St. John's University of Tanzania

MAT210 NUMERICAL ANALYSIS2013/14 Semester II

INTERPOLATIONNewton Divided Difference

Kaw, Chapter 5.03

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● Interpolation of functions● Approximating a complex or unknown function

with a simpler function– simpler function usually a polynomial

● Direct method● Solving n+1 simultaneous equations for an nth

order polynomial fit● Newton Divided Difference

● A combination of Taylor Series approximation and numerical differentiation

Introduction

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Linear NDD from Taylor Series

f (x)= f (x0)+(x−x0) f ' (x0)+O ((Δ x)2)

f (x)≈ f (x0)+(x−x0)[ f (x1)− f (x0)

x1−x0]

⇒ f 1(x)=b0+b1(x−x0)

where b0= f (x0) ,b1=[ f (x1)− f (x0)

x1−x0]

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Another perspective● For Linear Direct Method

● Equation has the slope-intercept form● For Newton Divided Difference

● Equation has the point-slope form● Methods are different, but equations are

simply different forms of the same line

f 1(x )=b0+b1(x−x0)=a0+a1 x

b0−b1 x0=a0, b1=a1

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Example: Velocity

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Linear

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Find v(16)

● Two closest points t=15 and t=20

Same as Direct

Same as Direct

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Quadratic

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NDD Quadratic● Just include another point and add another

term to the linear NDD

● b0 and b1 unchanged

f 2(x )=b0+b1(x−x0)+b2(x−x0)(x−x1)

b2=

f (x2)− f (x1)

x2−x1

−f (x1)− f (x0)

x1−x0

x2−x0

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Now apply Forward Difference Approximations for f '(x

1) and f '(x

0)

Whence b2?

● b2 is simply the coefficient of the O((Δx)²) term of the Taylor Series!

b2≈12

f ' ' (x0)≈f ' (x1)− f ' (x0)

2Δ xHow?b2=

f (x2)− f (x1)

x2−x1

−f (x1)− f (x0)

x1−x0

x2−x0

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Quadratic v(16) ● Use t=10,15,20 as before

Same as Direct

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The Error is the Error● The equations are the same, just found in

different ways and kept in different forms

|ϵa|=| vquadratic−v linear

vquadratic|

Same as Direct

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So What?● If Newton's Divided Difference produces

that same result as Direct, then why use it?● Isn't a matrix inversion easier?● Not really

● For Direct, each level is a new calculation, a new inversion. Doing an inversion many times is not efficient

● For NDD, each level builds on the last

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General Form

The divided differences are calculated recursivelyThis is more efficient

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The Big Picture

The process is a sequence of simple equations

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Going Deeper● This is an example of using multiple criteria

to evaluate a method● Accuracy and Efficiency … one is not enough

● The velocity is not all you have● Calculate the acceleration at t=16s in the linear,

quadratic and cubic cases. – Estimate the accuracy

● Calculate the distance traveled from 15s to 20s in each case– Estimate the accuracy