Finding Roots of Functions - PowerPoint PPT Presentation

COT 4810

26 February 2008

• “relation between two sets in which one element of the second set is assigned to each element of the first set”
• Example: y = x^2
• Bad example: y^2 = x^2

• 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

• Mathematicians
• Physicians
• Lots of people
• Computer Scientists

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

• If y=f(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.

• 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

• 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:

• 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:

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

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

• 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

• Combination of bisection, secant, and inverse quadratic interpolation
• Reliability of bisection, speed of secant/interpolation
• Will take at most N^2 iterations, where N is the number of iterations when using bisection

• 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.

• 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.

• 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:

• f(x) = x^3 – x – 3
• f'(x) = 3x^2 – 1
• Using x0 = 0, will start to cycle between xm=-3 and xn=0

• 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”

• For degrees less than five, the quadratic formula is generally the best. Can sometimes rewrite higher degree functions (x^6 – 4x^3 + 8 » u^2 – 4u + 8)‏
• Sturm's Theorem for finding number of real roots.
• Laguerre's method has cubic convergence.
• Problems with polynomials: Wilkinson

• 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.

• “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)‏

• 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:

• 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/

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