Transcript PPT

Forward Divided Difference
Major: All Engineering Majors
Authors: Autar Kaw, Sri Harsha Garapati
http://numericalmethods.eng.usf.edu
Numerical Methods for STEM undergraduates
7/12/2016
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1
Definition
lim f  x  Δx   f  x 
f  x  
Δx  0
Δx
.
Slope at xi
y
f(x)
xi
2
x
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Forward Divided Difference
f  x  x   f  x 
f  x  
x
f (x )
x
3
x  x
f  xi  x   f  xi 
f  xi  
x
x
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Example
Example:
The velocity of a rocket is given by


14  10 4
 t   2000 ln 
  9.8t ,0  t  30
4
14

10

2100
t


where

given in m/s and
the first derivative of
t  2s.
t is given in seconds. Use forward difference approximation of
νt  to calculate the acceleration at t  16s. Use a step size of
Solution:
νti 1   νti 
ati  
Δt
ti  16
4
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Example (contd.)
Δt  2
t i 1  t i  Δt  16  2  18
a 16  
 18   16
2


14  104
 18  2000 ln 
 9.818  453.02m/ s

4


14  10  2100 18 


14 104
 392.07m/ s
 16  2000 ln 
  9.816
4
14 10  210016
5
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Example (contd.)
Hence
a16 
 (18)  (16)
2
The exact value of
 453.02  392.07  30.475m / s 2
a16 can be calculated by differentiating


14  104
 t   2000 ln 
  9.8t
4
14

10

2100
t


as
d
 (t )   4040  29.4t
dt
 200  3t
a(16)  29.674m / s 2
a(t ) 
6
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Example (contd.)
The absolute relative true error is
t 
TrueValue  ApproximateValue
100
TrueValue
29.674  30.475
100
29.674
 2.6993%

7
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Effect Of Step Size
f ( x)  9e
Value of
h
0.05
0.025
0.0125
0.00625
0.003125
0.001563
0.000781
0.000391
0.000195
9.77E-05
4.88E-05
8
4x
f ' (0.2) Using forward difference method.
f ' (0.2)
88.69336
84.26239
82.15626
81.12937
80.62231
80.37037
80.24479
80.18210
80.15078
80.13512
80.12730
Ea
-4.430976
-2.106121
-1.0269
-0.507052
-0.251944
-0.125579
-0.062691
-0.031321
-0.015654
-0.007826
a %
5.258546
2.563555
1.265756
0.628923
0.313479
0.156494
0.078186
0.039078
0.019535
0.009767
Significant
digits
0
1
1
1
2
2
2
3
3
3
Et
t %
-8.57389
-4.14291
-2.03679
-1.00989
-0.50284
-0.25090
-0.12532
-0.06263
-0.03130
-0.01565
-0.00782
10.70138
5.170918
2.542193
1.260482
0.627612
0.313152
0.156413
0.078166
0.039073
0.019534
0.009766
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Effect of Step Size in Forward
Divided Difference Method
Initial step size=0.05
92
f'(0.2)
88
84
80
76
0
1
2
3
4
5
6
7
8
9
10
11
12
Num ber of tim es step size halved, n
9
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Effect of Step Size on
Approximate Error
Number of times step size halved, n
0
0
2
4
6
8
10
12
-1
Ea
-2
-3
-4
Initial step size=0.05
-5
10
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Effect of Step Size on Absolute
Relative Approximate Error
6
5
Initial step size=0.05
|Ea| %
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
Num ber of tim es step size halved, n
11
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Effect of Step Size on Least
Number of Significant Digits
Correct
Least number of significant digits
correct
4
Initial step size=0.05
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
Num ber of tim es step size halved, n
12
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Effect of Step Size on True Error
Num ber of tim es step size halved, n
0
0
2
4
6
8
10
12
Et
-3
-6
-9
Initial step size=0.05
Initial step size=0.05
13
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Effect of Step Size on Absolute
Relative True Error
Initial step size=0.05
12
10
|Et| %
8
6
4
2
0
1
2
3
4
5
6
7
8
9
10
11
Number of times step size halved, n
14
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