Transcript PPT

Central 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   f x  x 
f x  
x  0
x
.
Slope at xi
y
f(x)
xi
2
x
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Central Divided Difference
f  x  x   f  x  x 
f  x  
2x
f (x )
x  x
3
x
x  x
x
f  xi  x   f  xi  x 
f  xi  
2 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 t is given in seconds. Use central difference approximation of
the first derivative of
t  2s.
Solution:
a ti  
νt  to calculate the acceleration at t  16s. Use a step size of
 ti 1   ti  1
2t
ti  16
4
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Example (contd.)
Δt  2
t i 1  t i  Δt  16  2  18
ti 1  ti  t
a16 
 16  2  14
 18  14  18  14
2(2)

4


14  104
 18  2000 ln 
 9.818  453.02m/ s

4
14  10  210018


14 104
 14  2000 ln 
  9.814  334.24m/ s
4
14 10  210014
5
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Example (contd.)
Hence
a16 
 (18)  (14)
4
The exact value of
 453.02  334.24  29.695m / 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  29.695
100
29.674
 0.070769%

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 Central Divided Difference difference method.
f ' (0.2)
80.65467
80.25307
80.15286
80.12782
80.12156
80.12000
80.11960
80.11951
80.11948
80.11948
80.11947
Ea
-0.4016
-0.100212
-0.025041
-0.00626
-0.001565
-0.000391
-9.78E-05
-2.45E-05
-6.11E-06
-1.53E-06
a %
0.500417
0.125026
0.031252
0.007813
0.001953
0.000488
0.000122
3.05E-05
7.63E-06
1.91E-06
Significant
digits
1
2
3
3
4
5
5
6
6
7
Et
t %
-0.53520
-0.13360
-0.03339
-0.00835
-0.00209
-0.00052
-0.00013
-0.00003
-0.00001
0.00000
0.00000
0.668001
0.16675
0.041672
0.010417
0.002604
0.000651
0.000163
4.07E-05
1.02E-05
2.54E-06
6.36E-07
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Effect of Step Size in Central
Divided Difference Method
f'(0.2)
95
Initial step size=0.05
85
75
1
3
5
7
9
11
Num ber of tim es the step size is halved, n
9
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Effect of Step Size on
Approximate Error
Num ber of steps involved, n
0
3
5
7
9
11
E(a)
1
-1
10
Initial step size=0.05
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Effect of Step Size on Absolute
Relative Approximate Error
0.6
Initial step size=0.05
0.5
|E(a)|,%
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
Num ber of steps involved, n
11
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Least number of significant digits
correct
Effect of Step Size on Least Number
of Significant Digits Correct
Initial step size=0.05
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
Num ber of steps involved, n
12
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Effect of Step Size on True Error
Num ber of steps involved, n
0.0
-0.1
0
2
4
6
8
10
12
E(t)
-0.2
-0.3
-0.4
-0.5
-0.6
13
Initial step size=0.05
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Effect of Step Size on Absolute
Relative True Error
0.8
0.7
Initial step size=0.05
|Et| %
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10
11
Num ber of tim es step size halved,n
14
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