N2+ … +cnNn

Copyleft 2005 by MediaLab

Higher Order Interpolation Polynomials

- Higher order interpolation polynomials
x = [ -2 -1 0 1 2 3 4],

y = [ -15 0 3 0 -3 0 15]

Copyleft 2005 by MediaLab

Review and Discussion

- In Lagrange interpolation polynomial, it always go through given points.
Think with equation below

- The Lagrange form of polynomial is convenient when the same abscissas may occur in different applications.
- It is less convenient than the Newton form when additional data points may be added to the problem.

Copyleft 2005 by MediaLab

Newton Interpolation Polynomials

- Newton form of the equation of a straight line passing through two points (x1, y1) and (x2, y2) is
- Newton form of the equation of a parabola passing through three points (x1, y1), (x2, y2), and (x3, y3) is
- the general form of the polynomial passing through n points (x1, y1), …,(xn, yn) is

Copyleft 2005 by MediaLab

Newton Interpolation Polynomials (cont’d)

- Substituting (x1, y1) into
- Substituting (x2, y2) into
- Substituting (x3, y3) into

Copyleft 2005 by MediaLab

Newton Interpolation Parabola

- Passing through the points (x1, y1)=(-2, 4), (x2, y2)=(0, 2), and (x3, y3)=(2, 8).
- The equations is
- Where the coefficients are
- thus

Copyleft 2005 by MediaLab

Newton Interpolation Parabola (cont’d)

- Passing through the points (x1, y1)=(-2, 4), (x2, y2)=(0, 2), and (x3, y3)=(2, 8).

Copyleft 2005 by MediaLab

Additional Data Points

- We extend the previous example, adding the points (x4, y4) = (-1, -1) and (x5, y5) = (1, 1)
- Divided-difference
table becomes

(with new entries

shown in bold)

- Newton interpolation
polynomial is

Copyleft 2005 by MediaLab

Higher Order Interpolation Polynomials

- Consider again the data from
Example 8.4 with Lagrange form.

x = [ -2 -1 0 1 2 3 4 ],

y = [ -15 0 3 0 -3 0 15]

- Do it again with Newton form.

That the polynomial is cubic is clear.

Copyleft 2005 by MediaLab

Higher Order Interpolation Polynomials (cont’d)

- If the y values are modified slightly, the divided-difference table shows the small contribution from the higher degree terms:

Copyleft 2005 by MediaLab

MATLAB functions – Finding the coefficients

>> x = [-2 -1 0 1 2 3 4];

>> y = [-15 0 3 0 -3 0 15];

>> Newton_Coef(x,y);

d =

15 -6 1 0 0 0

3 -3 1 0 0 0

-3 0 1 0 0 0

-3 3 1 0 0 0

3 6 0 0 0 0

15 0 0 0 0 0

>>

function a = Newton_Coef(x, y)

n = length(x);

%Calculate coeffiecients of

Newton interpolating polynomial

a(1) = y(1);

for k=1 : n-1

d(k,1) = (y(k+1) - y(k))/(x(k+1) - x(k));

%1st divided diff

end

for j=2 : n-1

for k=1 : n-j

d(k,j) = (d(k+1,j-1) - d(k,j-1))/(x(k+j) - x(k));

%jth divided diff

end

end

d

for j=2:n

a(j) = d(1, j-1);

end

Copyleft 2005 by MediaLab

MATLAB functions – Evaluate the polynomials

>> a = [-15 15 -6 1 0 0 0];

>> x = [-2 -1 0 1 2 3 4];

>> t = [0 1 2];

>> Newton_Eval(t, x, a);

t =

0 1 2

p =

3 0 -3

function p = Newton_Eval(t,x,a)

% t : input value of the polynomial (x)

% x : x values of interpolating points

% a : answer of previous MATLAB function, i.e,

the coefficients of Newton polynomial.

n = length(x);

hold on;

for i =1 : length(t)

ddd(1) = 1; %Compute first term

c(1) = a(1);

for j=2 : n

ddd(j) = (t(i) - x(j-1)).*ddd(j-1);

% Compute jth term

c(j) = a(j).*ddd(j);

end;

p(i) = sum(c);

%plot(t(i),p(i)); grid on;

end

t

p

Copyleft 2005 by MediaLab

Humped and Flat Data

- The data
- x = [ -2 -1.5 -1 -0.5 0 – 0.5 1 1.5 2]
- y = [ 0 0 0 0.87 1 0.87 0 0 0]
illustrate the difficulty with using higher order polynomials to interpolate a moderately large number of points.

