You've probably heard of Newton's Method from your
calculus course. It can be used to locate zeros of
realvalued functions. But did you know that it is
possible to define a
multidimensional version of Newton's Method for
functions from R^{n} to R^{n}?
Here's how it goes. The derivative Df(x) of
the function f is the linear transformation that
best approximates f near the point x.
It can be represented by a matrix A whose entries are
the partial derivatives of the components of f
with respect to all the coordinates.
The best linear approximation to f is given by
the matrix equation:
yy_{0} = A (xx_{0})
So, if x_{0} is a good "guess" for the zero of f,
then solving for the zero of this linear approximation will
give a "better guess" for the zero of f.
Thus let y=0, and since y_{0}=f(x_{0})
one can solve the above equation for x. This leads to
the Newton's method formula:
x_{n+1} = x_{n}  A^{1} f(x_{n})
where x_{n+1} denotes the (n+1)st guess,
obtained from the nth guess x_{n} in the
fashion described above.
Iterating this will give better and better
approximations if you have a "good enough" initial guess.
Presentation Suggestions:
Point out how this generalizes the
usual Newton's method formula that they have learned.
(The inverse of A is analogous to dividing by f'.)
The Math Behind the Fact:
The set of all initial guesses (called seeds) that
converge to a given root is called the
basin of attraction for that root. This set can
often be fractal, and this idea is often the basis for many
of the pictures found in popular books on fractals.
You can learn about the multidimensional Newton's method
in a numerical analysis course, or
an advanced analysis course (since it may be used as a basis
for a proof of the Inverse Function Theorem), or
an operations research course called nonlinear programming.
The basics of linear transformations are covered in a course on
linear algebra.
How to Cite this Page:
Su, Francis E., et al. "Multidimensional Newton's Method."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
