W12_Interpolation0

8/2/2019 W12_Interpolation0

1/27

Centre for Computer Technology

ICT114Mathematics for

Computing

Week 12

Interpolation


2/27

March 20, 2012 Copyright Box Hill Institute

Curve Fitting

A relation between two variables isexpressed mathematically by an equation

connecting both. Finding equations approximating curves

that fit the given set of data is called curve

fitting.


3/27


Objectives

Review week 11

Problem of Interpolation

Finite Differences Diagonal differencetable

Newtons Forward Interpolation

Newtons Backward Interpolation

Lagranges Interpolation


4/27


Method of Least Squares (1)

For the data points (x1, y1), (x2,y2),.(xn,yn)

Let y = a+bx is the least squares line a and b are determined by solving the

(normal) equations

y = an + b x xy = a x + b x2


5/27


Method of Least Squares (2)

( y) (x2)( x) (xy)

a = ---------------------------------------------

n x2(x)2

n xy ( x)( y)

b = ----------------------------------------------n x2(x)2


6/27


NewtonsForward/BackwardInterpolation


7/27


Problem of Interpolation

The following table givesthe value of logx for certain

values of x To determine the value of

logx for x = 96.45 (which is

not given in the table), weuse the method ofInterpolation.

x y=logx

9596

97

9899

1.97771.9823

1.9868

1.99121.9956


8/27


Finite Differences Diagonal

Difference TableIn the table in the next slide

a any argument (for x)h any positive constant

f(x) = f(x+h) f(x) = first difference

f(a) is called the leading termf(a),2f(a),3f(a) are called the leadingdifferences


9/27


Argument

x

Entry

f(x)

1st

Differencef(x)

2nd

Difference2f(x)

3rd

Difference3f(x)

a f(a)

f(a)

a+h f(a+h) 2f(a)

f(a+h) 3f(a)

a+2h f(a+2h) 2f(a+h)

f(a+2h)

a+3h f(a+3h)


10/27


Newton's Forward Interpolation

u = (x x0)/h

(x)= (x0+uh)= y0 + u.y0 + u(u-1)/2! .

2y0+ ..

.+ u(u-1)(u-2).(u-n+1)/n! .ny0

y0 = f(x0)


11/27


Newtons Backward Interpolation

u = (x xn)/h

(x)= (xn+uh)= yn + u.yn-1 + u(u+1)/2! .

2yn-2+ ..

.+ u(u+1)(u+2).(u+n-1)/n! .ny0

yn = f(xn)


12/27


Example

From the followingtable determine thevalue of e0.1245 and

e0.1895

The first step is toconstruct the diagonal

difference table

x y=ex

0.12

0.13

0.140.15

0.16

0.17

0.18

0.19

1.127497

1.138828

1.1502741.161834

1.173511

1.185305

1.197217

1.209250


13/27

X y=ex y 2y

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

1.127497

1.138828

1.150274

1.161834

1.173511

1.185305

1.197217

1.209250

0.011331

0.011446

0.011560

0.011677

0.011794

0.011912

0.012033

0.000115

0.000114

0.000117

0.000117

0.000118

0.000121


14/27


To determine e0.1245 we have to interpolate at thebeginning of the table.

h = 0.01, x0 = 0.12, x = 0.1245

so, u = (0.1245 0.12)/0.01 = 0.45

therefore,f(0.1245) =y0 + u.y0 + u(u-1)/2! .

2y0

= 1.127497 + 0.45 x 0.011331

+ 0.45 x (0.45-1) /2! x 0.000115

= 1.13258172

Actual value = 1.13258202


15/27

March 20, 2012Copyright Box Hill Institute

To determine e0.1895 we have to interpolate at theend of the table.

h = 0.01, xn = 0.19, x = 0.1895

so, u = (0.1895 0.19)/0.01 = -0.05

therefore,

(0.1895) =yn + u.yn-1+ u(u+1)/2! .2yn-2= 1.209250 + (-0.05) x 0.012033

+(-0.05)(-0.05+1) / 2! x 0.000121

= 1.20864548

Actual value = 1.208645124


16/27


Question

Find the value of

log(96.45) and

log(98.75)using Newton's interpolation formulae fromthe table given in the second slide.

Compare them with the actual value.


