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Chapter 2
Solution of Nonlinear Equations:
Lecture (II)
Dr. Jie Zou PHY 3320
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Outline

Numerical methods
(2) Newton-Raphson (or simply
Newton’s) method
Dr. Jie Zou PHY 3320
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Newton-Paphson method

Newton’s method
algorithm:
f  xi 
xi 1  xi  '
: i  1, 2, 
f  xi 


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Dr. Jie Zou PHY 3320
x1: The initial guess for the
root of f(x) = 0.
x2: The next approximation
to the root. The point of
intersection of the tangent
to the curve at x1 with the x
axis gives x2.
The iterative procedure
stops when meeting a
convergence criterion:
|f(xi)|  , |xi –xi-1|  , or
|(xi-xi-1)/xi|  .
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Derivation of the Newton’s
method

Taylor’s series expansion of the function f(x) about an
arbitrary point x1:
1
2 ''
f x   f x1   x  x1  f x1   x  x1  f x1   
2!
'



Considering only the first two terms in the expansion:
f(x)  f(x1) + (x – x1) f ’(x1)
Set f(x)  f(x1) + (x – x1) f ’(x1) = 0, and solve for the root:
f  x1 
x  x2  x1  '
f  x1 
To further improve the root, replace x2 with x1 to obtain x3,
and so on.
Dr. Jie Zou PHY 3320
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Notes on Newton’s method



Dr. Jie Zou PHY 3320
Newton’s method requires
the derivative of the
function, f’ = df/dx; some
may be quite complicated.
f(x) may not be available in
explicit form, in which case
numerical differentiation
techniques are required.
Newton’s method
converges very fast in most
cases. However, it may not
converge (see examples on
the left).
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Example: Newton’s method

Example 2.8: Find the root of the equation
f x  
1.5 x
1  x 
2 2
 1  0.65 x
 0.65 tan   
0
2
 x  1 x
1
using the Newton-Raphson method with
starting point x1 = 0.0, and the convergence
criterion, |f(xi)|   with  = 10-5. Note: The
derivative of tan-1(u) is given by
d
1 du
1
tan u 
dx
1  u 2 dx
Dr. Jie Zou PHY 3320
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Plot function f(x)

Let’s first plot the function f(x) from x = 0 to
1 to gain some insight on the behavior of the
function.
Root
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Flowchart
x1=0.0, =10-5, i=0
i=i+1
|f(xi)|  
T
x_Root=xi
Dr. Jie Zou PHY 3320
end
F
xi 1
f  xi 
 xi 
f  xi 
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Implement Newton’s method:
by hand
• Show work step by step.
• Also, summarize the results in the Table below.
i
xi
f(xi)
Is |f(xi)|  ? f’(xi) (answer if
the previous
column is No
1
2
3
4
…
Dr. Jie Zou PHY 3320
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Implement Newton’s method:
write an M-file

For the Example given on slide #6,
write an M-file to compute the root of
the equation using Newton-Raphson
method.
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
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Follow the flowchart provided previously.
Save the M-file as myNewton-Raphson.m.
A copy of the M-file will be handed out
later.
Dr. Jie Zou PHY 3320
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