Copyleft 2005 by MediaLab

Noisy Straight Line

- The data
- x = [ 0.00 0.20 0.80 1.00 1.20 1.90 2.00 2.10 2.95 3.00]
- y = [ 0.01 0.22 0.76 1.03 1.18 1.94 2.01 2.08 2.90 2.95]

- Not well suited with noisy straight line.

Copyleft 2005 by MediaLab

Runge Function

- The function
is a famous example of the fact that polynomial interpolation does not produce a good approximation for some functions and that using more function values (at evenly spaced x values) does not necessarily improve the situation.

- Example 1.
- x = [ -1 -0.5 0.0 0.5 1.0 ]
- y = [0.0385 0.1379 1.0000 0.1379 0.0385]

Copyleft 2005 by MediaLab

Runge Function (cont’d)

- Example 2.
- x = [-1.000 -0.750 -0.500 -0.250 0.000 0.250 0.500 0.750 1.000 ]
- y = [0.0385 0.0664 0.138 0.3902 1.000 0.3902 0.138 0.0664 0.0385]

- The interpolation polynomial overshoots the true polynomial muchmore severely than the polynomial formed by using only five points.

Copyleft 2005 by MediaLab

8.2Hermite Interpolation

Copyleft 2005 by MediaLab

Hermite Interpolation

- Hermite interpolation allows us to find a ploynomial that matched both function value and some of the derivative values

Copyleft 2005 by MediaLab

More data for Product concentration

Copyleft 2005 by MediaLab

MATLAB Code for Hermite interpolation

Copyleft 2005 by MediaLab

Difficult Data

- As with lower order polynomial interpolation, trying to interpolate in humped and flat regions cause overshoots.

Copyleft 2005 by MediaLab

8.3Rational-Function Interpolation

Copyleft 2005 by MediaLab

Rational-Function Interpolation

- Why use rational-function interpolation?
- Some functions are not well approximated by polynomials.(runge-function)
- but are well approximated by rational functions, that is quotients of polynomials.

Copyleft 2005 by MediaLab

Bulirsch-Stoer algorithm

- Bulirsch-Stoer algorithm
- The approach is recursive, based on tabulated data(in a manner similar to that for the Newton form of polynomial interpolation).
- Given a set of m+1 data points (x1,y1), … , (xm+1, ym+1), we seek an interpolation function of the form

<Bulirsch-Stoer algorithm general pattern>

The proof is in J.Stoer and R.Bulirsch, 'Introduction to Numerical Analysis'

Copyleft 2005 by MediaLab

Bulirsch-Stoer algorithm(cont’d)

- Bulirsch-Stoer method for three data points

Copyleft 2005 by MediaLab

Second stage

data

First stage

x1 y1

R1= y1

x2 y2

R2= y2

x3 y3

R3=y3

x4 y4

R4=y4

x5 y5

R5=y5

Bulirsch-Stoer algorithm(cont’d)- Bulirsch-Stoer method for five data points

Copyleft 2005 by MediaLab

Bulirsch-Stoer Rational-Function(cont’d)

Copyleft 2005 by MediaLab

Example 8.13 rational-function interpolation

data points:

x = [-1 -0.5 0.0 0.5 1.0]

y = [0.0385 0.1379 1.0000 0.1379 0.0385]

Copyleft 2005 by MediaLab

Example 8.13 rational-function interpolation(cont’d)

Copyleft 2005 by MediaLab

Download Presentation

Connecting to Server..