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ECIV 301

ECIV 301. Programming & Graphics Numerical Methods for Engineers Lecture 8 Roots of Equations Open Methods. c must satisfy. Last Time The Problem. Define Function. c is the ROOT of the equation. Bracketing. Open. Fixed Point Iteration Newton-Raphson Secand. Graphical

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ECIV 301

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  1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 8 Roots of Equations Open Methods

  2. c must satisfy Last Time The Problem Define Function c is the ROOT of the equation

  3. Bracketing Open • Fixed Point Iteration • Newton-Raphson • Secand • Graphical • Bisection Method • False Position Last Time Classification Methods

  4. xr=0.5(xl+xu) xl xu Last Time Bisection Method Repeat until convergence

  5. f(xl) xr xu xl f(xu) Last TimeFalse Position Method

  6. Root = Last Time Bisection Method Check Convergence If Error

  7. Last Time Convergence

  8. Objectives • OPEN Methods • Fixed Point Iteration • Newton Raphson • Secant

  9. Open Methods Bracketing Methods Two Initial Estimates Needed that bracket the root Always Converge Open Methods ONE Initial Estimate Needed Sometimes Diverge

  10. root Fixed Point Iteration X x is a root if f(x) = 0

  11. root f1(X) f2(X) f1(X) f2(X) X root Fixed Point Iteration X + x + x

  12. f1(X) f2(X) f1(X) f2(X) X root Fixed Point Iteration x is a root if f1(x) = f2(x)

  13. New Guess New Guess Initial Guess Fixed Point Iteration f1(X) f2(X) root X

  14. New Guess New Guess Initial Guess Fixed Point Iteration f2(X) f1(X) Method Diverges root X

  15. New Guess Condition for Convergence f1(X) f2(X) X

  16. g’(xi) g’(xi) New Guess New Guess Initial Guess Newton Raphson g(x) X

  17. Newton Raphson

  18. Newton Raphson Inflection Point in Vicinity of Root

  19. Newton Raphson Persistent Oscillations near local max or min

  20. Newton Raphson Initial guess close to root jumps several roots away

  21. Newton Raphson Zero Derivative

  22. Newton Raphson It converges very fast!! (when it does) • No Convergence Criteria • Depends on Nature of Function • Depends on Initial Guess • Use Initial Guess Sufficiently Close to Root

  23. Homework • 5.7 • 5.19 • 6.4 • 6.8 (a),(b),(c) Due Date: September 22

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