Finding Roots of Functions

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Finding Roots of Functions

• Andrew Tomko
• COT 4810
• 26 February 2008

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Functions

• relation between two sets in which one element
  of the second set is assigned to each element of
  the first set
• Example y x2
• Bad example y2 x2

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/root

• The value for 'x' for which 'y' will equal zero
• If there is no 'y', can rearrange function so
  that it equals zero
• Functions can have none, one, or many roots

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Who cares?

• Mathematicians
• Physicians
• Lots of people
• Computer Scientists

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Methods for finding a root

• Bisection
• Newton-Raphson
• Secant
• False Position (regula falsi)?
• Müller
• Inverse Quadratic Interpolation
• Brent

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Intermediate Value Theorem

• If yf(x) is continuous on a,b, and N is a
  number between f(a) and f(b), then there is at
  least one c ? a,b such that f(c) N.

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Bisection Method

• Easiest
• Positive/Negative values chosen, then bisection
• Repeats until root is found
• Absolute error is halved each step
• Runs in linear time
• Has problems with multiple roots

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Bisection Method cont'd
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Secant Method

• Similar to bisection
• Takes secant of two points on the function and
  then moves points closer until root is found
• Does not always converge, runs in superlinear
  time, faster than Newton's method in practice
• Based on recurrence relation

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Secant Method cont'd
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False Position (regula falsi) Method

• Combination of bisection and secant
• Takes to points one above/below root
• Runs in superlinear time
• First recorded use in 3rd century BC
• Based on

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Müller's Method

• Based on secant method, but takes 3 points
• Faster than the secant method, slower than
  Newton's method
• Quadratic formula can yield complex values

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Polynomial Interpolation

• Given a series of points, find a polynomial
  function that satisfies these points.
• Useful for transforming more difficult functions
  (logarithmic, trigonometric, etc)?

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Inverse Quadratic Interpolation

• Rarely used on its own, because it can fail
  easily if points not chosen close to root.
• Tries to create a quadratic interpolation for the
  function's inverse.
• Using linear interpolation secant method
• Interpolating f instead of the inverse Müller's
  method

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(No Transcript)
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Brent's Method

• Combination of bisection, secant, and inverse
  quadratic interpolation
• Reliability of bisection, speed of
  secant/interpolation
• Will take at most N2 iterations, where N is the
  number of iterations when using bisection

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Householder's Methods

• Methods for finding a root in a function with one
  real variable and has a continuous derivative.
• Derives from geometric series.
• The iterations will converge to a zero if the
  first guess is close enough.

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Newton-Raphson History

• First described by Issac Newton in 1669
• First published in 1685 in A Treatise of Algebra
  both Historical and Practical
• Joseph Raphson published a simplified version in
  1690.
• Newton most likely got the formula from French
  mathematician François Viète seigneur de la
  Bigotière.

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Newton-Raphson Method

• Very efficient method for real functions
• Quadratic convergence the number of correct
  digits doubles every iteration
• Possibility to not converge
• Requires the calculation of the function's
  derivative

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Newton-Raphson Method cont'd
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Newton-Raphson example
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Newton-Raphson fail to converge

• f(x) x3 x 3
• f'(x) 3x2 1
• Using x0 0, will start to cycle between xm-3
  and xn0

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Newton-Raphson etc.

• Can be used to find multi-dimensional roots.
• Can be used to find roots in systems of
  non-linear, multivariable functions.
• Can be used to find the local minimums / maximums
  in a given function.
• Can be used to find complex roots, creates
  Newton Fractals

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x8 15x4 - 16
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p(z) z3 - 2z 2
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x5 - 1 0
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Roots of Polynomials

• For degrees less than five, the quadratic formula
  is generally the best. Can sometimes rewrite
  higher degree functions (x6 4x3 8 u2
  4u 8)?
• Sturm's Theorem for finding number of real roots.
• Laguerre's method has cubic convergence.
• Problems with polynomials Wilkinson

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Sturm's Theorem

• Used to determine the number of unique roots of a
  polynomial.
• Applies Euclid's algorithm to X and X'
• The final polynomial, Xr, is the GCD of X and X'
• Number of unique roots found by counting the
  number of sign changes.

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Fundamental Theorem of Algebra

• Every non-constant single-variable polynomial
  with complex coefficients has at least one
  complex root
• This can be reworked to show that every nth
  degree polynomial can be written in the form
  f(x) C(x-x1)(x-x2)...(x-xn)?

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Laguerre's Method

• Pretty much guaranteed a convergence on root,
  regardless of initial value.
• Cubic convergence (!)?
• Takes the natural log of both sides of formula on
  last slide, then takes two derivatives. (First
  derivative G, Second derivative H)?
• Where a the distance from your guess to the
  next root, and n which root you're looking for

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Wilkinson polynomial
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Sources

• Burden, Richard L. Faires, J. Douglas (2000),
  Numerical Analysis (7th ed.), Brooks/Cole
• Dewdney, A. K. The New Turing Omnibus. New York
  Henry Holt and Company, LLC, 1993.
• http//www.wikipedia.org/

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Questions

• 1) Do the next iteration of the first
  Newton-Raphson example.
• 2) What theorem do many of these methods depend
  on?