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OR682/Math685/CSI700. Lecture 5 Fall 2000. Nonlinear Equations. Solving f ( x ) = 0 (1 equation, n equations) Assume that [# of equations] = [# of variables] Closely related to: minimize F ( x ) Solve:  F ( x ) = 0

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Or682 math685 csi700

OR682/Math685/CSI700

Lecture 5

Fall 2000


Nonlinear equations
Nonlinear Equations

  • Solving f (x) = 0 (1 equation, n equations)

  • Assume that [# of equations] = [# of variables]

  • Closely related to: minimize F (x)

    • Solve: F(x) = 0

    • “Always” better to use optimization software to solve optimization problems

  • Applications:

    • Nonlinear differential equations

    • Design of integrated circuits

    • Data fitting with nonlinear models (e.g., exponential terms)


Examples
Examples

  • 1-variable: x2 = 4 sin(x)

  • 2-variable:


Solutions of nonlinear equations
Solutions of Nonlinear Equations

  • Nonlinear equations can have any number of solutions:

    • No solution: exp(x) + 1 = 0

    • 1 solution: exp(–x) – x = 0

    • 2 solutions: x2 – 4 sin(x) = 0

    • Infinitely many solutions: sin(x) = 0

  • Iterative methods are necessary: no general exact formulas exist, even for polynomials

  • Terminology: solution = root = zero


Multiple roots
Multiple Roots

  • A nonlinear equation can have a multiple root: f (x) = 0 and f(x) = 0

  • Examples: (x – 1)k = 0

  • It is impossible to determine a multiple root to full machine accuracy

  • It is harder computationally to determine a multiple root, especially one with even multiplicity


Accuracy of solutions
Accuracy of Solutions

  • We can measure if the residual is small:

  • Or if the error is small (x* is solution):

  • These are related, but not equivalent


Conditioning
Conditioning

  • Mathematically: x* = f –1(0)

  • If computing f (x) is insensitive, then computing the root is sensitive

  • If computing f (x) is sensitive, then computing the root is insensitive

  • If we define F( y) f –1( y) thenF (0) = 1 / f (x*)


Convergence rate
Convergence Rate

  • Measuring speed of an iterative method

  • Define error: ek = xk– x*

    • For some algorithms, error will be the length of an interval containing x*

  • The sequence converges to zero with rate r if:


Convergence rate continued
Convergence Rate (continued)

  • Some important cases:

    • Linear (r = 1): requires C < 1

    • Superlinear (r > 1): # of digits gained per iteration increases at each iteration

    • Quadratic (r = 2): # of accurate digits doubles at each iteration

  • Convergence rates refer to asymptotic behavior (close to the solution); early iterations of the algorithm may produce little progress


Bisection simple safe
Bisection: Simple & Safe

  • Require [a,b] with f (a)  f (b) < 0

  • Reduce interval until error is “small”

  • While ((b – a) > tol1)

    Compute midpoint m = a + (b– a)/2

    If | f (m)| < tol2, stop

    If f (a) f (m) < 0 then b = m, else a = m

    end

Matlab m-files: bisect.m


Bisection continued
Bisection, Continued

  • Interval reduced by ½ each iteration

  • Linear convergence (r = 1, C = ½)

  • Bisection approximates f (x) by the line through [a,sign( f (a))] and [b,sign( f (b))] and determines the point m where this line is zero

  • This is a crude model of f (x)

  • What about multiple roots?

Matlab m-files: bisect_model.m


Newton s method
Newton’s Method

  • Approximate f (x) by its Taylor series:

  • Find point where line is zero:

  • Repeat this computation to get Newton’s method:

Matlab m-files: newton_model.m, newton.m


Newton s method convergence
Newton’s Method: Convergence

  • Note: ek = xk– x* so x* = xk – ek. Thus

  • Quadratic convergence (r = 2) if f(x*)  0


Secant method
Secant Method

  • Goal: reduce iteration cost of Newton’s method

  • Approximate f(x) by finite difference:

  • Superlinear convergence (r 1.6)


Safeguarded methods
Safeguarded Methods

  • Newton, secant methods:

    • Fast close to solution

    • Potentially unreliable (esp. away from solution)

  • Bisection (and other) methods:

    • Slow to converge

    • Reliable

  • Safeguarded method:

    • Monitor performance of fast method

    • Use slow, safe method to guarantee convergence

    • Near solution, the slow method usually not needed


Systems of nonlinear equations
Systems of Nonlinear Equations

  • Much more difficult than scalar case

  • Theoretical analysis harder, behavior of roots potentially stranger

  • No absolutely safe, reliable method

  • Costs rise rapidly with # of variables

  • Can only guarantee that algorithm converges to a solution of:


Newton s method1
Newton’s Method

  • In n dimensions:

    where (J = Jacobian matrix)

  • Quadratic convergence rate (if assumptions satisfied)

Matlab m_files: newton_s.m


Newton s method continued
Newton’s Method (continued)

  • Computational costs

    • O(n2) to compute Jacobian

    • O(n3) to solve Newton equations

  • Alternative methods

    • Analogs of secant method

  • Safeguards

    • Essential to guarantee convergence

    • “line search” or “trust region”


Matlab software
Matlab Software

  • 1-variable: fzero

  • n-variable: fsolve


For next class
For Next Class

  • Homework: see web site

  • Reading:

    • Heath: chapter 7


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