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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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