Assignments

1. The function in the bisect.f evaluates . Run the code and check your answer (try the xcalc utility on the mouse). Modify the function to compute . Find the three real roots of . (You might want to use matlab's graphics routine to get a picture of the cubic, to get estimates of where the roots lie.) Then find the first 4 positive x-values where . (Zero is obvisouly a solution; find the next 4 positive solutions.)

2. Repeat (1) using false positions.

3. Repeat (1) using Newton's method. For the largest solution of , try initial guesses 13.0, 13.5, and 14.0. Can you explain what is happening?

4. Repeat (1) using the secant algorithm. You will need a good secant method for a later exercise.

5. Write a code using centered differences to compute the approximate derivative of a function . Test it thoroughly. Then incorporate it as a subroutine into your Newton's method code, instead of the function routine for fprime. Use this new code to repeat exercise (1).

6. Change the integration code so that it uses the trapezodial rule. Check that this new method gives exact answers if f is a constant or a line. Then run the trapezodial algorithm for , using 10, then 20, then 40, then 80 subintervals. Check the order of convergence; is it right? Now compute .

7. A more challenging problem. Define . For what x is ?

8. Write a code implementing the trapezodial method to solve the ODE , up to t=1 A timestep of 1/20 ought to suffice. But to check yourself, run the code using 1/20, 1/40 and 1/80 and make sure you are getting (essentially) second order convergence.

9. Write a code to implement a trapezodial method for the preditor-prey equations, for 0<t<10. Use a1=2.0, b1=0.1, c1=3.0, d1=0.1, W(t=0) = 20.0, D(t=0) = 20.0. Output your results in two files, listing t and D in one file, t and W in the other. Use Matlab to (1) on same plot, show D and W as functions of t (2) show W vs. D. Now try changing W0 and D0 (e.g. (30,30),(20,30),(30,20)). Can you tell what is going on?