You've probably heard of Newton's Method from your
calculus course. It can be used to locate zeros of
real-valued functions. But did you know that it is
possible to define a
multi-dimensional version of Newton's Method for
functions from Rn to Rn?
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:
y-y0 = A (x-x0)
So, if x0 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 y0=f(x0)
one can solve the above equation for x. This leads to
the Newton's method formula:
xn+1 = xn - A-1 f(xn)
where xn+1 denotes the (n+1)-st guess,
obtained from the n-th guess xn in the
fashion described above.
Iterating this will give better and better
approximations if you have a "good enough" initial guess.
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 multi-dimensional 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 non-linear programming.
The basics of linear transformations are covered in a course on
How to Cite this Page:
Su, Francis E., et al. "Multidimensional Newton's Method."
Math Fun Facts.