Central Divided Difference
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Central Divided Difference - Numerical Methods - Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization
Main points are: Central Divided Difference, Velocity of Rocket, Central Difference Approximation, Calculate Acceleration, First Derivative, Absolute Relative True Error, Effect of Step Size, Approximate Error
Typology: Slides
2012/2013
1 / 14
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Central Divided Difference
Definition
x i
i
x
.
x
f x f x x
x
f x
lim
f(x)
Slope at
x
y
Example
ti = 16
Example :
The velocity of a rocket is given by
( ) 9. 8 , 0 30 14 10 2100
14 10 2000 ln 4
4
− ≤^ ≤
× −
× = t t t
ν t
where ν given in m/s and (^) t is given in seconds. Use central difference approximation of
the first derivative of (^) ν ( ) t to calculate the acceleration at t = 16 s .Use a step size of
∆ t = 2 s.
( )
( ) ( )
t
t t a t
i i i ∆
2
ν 1 ν 1
Solution:
Δ t = 2
t i + 1 = ti +Δ t =^16 +^2 =^18
( )
( ) ( ) ( ) ( )
ν ν ν − ν
a =
( ) ( )
- 8 ( 18 ) 14 10 210018
18 2000 ln 4
4
−
× −
×
ν = = 453. 02 m / s
( ) ( )
- 8 ( 14 ) 14 10 210014
14 2000 ln 4
4
^ −
× −
×
ν = = 334. 24 m / s
Example (contd.)
t i − 1 = ti −∆ t =^16 −^2 =^14
The absolute relative true error is
Example (contd.)
× 100
TrueValue
TrueValue ApproximateValue
ε t
×
Effect Of Step Size
h (^) f '( 0. 2 ) E (^) a ε (^) a % Significant digits
Et (^) ε (^) t %
0.05 80.65467 - 0.53520 0.
0.025 80.25307 - 0.4016 0.500417 1 - 0.13360 0.
0 .0125 80.15286 - 0.100212 0.125026 2 - 0.03339 0.
0.00625 80.12782 - 0.025041 0.031252 3 - 0.00835 0.
0.003125 80.12156 - 0.00626 0.007813 3 - 0.00209 0.
0.001563 80.12000 - 0.001565 0.001953 4 - 0.00052 0.
0.000781 80.11960 - 0.000391 0 .000488 5 - 0.00013 0.
0.000391 80.11951 - 9.78E- 05 0.000122 5 - 0.00003 4.07E- 05
0.000195 80.11948 - 2.45E- 05 3.05E- 05 6 - 0.00001 1.02E- 05
9.77E- 05 80.11948 - 6.11E- 06 7.63E- 06 6 0.00000 2.54E- 06
4.88E- 05 80.11947 - 1.53E- 06 1.91E- 06 7 0.00000 6.36E- 07
x f x e
4 ( ) = 9
Value of (^) ( 0. 2 ) ' f Using Central Divided Difference difference method.
Effect of Step Size on Approximate
Error
0
1 3 5 7 9 11
Num ber of steps involved, n
E(a)
Initial step size=0.
Effect of Step Size on Absolute Relative
Approximate Error
0
0 1 2 3 4 5 6 7 8 9 10 11 12
Num ber of steps involved, n
|E(a)|,%
Initial step size=0.
Effect of Step Size on True Error
-0.
-0.
-0.
-0.
-0.
-0.
0 2 4 6 8 10 12
Num ber of steps involved, n
E(t)
Initial step size=0.
Effect of Step Size on Absolute Relative
True Error
0
1 2 3 4 5 6 7 8 9 10 11
Num ber of tim es step size halved,n
|E
| %t
Initial step size=0.