CSE 541 Numerical Methods

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CSE 541Numerical Methods

• Root Finding Bisection, Regula Falsi

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Root Finding

• Where does the function cross the x-axis?
• Set f(x) 0, solve for x
• More than once maybe???
• More generally, set f(x, y, z, ) 0
• Phrase it differently f(x,y,z,) g(x,y,z,)

roots
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Motivation

• Take well known functions such as polynomials
• For example
• f(x) x2 2x 3
• Two roots at r -1 and r 3
• f(-1) 1 2 3 0
• f(3) 9 6 3 0
• We can also look at f in its factored form
• f(x) x2 2x 3 (x 1)(x 3)

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Factored Form of Functions

• The factored form is not limited to polynomials
• Consider
• f(x) x sin x sin x
• A root exists at x 1
• f(x) (x 1) sin x
• And
• sin x sin kp 0 Þ (x p) x (x p) (x 2p)
  (x 3p)

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Examples

• Compute Ö2?
• x Ö2
• x2 2
• In general,
• xp c ? xp c 0

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Root Finding Algorithm

• Limited information about the function
• Assume we are allowed to evaluate the function
  anywhere and as many times as we want
• How do we find roots?

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Domain Interval

• Interval Two locations on the domain, a and b,
• where x Îa, b, where a lt b
• Lets define a couple of intervals
• How do we know if there is a root in the interval?

?
?
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Observation

• Evaluate the function at endpoints a and b
• f (a) gt 0 and f (b) gt 0

a
b
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Observation

• Evaluate the function at endpoints a and b
• f (a) gt 0 and f (b) lt 0

a
b
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Bracketing Techniques

• Surround the root (solution)
• Find an interval a, b that contains the root
• So f (a) f (b) lt 0
• What is a good approximation to our solution?
• Can we improve our approximation?

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Bisection Method

• c(ab)/2

f(a)gt0
f(c)gt0
a
b
c
f(b)lt0
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Bisection Method
a c f(a)gt0
b
c
a
f(c)lt0
f(b)lt0
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Bisection Method
b c f(b)lt0
b
a
c
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Convergence Error

• Will this always converge?
• Intermediate Value Theorem
• (Continuous functions)
• At any step in the algorithm, what is the error
  of our approximation?
• I.e., root c

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Error Estimation

• Use error bound and relative error

c
x
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Error Estimate

• What does this mean in binary mode?
• err0 ? b-a
• err1 ? b-a/2
• erri1 ? erri/2 b-a/2i1
• We gain an extra bit each iteration!!!

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Convergence

• The bisection method converges linearly
• r ci1 lt ½ r ci C r ci
• Linear Convergence Theorem
• Not bad, why do I need anything else?
• Can we do better?

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A Note on Functions

• Functions can be simple, but I may need to
  evaluate it many many times.
• Or, a function can be extremely complicated.
  Consider
• Interested in the configuration of air vents
  (position, orientation, direction of flow) that
  makes the temperature in the room at a particular
  position (teachers desk) equal to 72.
• Is this a function?

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A Note on Functions

• This function may require a complex
  three-dimensional heat-transfer coupled with a
  fluid-flow simulation to evaluate the function
• ? hours of computational time on a
  supercomputer!!!
• May not necessarily even be computational

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Bisection Method

• Given a and b, such that f(a)f(b)lt0
• c (a b)/2.0 // Find the midpoint
• while( ????? )
• if( f(a)f(c) lt 0 ) // root in the left half
• b c
• else // root in the right half
• a c
• c (a b)/2.0 // Find the new midpoint
• return c
• Stopping criteria?

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Regula Falsi

• New Idea
• Assume the function is linear within the bracket
• Construct a line and use it to find the
  intersection with the x-axis

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Regula Falsi
f(c)lt0
c
a
b
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Regula Falsi

• Benefits
• If the function is close to linear in the
  interval
• If the root is much closer to one side

b
a
c
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Bracketing Methods

• Guaranteed convergence
• Convergence typically slower than open methods
• Relies on identifying two points a and b
  initially such that
• f(a) f(b) lt 0
• Use to find approximate location of roots
• Polish with open methods