Newton's method

You should have seen Newton's method in your calculus class. The basic idea of the algorithm is this: given an initial guess, call it to a root of f(x)=0, a refined guess, is computed based on the x-intercept of the line tangent to f at . That is, consider the equation of the line tangent to f at ,

This is the point-slope equation of a line, with being the point and the slope. Now calculate that x at which y=0. That is,

This value of x is the new guess at the root - call it . We repeat the process, computing an , until we compute an at which , where tol is a prescribed tolerance.

If you examine the formula for the new iterates you can see one potential for failure of Newton's method, namely could be zero - or, what is essentially the same, very small. This may lead to wild divergences in the iteration procedure. There are other possible failures in the method that we will uncover in the exercises. For now, here is a skeleton code for Newton's method. Note: To finish the coding, you need formulae for both f and f'.

• newton.f
• Tolerance
• Convergence
• Odds and ends