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Notes on NewtonRaphson – A rootsolving technique
(Updated 11/3/2010)
(Dr. Shoane)
1.
Introduction to NewtonRaphson
Based on geometric interpretation or Taylor series expansion of the function, f(x) at x
i
.
From the geometry of the situation seen in the graph, we have
1
0
)
(
)
(
+


=
′
i
i
i
i
x
x
x
f
x
f
Hence
)
(
)
(
1
i
i
i
i
x
f
x
f
x
x
′

=
+
The NewtonRaphson algorithm attempts to minimize the difference between x
i
and x
i+1
by iteratively updating the next x
i
value.
This continues until x
i
and x
i+1
are very close to
each other.
The critical point occurs when the function, f, crosses the zero line, which is
where the equation f(x)=0 (e.g., x
3
+2x
2
+1=0) is satisfied.
It can be shown that at this
critical point, x
i
≈
x
i+1.
Importantly, the value of x at this point is a root (or solution) of
the equation f(x)=0.
1
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View Full Document 2. Examples
Example 1
(
29 (
29 (
29
7
10
3
3
7
5
7
10
3
)
(
0
3
7
5
1
1
3
)
(
2
2
3
1
2
2
3
+


+


=
+

=
′
=

+

=



=
+
i
i
i
i
i
i
i
x
x
x
x
x
x
x
x
x
x
f
x
x
x
x
x
x
x
f
Example 2
dy
y
x
df
y
x
f
x
x
xy
x
y
x
f
i
i
/
)
,
(
)
,
(
0
10
)
,
(
1
2

=
=

+
=
+
3.
Video
http://www.youtube.com/watch?v=lFYzdOemDj8
4. Code for function “newton”
http://math.fullerton.edu/mathews/n2003/newtonsmethod/New
ton%27sMethodProg/Links/Newton%27sMethodProg_lnk_3.html
2
Modified Program Demonstrating NewtonRaphson Technique (Dr. Shoane debugged
the above program, and
it should run “as is”, after cleaning up any comments that were
moved out of position.)
******** Copy the entire code starting from
here to the end of the program **********
function
newton_test()
% From http://math.fullerton.edu/mathews/n2003/newtonsmethod/Newton
%27sMethodProg/Links/Newton%27sMethodProg_lnk_3.html
% 1) Modified by Prof. George Shoane 11/2/2010: positions of functions, plus
comments at each pause.
% 2) The test functions have been put at the end of the program.
% 3) It is suggested that the students learn how the function "newton"
%
works, and then write their own functions to solve any given problem.
%
This is probably more efficient than trying to input a complicated
%
nonlinear system equation with two variables to fit the format of the
function "newton".
% 4) If the nonlinear function consists of 2 variables where one of them
%
is an indpendent variable, then one technique is to update the
%
independent variable and set it as a constant at each iteration.
Then
%
the "newton" function can be used directly. The output after the function
call
%
provides the estimated value of the dependent variable for that particular
%
independent variable value.
(Updated 11/3/2010)
% 5) Hint: Use symbolic representation to define the function and obtain the
%
derivative of the function.
But do not use the symbolic variables in the
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This note was uploaded on 02/12/2011 for the course 125 305 taught by Professor Madabhushi during the Fall '08 term at Rutgers.
 Fall '08
 Madabhushi

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