17/27




18/27



Newtons formulae can only be used for

equidistant values of the argument

But sometimes it is difficult to obtaintabulated values of a function forequidistant values of the argument.

To deal with such a situation we useLagranges interpolation formula


19/27


The Lagranges Interpolation formula is given below

(x-x1)(x-x2)..(x-xn)(x) = --------------------------------- y0

(x0-x1)(x0-x2)(x0-xn)

(x-x0)(x-x2)..(x-xn)+ ---------------------------------- y1 +

(x1-x0)(x1-x2)(x1-xn)

(x-x0)(x-x2)..(x-xn-1)+ --------------------------------- yn

(xn-x0)(xn-x2)(xn-xn-1)


20/27


Example

From the given table find the value of y for x = 3.743

x y

3.7428037

3.7428822

3.7429607

3.74303923.7431176

5531

5532

5533

55345535

x0 = 3.7428037, y0 = 5531

x1 = 3.7428822, y1 = 5532x2 = 3.7429607, y2 = 5533

x3 = 3.7430392, y3 = 5534

x4 = 3.7431176, y4 = 5535

x = 3.743


21/27


(x-x1)(x-x2)(x-x3) (x-x4)(x) = ---------------------------------- y0 (0.0234250 x 5531)

(x0-x1)(x0-x2)(x0-x3)(x0-x4)

(x-x0)(x-x2)(x-x3)(x-x4)+ ------------------------------------ y1 + (-0.156157 x 5532)

(x1-x0)(x1-x2)(x1-x3)(x1-x4)

(x-x0

)(x-x1

)(x-x3

)(x-x4

)+ ------------------------------------ y2 + (0.702259 x 5533)

(x2-x0)(x2-x1)(x2-x3)(x2-x4)

(x-x0)(x-x1)(x-x2)(x-x4)+ ------------------------------------ y3 + (0.469666 x 5534)

(x3-x0)(x3-x1)(x3-x2)(x3-x4)

(x-x0)(x-x1)(x-x2)(x-x3)+ ------------------------------------ y4 (-0.0391929 x 5535)

(x4-x0)(x4-x1)(x4-x2)(x4-x3)


22/27


y = (3.743)

= (0.0234250 x 5531)

+ (-0.156157 x 5532)

+ (0.702259 x 5533)+ (0.469666 x 5534)+ (-0.0391929 x 5535)

= 5533.501


23/27


Question

From the given table find the value of y for x = 0.135

x y

0.12

0.13

0.14

0.15

1.127497

1.138828

1.150274

1.161834


24/27


Summary

Newtons Forward Interpolation formula

u = (x x0)/h

(x) = (x0+uh)

= y0

+ u.y0

+ u(u-1)/2! .2y0

+ ..

.+ u(u-1)(u-2).(u-n+1)/n! .ny0

y0 = f(x0)


25/27


Summary

Newtons Backward Interpolation formula

u = (x xn)/h

(x)= (xn+uh)= yn + u.yn-1+ u(u+1)/2! .

2yn-2+ ..

.+ u(u+1)(u+2).(u+n-1)/n! .ny0

yn = f(xn)


26/27

March 20, 2012

Copyright Box Hill Institute

SummaryThe Lagranges Interpolation formula is given below

(x-x1)(x-x2)..(x-xn)(x) = --------------------------------- y0

(x0-x1)(x0-x2)(x0-xn)

(x-x0)(x-x2)..(x-xn)+ ---------------------------------- y1 +

(x1-x0)(x1-x2)(x1-xn)

(x-x0)(x-x2)..(x-xn-1)+ --------------------------------- yn

(xn-x0)(xn-x2)(xn-xn-1)


27/27

March 20, 2012

Copyright Box Hill Institute

References

H L Verma and C W Gross : Introduction toQuantitative Methods,John Wiley

JB Scarborough : Numerical Mathematical

Analysis, Jon Hopkins Hall, New Jersey Gerald W. Recktenwald, Numerical Methods

with MATLAB, Implementation and Application,Prentice Hall

Murray Spiegel, John Schiller, Alu Srinivasan,Probability and Statistics, Schaums easyOutlines

http://mathworld.wolfram.com

W12_Interpolation0

Documents

Transcript of W12_Interpolation0