7/2/2015 1 backward divided difference major: all engineering majors authors: autar kaw, sri harsha...
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04/19/23http://
numericalmethods.eng.usf.edu 1
Backward Divided Difference
Major: All Engineering Majors
Authors: Autar Kaw, Sri Harsha Garapati
http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM
Undergraduates
Backward Divided Difference
http://numericalmethods.eng.usf.edu
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Definition
ix
ix
.
x
xxfxf
xxf
0
lim
Slope at
f(x)
y
x
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Backward Divided Difference
xx x
x
x
xxfxfxf
x
xxfxfxf iii
)(xf
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Example
Example:
The velocity of a rocket is given by
300,8.921001014
1014ln2000
4
4
tt
tt
wher
e given in m/s
and t is given in seconds. Use backward difference
approximation Of the first
derivative of tν to calculate the acceleration at .16st Use a step size
of .2st
t
ttta iii
1
Solution:
16it
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2Δ t
ttt ii 1 14216
2
141616
a
168.91621001014
1014ln200016
4
4
sm /07.392
148.91421001014
1014ln200014
4
4
sm /24.334
Example (contd.)
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Hence
The exact value of 16a can be calculated by differentiating
tt
t 8.921001014
1014ln2000
4
4
as
Example (contd.)
2/915.2824.33407.3922
)14()16(16 sma
2/674.29)16(
3200
4.294040)()(
sma
t
tt
dt
dta
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The absolute relative true error is
Example (contd.)
100
TrueValue
eValueApproximatTrueValuet
%557.2
100674.29
915.28674.29
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Effect Of Step Sizexexf 49)(
Value of )2.0('f Using backward Divided difference method.
h )2.0('f aE a % Significant
digits tE
t %
0.05 72.61598 7.50349 9.365377 0.025 76.24376 3.627777 4.758129 1 3.87571 4.837418 0.0125 78.14946 1.905697 2.438529 1 1.97002 2.458849 0.00625 79.12627 0.976817 1.234504 1 0.99320 1.239648 0.003125 79.62081 0.494533 0.62111 1 0.49867 0.622404 0.001563 79.86962 0.248814 0.311525 2 0.24985 0.31185 0.000781 79.99442 0.124796 0.156006 2 0.12506 0.156087 0.000391 80.05691 0.062496 0.078064 2 0.06256 0.078084 0.000195 80.08818 0.031272 0.039047 3 0.03129 0.039052 9.77E-05 80.10383 0.015642 0.019527 3 0.01565 0.019529 4.88E-05 80.11165 0.007823 0.009765 3 0.00782 0.009765
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Effect of Step Size in Backward Divided Difference
Method
75
85
95
1 3 5 7 9 11
Number of times the step size is halved, n
f'(0
.2)
Initial step size=0.05
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Effect of Step Size on Approximate Error
0
1
2
3
4
1 3 5 7 9 11
Number of steps involved, n
E a
Initial step size=0.05
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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
Number of steps involved, n
|Ea|,%
Initial step size=0.05
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Effect of Step Size on Least Number of Significant Digits
Correct
0
0.5
1
1.5
2
2.5
3
3.5
1 2 3 4 5 6 7 8 9 10 11
Number of steps involved, n
Least
nu
mb
er
of
sig
nif
ican
t
dig
its c
orr
ect
Initial step size=0.05
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Effect of Step Size on True Error
0
1
2
3
4
5
6
7
8
0 2 4 6 8 10 12
Number of steps involved, n
E t
Initial step size=0.05
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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
Number of steps involved, n
|Et|
%
Initial step size=0.05
Additional ResourcesFor all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit
http://numericalmethods.eng.usf.edu/topics/continuous_02dif.html
