Newton-Raphson Method
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Newton-Raphson Method - Numerical Methods - Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization
Main points are: Newton’s Method, One-Dimensional Optimization, Open Search Method, Golden Section Search Method, Newton-Raphson Method, Minima of Function, Cross-Sectional Area, Summary of Iterations, Actual Solution
Typology: Slides
2012/2013
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Newton-Raphson Method
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Newton-Raphson Method
f (x )
f(x )
x = x -
i
i i i
- 1
f(x)
f(xi)
f(xi- 1 )
xi+2 xi+1 xi X
θ
[ xi , f ( xi )]
Figure 1 Geometrical illustration of the Newton-Raphson method.
2 http://numericalmethods.eng.usf.eduDocsity.com
Algorithm for Newton-Raphson
Method
http://numericalmethods.eng.usf.eduDocsity.com
Step 1
Evaluate f ′( x ) symbolically.
5 http://numericalmethods.eng.usf.eduDocsity.com
Step 3
1
1
x
x - x
i
i i a
∈ ×
Find the absolute relative approximate error ∈ a as
7 http://numericalmethods.eng.usf.eduDocsity.com
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
∈ a >∈ s
8 http://numericalmethods.eng.usf.eduDocsity.com
Example 1 Cont.
The equation that gives the depth x in meters to
which the ball is submerged under water is given by
3 2 4
-
f x = x -. x +. ×
Use the Newton’s method of finding roots of equations to find
a) the depth ‘x’ to which the ball is submerged under water. Conduct three
iterations to estimate the root of the above equation.
b) The absolute relative approximate error at the end of each iteration, and
c) The number of significant digits at least correct at the end of each
iteration.
10 http://numericalmethods.eng.usf.edu
Figure 3 Floating ball problem.
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Example 1 Cont.
This image cannot currently be displayed.
3 2 4
-
f x = x -. x +. ×
To aid in the understanding
of how this method works to
find the root of an equation,
the graph of f(x) is shown to
the right,
where
Solution
Figure 4 Graph of the function f(x)
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Example 1 Cont.
( )
( )
( ) ( )
( ) ( )
( )
3
4
2
(^324)
0
0 1 0
− ×
×
− + ×
−
−
−
f x
f x x x
Iteration 1
The estimate of the root is
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Example 1 Cont.
Figure 5 Estimate of the root for the first iteration.
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Example 1 Cont.
5
3
7
2
(^324)
1
1 2 1
= − ×
− ×
− ×
− + ×
−
−
−
−
f x
f x
x x
Iteration 2
The estimate of the root is
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Example 1 Cont.
Figure 6 Estimate of the root for the Iteration 2.
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Example 1 Cont.
9
3
11
2
(^324)
2
2 3 2
= − − ×
− ×
×
− + ×
−
−
−
−
f x
f x
x x
Iteration 3
The estimate of the root is
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Example 1 Cont.
Figure 7 Estimate of the root for the Iteration 3.
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