Backward Divided Difference
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Backward Divided Difference - Numerical Methods - Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization
Main points are: Backward Divided Difference, Velocity of Rocket, First Derivative, Difference Approximation, Calculate Acceleration, Absolute Relative True Error, Effect of Step Size, Divided Difference Method
Typology: Slides
2012/2013
1 / 14
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Backward Divided Difference
Definition
x i
i
x
.
x
f x f x x
x
f x
lim
Slope at
f(x)
y
x
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 backward 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 ∆
− ≅
− 1 ν ν
Solution:
Δ t = 2
t i − 1 = ti −∆ t = 16 − 2 = 14
( )
( ) ( )
ν − ν
a =
( ) ( )
- 8 ( 16 ) 14 10 210016
16 2000 ln 4
4
^ −
× −
×
ν = = 392. 07 m / s
( ) ( )
- 8 ( 14 ) 14 10 210014
14 2000 ln 4
4
^ −
× −
×
ν = =^334.^24 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 72.61598 7.50349 9.
0.025 76.24376 3.627777 4.758129 1 3.87571 4.
0 .0125 78.14946 1.905697 2.438529 1 1.97002 2.
0.00625 79.12627 0.976817 1.234504 1 0.99320 1.
0.003125 79.62081 0.494533 0.62111 1 0.49867 0.
0.001563 79.86962 0.248814 0.311525 2 0.24985 0.
0.000781 79.99442 0.124796 0.156006 2 0.12506 0.
0.000391 80.05691 0.062496 0.078064 2 0.06256 0.
0.000195 80.08818 0.031272 0.039047 3 0.03129 0.
9.77E- 05 80.10383 0.015642 0.019527 3 0.01565 0.
4.88E- 05 80.11165 0.007823 0.009765 3 0.00782 0.
x f x e
4 ( ) = 9
Value of (^) ( 0. 2 ) ' f Using backward Divided difference method.
Effect of Step Size on Approximate Error
0
1
2
3
4
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
1
2
3
4
5
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
1
2
3
4
5
6
7
8
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
1 2 3 4 5 6 7 8 9 10 11
Num ber of steps involved, n
|E
| %t
Initial step size=0.