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Newton’s Method
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NEWTON, Sir Isaac. Arithmetica Universalis; sive de Compositione et Resolutione Arithmetica Liber. Ciu accessit Helleiana Aequationum Radices Arithmetice Inveniendi Methodus. Edited by William Whiston. Cambridge: Typis Academicus; London: Benjamin Tooke, 1707.
Actually written by Joseph Raphson
Remarks
One of the common problems in mathematics is to solve equations and find zeros
We have methods to solve simple equations like quadratics, but must resort to numerical solutions to solve more complicated equations like sin(x) +ex = x
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RemarksA common numerical approach is bisection which can be applied on a closed interval
If we know f(a) < 0 and f(b) > 0, we know there is a number c where a < c < b and f(c) = 0 by the Intermediate Value Theorem
We can then test f(x) at the midpoint (a + b)/2 and if greater than zero we set a = midpoint and if less than zero we set b = midpoint
We continue this process until we achieve the desired degree of accuracy
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Remarks
The disadvantage of the bisection method is that it converges slowly
The advantage of the bisection method is that it the function is continuous we expect it to converge, because we are promised there is a zero by the Intermediate Value Theorem
The bisection method is less efficient than Newton's Method but it is less prone to odd behavior
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Remarks
The error when using the bisection method is after n iterations
This section is about Newton’s Method which can converge quickly even though it does not always work
12n
b a
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Remarks
►Around 1669, Isaac Newton (1643-1727) gave a new algorithm to solve a polynomial equations
►He applied his method to polynomial equations, viewed his method as purely algebraic and apparently did not notice the connection with calculus
►In 1690, Joseph Raphson (1678-1715) proposed a method which avoided the substitutions used by Newton
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Newton probably
derived his method from a similar but less precise method by Francois Viète
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Heron of Alexandria essentially used the same method to determine the square root of 72
For polynomials Newton's method is essentially the same as Heron's method
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Joseph RaphsonBorn in Middlesex in 1648, very little is known about his life, not even an obituary has been found
By the records of the University of Cambridge, all that is known that he attended Jesus College and graduated with a Master of Arts degree in 1692
Raphson used ideas of the Calculus to generalize this ancient method to find the zeros of an arbitrary equation ( ) 0f x
Their underlying idea is the approximation of the graph of the function f (x) by the tangent lines
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Joseph Raphson
• Had an important relationship to Newton, although it is not well understood by historians
• Was one of the very few people of whom Newton would allow to see his papers
• Wrote up some of Newton's work into books
• As well as his work on calculus and analysis, he also had ideas on space and philosophy, based on Cabala, which is a Jewish Mysticism
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Joseph Raphson
Raphson's ideas of space and philosophy were based on Cabalist ideas
Cabala was a Jewish mysticism which was influential from the 12th century on
The doctrines included the withdrawal of the divine light, thereby creating primordial space, the sinking of luminous particles into matter and a "cosmic restoration" that is achieved by Jews through living a
mystical life
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0f x
0x1x
1 0
0
1 0
to find zero, let 0
y f xm f x
x x
y
The Idea
We use the tangent line to estimate the zero13
We defined the linear approximation of at asf x x a
L x f a f a x a
Since we trying the find the zero we let 0L x
0
0 0 0 1
We let our first quess be and have
0 and let the next guess be
a x
f x f x x x x
0 0 1 0then 0 f x f x x x
01 1 0
0
( )solving for we have
( )
f xx x x
f x
Discussion
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The Method
We are trying to find the zeroes of ( ) 0f x
We make a reasonable first guess called our seed x0
01 0
0
( )
( )
f xx x
f x
We obtain our second guess by
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The Method
We repeat this procedure recursively using the formula
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( )
( )
nn n
n
f xx x
f x
As our guess come closer and closer to the root r, we
say
We then say it converges
lim nx
x r
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Example
01 0
0
( ) 1.29Then 2.3 2.0196
( ) 4.6
f xx x
f x
We take the derivative 2f x x
0We begin with an intial guess Let 2.3x
2Find zero of 4f x x
2
We evaluate 2.3 2.3 4 1.29f
We evaluate 2.3 2 2.3 4.6f
We can find the next by repeating this process
until we achieve our desired accuracy
nx
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Example 2Find zero of 4f x x
2
1
We can create a recrusive function to obtain the next guess
( ) 4
( ) 2
n nn n n
n n
f x xx x x
f x x
We can then find the next xn
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Example 2Find zero of 4f x x
n xn Error
0 2.3000000000 1.29000000
1 2.0195652174 0.07878416
2 2.0000951079 0.00038044
3 2.0000000023 0.00000001
Our result after three iterations
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Example 2Find zero of 4f x x
The third iteration using the TI-89
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n=0
xn
xn+1 = xn – f(xn)/f’(xn)
|xn-xn+1|<En+1n
YesNo
Flowchart
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Example2
2
We want to solve 5
We begin by creating a function which is
zero when 5
x
x
Find the square root of 5
2
We let LHS RHS
In our example,
we let 5
f x
f x x
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2Let ( ) 5f x x ( ) 2f x x
0Take 2 (2) 1 and (2) 4x f f
1
1 92 2.25
4 4x
9 1 9 9 and
4 16 4 2f f
2
9 1 16 1612.236
4 9 2 72x
Example
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Find the square root of 5
TI-84 Solution
Graph
x1
x2
or fromHome
Example
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Find the square root of 5
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Example 3Find roots of 1f x x x
The sketch of the graph will tell us that this polynomial has exactly one real root
We first need a good guess as to where it is
We see f(-2) < 0 and f (-1) > 0. By the IVT we know there is a zero in the interval (-2,-1)
We could choose x0 = -125
3 1f x x x We can take the derivative
2 3 1f x x
1
n
n
n n
f xx x
f x
To perform the iteration
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2
1
3 1
n nn
n
x xx
x
Example 3Find roots of 1f x x x
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x0 -1
x1 -1.500000
x2 -1.347826
x3 -1.325200
x4 -1.324718
x5 -1.324717
x6 -1.324717
x7 -1.324717
Notice the values become closer and closer to the same value.
