1. MAT 4725 Numerical Analysis Section 2.1 The Bisection Method http://myhome.spu.edu/lauw

2. MCM Teams Any progress?

3. Homework Download the homework Read 2.2 (Burden) You may skip all the proofs unless specified

4. Preview Find the solutions of an equation in one variable. Repeatly cut the intervals that contain the solution in half.

5. Population Model 1 N(t) = size of a population = birth rate Why?

6. Population Model 2 N(t) = size of a population = birth rate v = Why?

7. Population Model 2 N 0 = 1,000,000, N(1) = 1,564,000 = ??? v = 435,000

8. Population Model 2 We want to find = such that

9. Population Model 2

10. In general We want to find the solutions of a equation in one variable.

11. IVT

12. IVT: Special Case

13. The Bisection Method Idea

14. Theorem 2.1 The bisection method generates a sequence {p n } approximating a zero p of f such that

15. Theorem 2.1 The bisection method generates a sequence {p n } approximating a zero p of f such that Thus, the method always converges to a solution

16. Algorithm 2.1 Pseudo code (description) of the algorithm will be given. Easy to translate it into a program

17. Algorithm 2.1

18. Example 1

19. Example 2 Theoretical Computations Find the number of iterations n needed such that

20. Classwork 1 Write a program to implement the bisection algorithm.

21. Remark #1 Bisect:=proc(f, aa, bb, tol, N0) local i, p, a, b, FA, FP; a:=aa; b:=bb; The function f is passed into the procedure, not the expression f(x)

22. Remark #2 Bisect:=proc(f, aa, bb, tol, N0) local i, p, a, b, FA, FP; a:=aa; b:=bb; The values of the parameters passed into a procedure cannot be changed

23. Remark #3 Use return() to stop the program

24. Homework From now on… Use the Maple program in your classwork to do all the computations Use Maple to plot all the graphs