Section Summary: 4.9

  1. Definitions

    Newton's method (or Newton-Raphson method): a method for finding the roots, or zeros, of a function (i.e., those values of x such that f(x)=0). The method is based on the linearization of a function: we find the tangent line to a function at a point tex2html_wrap_inline128 , and find where it has a root: we hope that this root is closer than tex2html_wrap_inline130 to the true root, r.

    Often these roots are simply approximations of the true roots, but may be good enough for our work. The method is based on the iterative formula

    displaymath120

    and a good initial guess (denoted tex2html_wrap_inline130 , pronounced ``x-not''). From tex2html_wrap_inline130 we calculate tex2html_wrap_inline138 ; from tex2html_wrap_inline138 we compute tex2html_wrap_inline142 ; etc. If the sequence converges to some finite number, i.e.

    displaymath121

    then we have a root r of the function (as a check, compute f(r)).

    To find a root to a given degree of accuracy (say five decimal places), we compute using the iterative formula until the successive iterates tex2html_wrap_inline148 and tex2html_wrap_inline150 agree to five decimal places.

  2. Theorems

    None.

  3. Properties/Tricks/Hints/Etc.

    Newton's method may fail for various reasons:

    Suppose that we seek the roots of f(x). The book suggests that the graph of f(x) and Newton's method may be used in tandom: the graph will provide a good starting guess tex2html_wrap_inline130 , and Newton's method will provide a good approximation.

  4. Summary

    We are introduced to Newton's method for computing roots of a function, based on the linearization of the function. Once again, we replace a function by a closely related linear function, hoping that the linear function will inform us about the function itself. Linear functions are the simplest interesting functions we have, and they pop up constantly in mathematics (e.g. the slant asymptotes described in section 4.5 are good replacements for the original functions far from the origin).

    Newton's method is an iterative method, meaning that we do the same thing over and over, and hope that results get successively better. This is quite likely the basic method on which your calculator's ``solve'' command is based.

    It is possible to find different roots from different starting points, in the event that there are multiple roots (e.g. a quadratic equation may have two, etc.). This method often gives a good approximation, but there are a variety of reasons for why Newton's method may fail.

Problems:

Problems, pp. 297-298, #1, 4, 6, 13, 14, 27-30



LONG ANDREW E
Sun Oct 21 23:10:48 EDT 2001