We have found the approximate solution to six decimal places
obtained after only five relatively painless steps.
Example 3Find roots of 1f x x x
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ExampleFind zero of 2 2 4 f x x x
using graphing calculator
With trigonometric functions, use nDeriv on the TI to force a numerical solution
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Remarks
With trigonometric functions
• Set your calculator angle mode is in radians, notdegrees
• Use radians for you calculations, not degrees
• If using a calculator to assist with taking thederivative, numerical derivatives are recommended
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Example Find a solution to cos 0x
Let cosy x
Then siny x
1n n
f xx x
f x
cos
sin
nn
n
xx
x
Define our function
Take the derivative
Find the integrative formula
Make an initial guess0 1x
cotn nx x
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ExampleOur first guess of 1
cos(x) = 0
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Our Tangent Line
Our second guess is x = 1.64209
Example cos(x) = 0
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Example cos(x) = 0
And the Next Tangent Line
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Our estimate is 1.5708
x x + cot x
1 1.64209
1.64209 1.57068
1.57068 1.5708
1.5708 1.5708
1.5708 1.5708
Example cos(x) = 0
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Example sinEstimate zero of 0xe x
sinLet xf x e x
sin cos 1xf x e x
sin
1 sin cos 1
x
n n x
e xx x
e x
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Example sinEstimate zero of 0xe x
0.5386
0Let us begin with a seed of 1x
sin 1
1 sin 1
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cos 1 1
ex
e
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Example sinEstimate zero of 0xe x
sin 0.5386
2 sin 0.5386
10.5386
cos 0.5386 1
ex
e
0.5783
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sin 0.5783
3 sin 0.5783
10.5783
cos 0.5783 1
ex
e
0.5787
We then have 0.5787 as our
estimate of the zero
Example sinEstimate zero of 0xe x
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Advantages
Applies to an equation of any degree
Applies to non-polynomial equations
Gives answers in numerical form
Continue computation until desired degree of accuracy
achieved
Converges quickly; we often need only two or iterations
to attain desired accuracy39
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Remarks
Newton's method will usually converge provided the initial guess is close enough to the unknown zero
Furthermore, for a zero of multiplicity 1, the convergence is quadratic, which intuitively means that the number of correct digits roughly doubles in every step
If the root being sought has multiplicity greater than
one, the convergence rate is reduced to linear
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Newton’s Method Fails If
1. If our first guess (or any guesses thereafter) is a point at which there is a horizontal tangent line, then this line will never hit the x-axis, and Newton's Method will fail to locate a root. If there is a horizontal tangent line then the derivative is zero, and we cannot divide by f '(x) as the formula requires.
2. If our guesses oscillate back and forth then Newton's method will not work.
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Newton’s Method Fails If
3. If there are two roots, we must have a first guess near the root that we are interested in, otherwise Newton's method will find the wrong root.
4. If there are no roots, then Newton's method will fail to find it.
5. It may fail for piecewise functions
6. If f(x) = 0 and f'(x) = 0 have a common root
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Example
0Let 1x
n xn
0 1
1 -2
2 4
3 -8
4 16
1 2n nx x
3Find zero of ( )f x x
1
1 3
2 32
1 3
n
n n
n
nn
n
f xx x
f x
xx x
x
We see our process does not converge43
Other Root Finding AlgorithmsBinary Search Method
Bairstow's Method
Brent's Method
Durand-Kerner Method
False Position
Inverse Quadratic Interpolation
Laguerre's Method
Lehmer-Schur Algorithm
Müller's Method
Principal nth Root
Ruffini's Rule
Secant Method
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Secant Method
1
1
1
Then the Secant Method is
( )n n n
n n
n n
f x x xx x
f x f x
We do not have to take the derivative
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( )From Newton's Method,
( )
nn n
n
f xx x
f x
1
1
but we have ( )n n
n
n n
f x f xf x
x x
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Remarks
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TI-83/84 and TI-89 program listings for Newton’s Methodhttp://www.tc3.edu/instruct/sbrown/ti83/newton.htm
Programs for the several calculators – download – or enter directly into calculatorhttp://math.arizona.edu/~krawczyk/calcul.html
Animations of Newton’s Methodhttp://www.ecs.fullerton.edu/~mathews/a2001/Animations/RootFinding/NewtonMethod/NewtonMethod.html
Fractal ImagesFractal images can be created by applying the Newton-Raphson Method to various real-valued polynomials with complex solutions
47http://www.youtube.com/watch?v=7RZn3B709wo
-x 3 + 9x 2 - 18x + 6 Example
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-x5 + 25x4 - 200x3 + 600x2 - 600x + 120Example
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x 5 - 1 Example
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Remarks
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There are many fractal programs available today
For example see fractalarts.com/ASF/download.html
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http://www.youtube.com/watch?feature=player_embedded&v=qB8m85p7GsU
Arthur Clarke - Fractals The Colors Of Infinity 1 of 6
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