2. Solving Equations of One Variable

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2. Solving Equations of One Variable

• Korea University Computer Graphics Lab.
• Lee Seung Ho / Shin Seung Ho Roh Byeong Seok /
  Jeong So Hyeon

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Contents

• Bisection Method
• Regula Falsi and Secant Method
• Newtons Method
• Mullers Method
• Fixed-Point Iteration
• Matlabs Method

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Bisection Method
4
Bisection Method
5
Finding the Square Root of 3 Using Bisection
How can we get ?
6
Approximating the Floating Depth for a Cork Ball
by Bisection(1/2)
Cork ball Radius 1 Density 0.25
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Approximating the Floating Depth for a Cork Ball
by Bisection(2/2)
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Discussion of Bisection Method
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Fixed-Point Iteration

• Solution of equation
• Convergence Theorem of fixed-point iteration

10
Fixed-Point Iteration to Find a Zero of a Cubic
Function
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Matlabs Methods(1/2)

• roots(p)
• p vector
• Example
• 
  
• EDUgt r roots(p) (p1 -7 14 -7)
• r 3.8019
• 2.445
• 0.75302

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Matlabs Methods(2/2)

• fzero( function name,x0 )
• function name string
• x0 initial estimate of the root
• Example
• function y flat10(x)
• y x.10 0.5
• z fzero(flat10,0.5)
• z 0.93303

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Regular Falsi and Secant Methods

• 2005. 3. 23
• Byungseok Roh

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Regula Falsi Method

• The regula falsi method start with two point, (a,
  f(a)) and (b,f(b)), satisfying the condition that
  f(a)f(b)lt0.
• The straight line through the two points (a,
  f(a)), (b, f(b)) is
• The next approximation to the zero is the value
  of x where the straight line through the initial
  points crosses the x-axis.

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Regula Falsi Method (cont.)

• If there is a zero in the interval a, c, we
  leave the value of a unchanged and set b c.
• On the other hand, if there is no zero in a, c,
  the zero must be in the interval c, b so we
  set a c and leave b unchanged.

• The stopping condition may test the size of y,
  the amount by which the approximate solution x
  has changed on the last iteration, or whether the
  process has continued too long.
• Typically, a combination of these conditions is
  used.

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Example

• Finding the Cube Root of 2 Using Regula Falsi

• Since f(1) -1, f(2)6, we take as our starting
  bounds on the zero a1 and b2.
• Our first approximation to the zero is
• We then find the value of the function
• Since f(a) and y are both negative, but y and
  f(b) have opposite signs

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Example (cont.)

• Calculation of using regula falsi.

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Secant Method

• The secant method, closely related to the regula
  falsi method, results from a slight modification
  of the latter.

• Instead of choosing the subinterval that must
  contain the zero, we form the next approximation
  from the two most recently generated points
• At the k-th stage, the new approximation to the
  zero is

• The secant method has converged with a tolerance
  of .

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Example

• Finding the Square Root of 3 by Secant Method
• To find a numerical approximation to , we
  seek the zero of
• .
• Since f(1)-2 and f(2)1, we take as our starting
  bounds on the zero and .
• Our first approximation to the zero is
• Calculation of using secant method.

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NEWTONS METHOD
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Newtons Method

• Newtons method uses straight-line approximation
  which is the tangent to curve.
• .
• Intersection point

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Example

• Finding Square Root of ¾
• approximate the zero of
  using the fact that .
• Continuing for one more step

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Finding Floating Depth for a Wooden Ball

• Volume of submerged segment of the Sphere
• To find depth at which the ball float, volume of
  submerged segment is time.
• Simplifies to

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Finding Floating Depth for a Wooden Ball (cont.)

• To find depth a ball, density is one-third of
  water float.

Calculation f(x) using Newtons Method
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Oscillations in Newton Method

• Newtons method give Oscillatory result for some
  funtions initial estimates.
• Ex)

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Mullers Method
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Mullers Method

• based on a quadratic approximation
• procedure
• Decide the parabola passing through (x1,y1), (x2,
  y2) and (x3,y3)
• Solve the zero(x4) that is closest to x3
• Repeat 1,2 until x converge to predefined
  tolerance
• advantage
• Requires only function values
• Derivative need not be calculated
• X can be an imaginary number.

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Mullers Method (Cont)

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Example

• Finding the sixth root of 2 using Mullers method
• , ,
  ,

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Example (Cont)
i x y
1 0.5 -1.9844
2 1.5 9.3906
3 1 -1
4 1.0779 -0.43172
5 1.117 -0.05635
6 1.1255 0.00076162
7 1.1255 -4.7432e-07
converge
Calculation of using Mullers method
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Another Challenging Problem
Tolerance 0.0001

step x Y
1 0 -0.5
2 1 0.5
3 0.5 -0.49902
4 0.80875 -0.38029
5 0.9081 -0.11862
6 0.94325 0.057542
7 0.93269 -0.0018478
8 0.93303 -6.3021e-06
9 0.93303 -3.1235e-10
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MATLAB function for Mullers Method

• P.6566 code