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.
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 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 linear algebra.
How to Cite this Page:
Su, Francis E., et al. “Multidimensional Newton’s Method.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.
References:
A classic text on real analysis is Walter Rudin’s Principles of Mathematical Analysis.
Fun Fact suggested by:
Francis Su