Upload
abner-floyd
View
235
Download
0
Embed Size (px)
Citation preview
Newton-Raphson Method
Chemical Engineering Majors
Authors: Autar Kaw, Jai Paul
http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM
Undergraduates
04/20/23 1http://
numericalmethods.eng.usf.edu
Newton-Raphson Method
http://numericalmethods.eng.usf.edu
Newton-Raphson Method
)(xf
)f(x - = xx
i
iii 1
f ( x )
f ( x i )
f ( x i - 1 )
x i + 2 x i + 1 x i X
ii xfx ,
Figure 1 Geometrical illustration of the Newton-Raphson method. http://numericalmethods.eng.usf.edu3
Derivation
f(x)
f(xi)
xi+1 xi
X
B
C A
)(
)(1
i
iii xf
xfxx
1
)()('
ii
ii
xx
xfxf
AC
ABtan(
Figure 2 Derivation of the Newton-Raphson method.4 http://numericalmethods.eng.usf.edu
Algorithm for Newton-Raphson Method
5 http://numericalmethods.eng.usf.edu
Step 1
)(xf Evaluate symbolically.
http://numericalmethods.eng.usf.edu6
Step 2
i
iii xf
xf - = xx
1
Use an initial guess of the root, , to estimate the new value of the root, , as
ix
1ix
http://numericalmethods.eng.usf.edu7
Step 3
0101
1 x
- xx =
i
iia
Find the absolute relative approximate error asa
http://numericalmethods.eng.usf.edu8
Step 4
Compare the absolute relative approximate error with the pre-specified relative error tolerance .
Also, check if the number of iterations has exceeded the maximum number of iterations allowed. If so, one needs to terminate the algorithm and notify the user.
s
Is ?
Yes
No
Go to Step 2 using new estimate of the
root.
Stop the algorithm
sa
http://numericalmethods.eng.usf.edu9
http://numericalmethods.eng.usf.edu10
Example 1
You have a spherical storage tank containing oil. The tank has a diameter of 6 ft. You are asked to calculate the height, h, to which a dipstick 8 ft long would be wet with oil when immersed in the tank when it contains 4 ft3 of oil.
h
r
Spherical Storage Tank
Dipstick
Figure 3 Spherical storage tank problem.
http://numericalmethods.eng.usf.edu11
Example 1 Cont.The equation that gives the height, h, of liquid in the spherical tank for the given volume and radius is given by
Use the Newton-Raphson method of finding roots of equations to find the height, h, to which the dipstick is wet with oil. Conduct three iterations to estimate the root of the above equation. Find the absolute approximate error at the end of each iteration and the number of significant digits at least correct at the end of each iteration.
0819739 23 .hhf(h)
http://numericalmethods.eng.usf.edu12
Example 1 Cont.
Figure 4 Graph of the function f(h).
08197.39 23 hhhf
0 1 2 3 4 5 6150
100
50
0
50
f(x)
3.8197
104.1803
0
f x( )
60 x
http://numericalmethods.eng.usf.edu13
Example 1 Cont.Solution
Iteration 1The estimate of the root is
72131.0
183
8197.391
'
02
0
20
30
0
001
hh
hhhf
hfhh
Figure 5 Graph of the estimated root after Iteration 1.
Entered function along given interval with current and next root and the tangent line of the curve at the current root
0 1 2 3 4 5 6
98.2
76.39
54.59
32.79
10.98
0
f(x)prev. guessnew guesstangent line
10.8197
104.1803
0
f x( )
f x( )
f x( )
tan x( )
60 x x 0 x 1 x
The absolute relative approximate error is
% 636.38
1001
01
h
hha
The number of significant digits at least correct is 0.
http://numericalmethods.eng.usf.edu14
Example 1 Cont.Iteration 2The estimate of the root is
67862.0
183
8197.39'
12
1
21
31
1
1
112
hh
hhh
hf
hfhh
The absolute relative approximate error is
The number of significant digits at least correct is 0.
%6.2907
1002
12
h
hha
Figure 6 Graph of the estimated root after Iteration 2.
