Department of Mathematics

Math 300: Mathematical Computing

Assignment 8

Recall that Newton's Method is an iterative way of approximating a zero of a function \(f\). The idea is that, given a starting guess \(x_0\) and an error tolerance \(\tau\), we compute new estimates of the zero of \(f\) using the formula \[ x_{n+1} = x_n-f(x_n)/f'(x_n) \] for \(n=0,1,\ldots\) We use this iteration until \(\vert x_n-x_{n-1}\vert\lt\tau\) or until we give up trying. Unfortunately, we do not know how to use Matlab to compute the derivative of \(f\) (it can, we just have not done it). Instead, we can choose some small number \(h\) and use the approximation \[f'(x_n) \approx \frac{f(x_n)-f(x_n-h)}{h} \] instead of \(f'\). If we use successive estimates of the root, then the formula becomes \[ x_{n+1}=\frac{x_{n-1}f(x_n)-x_nf(x_{n-1})}{f(x_n)-f(x_{n-1})}. \] This is called the secant rule. Write a Matlab function secant(f,initGuess,tolerance) that finds a zero of a function using this secant formulation. Note that the first argument is a function handle - not an ordinary variable. The other two arguments are scalars. Be sure to put helpful comments at the top of your file, so that when we type help secant we get instructions on how to use your function. Note that you need to figure out how to use the given initial guess to start the iteration, inasmuch as the formula actually uses two points to progress to a third. Be sure to explain that in the program comments. Use that function to find the zero of your name function that lies nearest the origin. Type comments in your function explaining how that is done.

This assignment is worth 35 points. The assignment is completed when your code appears as an attachment to an email message to the instructor. The assignment is due at 9 AM on Thursday, 3 November.

The last test will take place at the final exam time, Monday 12 December, 1:30-3:30. It will be written as a one-hour in-class on-line test. As usual, you can use any resources you want. A Sample Exam is available. Note that the URL for the exam will not be emailed to all. If you need to take the exam from outside class, contact the instructor to get on the email list.

Assignment B is posted.

Most recent scores are on-line at My.math. Check the Info page. Let the instructor know if you find a discrepancy.

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