Lecture slides based on Autar Kaw's Numerical Methods Section on Interpolation with Newton Divided Difference, http://nm.mathforcollege.com

1. St. John's University of Tanzania
MAT210 NUMERICAL ANALYSIS
2013/14 Semester II
INTERPOLATION
Newton Divided Difference
Kaw, Chapter 5.03

2. MAT210 2013/14 Sem II 2 of 16
● 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
]

4. MAT210 2013/14 Sem II 4 of 16
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

9. MAT210 2013/14 Sem II 9 of 16
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

10. MAT210 2013/14 Sem II 10 of 16
Now apply Forward Difference
Approximations for f'(x1
) and f'(x0
)
Whence b2
?
● b2 is simply the coefficient of the O((Δx)²) term of
the Taylor Series!
b2≈
1
2
f ' ' (x0)≈
f '(x1)− f ' (x0)
2Δ x
H
o
w
?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−vlinear
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 recursively
This 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