Entered function along given interval with current and next root and the tangent line of the curve at the current root
0 1 2 3 4 5 6
98.71
77.42
56.12
34.83
13.54
0
f(x)prev. guessnew guesstangent
7.75175
104.1803
0
f x( )
f x( )
f x( )
tan x( )
60 x x 1 x 2 x
http://numericalmethods.eng.usf.edu15
Example 1 Cont.Iteration 3The estimate of the root is
67747.0
183
8197.39'
22
2
22
32
2
2
223
hh
hhh
hf
hfhh
The absolute relative approximate error is
The number of significant digits at least correct is 2.
%0.17081
1002
12
h
hha
Figure 7 Graph of the estimated root after Iteration 3.
0 1 2 3 4 5 6
98.78
77.55
56.33
35.11
13.88
0
f(x)prev. guessnew guesstangent
7.33941
104.1803
0
f x( )
f x( )
f x( )
tan x( )
60 x x 2 x 3 x
Advantages and Drawbacks of Newton
Raphson Method
http://numericalmethods.eng.usf.edu
16 http://numericalmethods.eng.usf.edu
Advantages
Converges fast (quadratic convergence), if it converges.
Requires only one guess
17 http://numericalmethods.eng.usf.edu
Drawbacks
18 http://numericalmethods.eng.usf.edu
1. Divergence at inflection pointsSelection of the initial guess or an iteration value of the root that is close to the inflection point of the function may start diverging away from the root in ther Newton-Raphson method.
For example, to find the root of the equation .
The Newton-Raphson method reduces to .
Table 1 shows the iterated values of the root of the equation.
The root starts to diverge at Iteration 6 because the previous estimate of 0.92589 is close to the inflection point of .
Eventually after 12 more iterations the root converges to the exact value of
xf
0512.01 3 xxf
2
33
113
512.01
i
iii
x
xxx
1x
.2.0x
Drawbacks – Inflection Points
Iteration Number
xi
0 5.0000
1 3.6560
2 2.7465
3 2.1084
4 1.6000
5 0.92589
6 −30.119
7 −19.746
18 0.2000 0512.01 3 xxf
19 http://numericalmethods.eng.usf.edu
Figure 8 Divergence at inflection point for
Table 1 Divergence near inflection point.
2. Division by zeroFor the equation
the Newton-Raphson method reduces to
For , the denominator will equal zero.
Drawbacks – Division by Zero
0104.203.0 623 xxxf
20 http://numericalmethods.eng.usf.edu
ii
iiii xx
xxxx
06.03
104.203.02
623
1
02.0or 0 00 xx Figure 9 Pitfall of division by zero or near a zero number
Results obtained from the Newton-Raphson method may oscillate about the local maximum or minimum without converging on a root but converging on the local maximum or minimum.
Eventually, it may lead to division by a number close to zero and may diverge.
For example for the equation has no real roots.
Drawbacks – Oscillations near local maximum and minimum
02 2 xxf
21 http://numericalmethods.eng.usf.edu
3. Oscillations near local maximum and minimum
Drawbacks – Oscillations near local maximum and minimum
22 http://numericalmethods.eng.usf.edu
-1
0
1
2
3
4
5
6
-2 -1 0 1 2 3
f(x)
x
3
4
2
1
-1.75 -0.3040 0.5 3.142
Figure 10 Oscillations around local minima for .
2 2 xxf
Iteration Number
0123456789
–1.0000 0.5–1.75–0.30357 3.1423 1.2529–0.17166 5.7395 2.6955 0.97678
3.002.255.063 2.09211.8743.5702.02934.9429.2662.954
300.00128.571 476.47109.66150.80829.88102.99112.93175.96
Table 3 Oscillations near local maxima and mimima in Newton-Raphson method.
ix ixf %a
4. Root JumpingIn some cases where the function is oscillating and has a number of roots, one may choose an initial guess close to a root. However, the guesses may jump and converge to some other root.
For example
Choose
It will converge to instead of
-1.5
-1
-0.5
0
0.5
1
1.5
-2 0 2 4 6 8 10
x
f(x)
-0.06307 0.5499 4.461 7.539822
Drawbacks – Root Jumping
0 sin xxf
23 http://numericalmethods.eng.usf.edu
xf
539822.74.20 x
0x
2831853.62 x Figure 11 Root jumping from intended location of root for
. 0 sin xxf
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/newton_raphson.html