Chem 302 Lab Assignment

1. Tuesday 2:30 – 4:00ColinChris MJohnathan PCraigJonathan EAlex Tuesday 4:00 – 5:30CurtisMattChris LJanetCoryKristen Chem 302 Lab Assignment

2. Chem 302 - Math 252 Chapter 1Solutions of nonlinear equations

3. Roots of Nonlinear Equations • Many problems in chemistry involve nonlinear equations • Linear and quadratic equations are trivial, can be solved analytically • Cubic and higher order solve numerically • Present a overview of basic methods • Not a complete discussion

4. Successive Approximations • Simplest method • Want to solvef(x)=0rearrange into the formx=g(x) • Use as iteration formulaxi+1=g(xi) • Use initial guess and iterate until self consistent

5. Successive Approximations

7. Successive Approximations Second root

8. Successive Approximations Second root

9. Successive Approximations Different formula Will find x=2, will not find x=1 Different formula – different results No one formula is best Slow to converge

10. Successive Approximations How to find initial guesses? Grid search Course to start Get finer

11. Analysis of Convergence • Each stage of iterationxi+1=g(xi) • For convergence • True root • Intersection of two functions: x & g(x)

12. Analysis of Convergence

13. Analysis of Convergence • Four possibilities • Monotonic convergence • Oscillating convergence • Monotonic divergence • Oscillating divergence

14. Monotonic convergence

15. Oscillating convergence

16. Monotonic divergence

17. Oscillating divergence

18. Analysis of Convergence • Key is g(x) • Mean Value Theorem • If g(x) and g(x) are continuous on the interval [a,b] then there exists an e (a<e<b) such that

19. Analysis of Convergence

20. Analysis of Convergence If M < 1 guaranteed to converge Sufficient but not necessary To converge LHS → 0

21. Speed of Convergence Taylor expansion about root For xi When close to convergence • Dominant term will be 1st nonzero derivative • Order of Convergence

22. Newton-Raphson Method • One of most common methods • Usually 2nd order convergence • Generally superior to simple iteration • Uses function and 1st derivative to predict root (assume f is linear)

23. Newton-Raphson Method

24. Newton-Raphson Method • Fairly robust • Need analytic expressions for f(x) and f'(x) • May be complicated or not available • Need to evaluate f(x) and f'(x) many times • Maximum efficiency • f'(x) close to zero will cause problems • Especially important for multiple roots • Need to checks in program • Value of f'(x) • Max iterations

25. Newton-Raphson Method • Test conditions Convergence

26. Secant Method • NR but use numeric derivative • Need two points to start

27. False-Position Method • Similar to Secant Method • Uses two points (one on each side of root) (need search) • Find where function would be zero if linear between two points • During each iteration one point is held fixed (pivot), other is moved • More stable but slower than NR • If pivot far from root then slow • If pivot close to root then denominator can be small

28. False-Position Method • Algorithm • Pick xL & xR (xL < < xR) • Evaluate • Calculate • Calculate • If then xM is the root • Replace xL or xR with xM (depends on sign of ) xM xL xR Repeat

29. False-Position Method

30. Bisection Method • Same as FP Method but xM is average of xL & xR • Drawbacks of both FP & Bisection • Two initial guesses on opposite sides of root • Multiple or close roots a problem • f always same sign a problem • Round-off error as xM gets close to root • Secant, FP & Bisection methods do not require analtyic expression of f'(x)

31. Roots of Polynomials • Want as efficient an algorithm as possible • Efficiency of operations • + • * • / • Power • Naive method (10 ×, 4 +) • Most efficient method for polynomial of order m requires m additionsand m multiplications • Nesting (Horner’s Method)

32. Horner’s Method • For polynomial of degree m • Divide by (x-r) • b0 is f(r)

33. Horner’s Method m×, m +

34. Horner’s Method 1 is a root

35. Horner’s Method 2 is a root

36. Horner’s Method • If b0 = 0 (i.e. r is a root) • Have factored (x-r) from equation (i.e. reduced order, called deflating) • Continue to find roots of reduced equation

37. Horner’s Method 2 is a root (i.e. a double root of original equation)

38. Birge-Vieta Method • NR method with f(x) and f'(x) evaluated using Horner’s method • Once a root is found, reduce order of polynomial

39. Birge-Vieta Method

40. Birge-Vieta Method

41. Birge-Vieta Method

42. Birge-Vieta Method

43. Birge-Vieta Method

44. Example

45. Example • NR • x0 4.33  3 • x0> 4.33  5 • Third root? Try it!!!

46. BV

47. Roots of Polynomials • What about complex roots? • Occur in pairs • Have form a+ ib & a – ib • Roots of quadratic equationf(x) = x2 – 2ax + a2 + b2 • BV method – we removed factors of x – r. • Quadratic can be solved analytically – therefore best to remove quadratic factors

48. Lin-Bairstow Method

49. Lin-Bairstow Method • Iteration Scheme

50. Lin-Bairstow Method • Algorithm • m > 3: determine quadratic roots, reduce order of problem by 2 • m = 3: determine linear root then quadratic roots • m = 2: determine quadratic roots • m = 1: determine linear root