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OR682/Math685/CSI700

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

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  1. OR682/Math685/CSI700 Lecture 5 Fall 2000

  2. 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)

  3. Examples • 1-variable: x2 = 4 sin(x) • 2-variable:

  4. 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

  5. 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

  6. 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

  7. 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*)

  8. 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:

  9. 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

  10. 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

  11. 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

  12. 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

  13. Newton’s Method: Convergence • Note: ek = xk– x* so x* = xk – ek. Thus • Quadratic convergence (r = 2) if f(x*)  0

  14. Secant Method • Goal: reduce iteration cost of Newton’s method • Approximate f(x) by finite difference: • Superlinear convergence (r 1.6)

  15. 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

  16. 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:

  17. Newton’s Method • In n dimensions: where (J = Jacobian matrix) • Quadratic convergence rate (if assumptions satisfied) Matlab m_files: newton_s.m

  18. 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”

  19. Matlab Software • 1-variable: fzero • n-variable: fsolve

  20. For Next Class • Homework: see web site • Reading: • Heath: chapter 7

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