Tutorial 5 - 2010 - Solution/Question 4

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Question 4 [1]

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Comment on the 3 approaches used so far. Are your calculations what you would expect from each method?

Solution

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From question 1 we observed that the bisection method is a robust yet slow root finding tool. As long as you know that your bounds contain a single root and that the function changes sign on either side of this root, the bisection method is guaranteed to converge to within any pre-specified tolerance given enough time. This was, of course, expected.

In question 2 we observed that fixed-point iteration methods converge much more rapidly than the bisection method but performance is highly dependent on the iterative function selected. Again this was to be expected.

Finally, we observed that the Newton-Raphson method wildly diverged on the first iteration but appeared to be rapidly coming back to the "root region" in the second and third iterations. It was expected that the Newton-Raphson method would be the fastest to converge and arguably it had the fastest iteration to iteration convergence. The problem in this case was the poor initial guess of 380 K. If we go back to the plot presented in the solution to question 1 we see that the function is near its minimum at 380 K. Thus it is no surprise that the function's derivative was near zero and so this caused the first Newton-Raphson iterate to explode (you are dividing by the derivative after all). A better initial guess would likely greatly improve convergence.

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