Forward Divided Difference
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Forward Divided Difference - Numerical Methods - Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization
Main points are: Forward Divided Difference, Velocity of Rocket, Difference Approximation, Calculate Acceleration, Step Size, First Derivative, Effect of Step Size, Approximate Error, Significant Digits Correct
Typology: Slides
2012/2013
1 / 14
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Forward Divided Difference
Definition
x i
i
x
.
x
f x x f x
x
f x
lim + −
f(x)
Slope at
x
y
Example
= 16 i
t
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 forward 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
ν t ν t a t
i i i Δ
1 − ≅
Solution:
Δ t = 2
t i + 1 = ti +Δ t = 16 + 2 = 18
( )
( ) ( )
ν − ν
a =
( ) ( )
- 8 ( 18 ) 14 10 210018
18 2000 ln 4
4
−
× −
×
ν = = 453. 02 m / s
( ) ( )
- 8 ( 16 ) 14 10 210016
16 2000 ln 4
4
^ −
× −
×
ν = =^392.^07 m^ / s
Example (contd.)
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 88.69336 - 8.57389 10.
0.025 84.26239 - 4.430976 5.258546 0 - 4.14291 5.
0.0125 82.15626 - 2.106121 2.563555 1 - 2.03679 2.
0.00625 81.12937 - 1.0269 1.265756 1 - 1.00989 1.
0.003125 80.62231 - 0.507052 0.628923 1 - 0.50284 0.
0.001563 80.37037 - 0.251944 0.313479 2 - 0.25090 0.
0.000781 80.24479 - 0.125579 0 .156494 2 - 0.12532 0.
0.000391 80.18210 - 0.062691 0.078186 2 - 0.06263 0.
0.000195 80.15078 - 0.031321 0.039078 3 - 0.03130 0.
9.77E- 05 80.13512 - 0.015654 0.019535 3 - 0.01565 0.
4.88E- 05 80.12730 - 0.007826 0.009767 3 - 0.00782 0.009 766
x f x e
4 ( ) = 9
Value of (^) ( 0. 2 ) ' f Using forward difference method.
Effect of Step Size on Approximate
Error
0
0 2 4 6 8 10 12
Number of times step size halved, n
E
a
Initial step size=0.
Effect of Step Size on Absolute Relative
Approximate Error
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8 9 10 11 12
Num ber of tim es step size halved, n
|E
| %a
Initial step size=0.
Effect of Step Size on True Error
0 0 2 4 6 8 10 12
Num ber of tim es step size halved, n
E
t
Initial step size=0.
Initial step size=0.
Effect of Step Size on Absolute Relative
True Error
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9 10 11
Number of times step size halved, n
|E
| %t
Initial step size=0.