Newton's Method Page: 1 of 7
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Newton's Method *
Juan C. Meza**
March 15, 2010
Newton's method is one of the most powerful techniques for solving systems
of nonlinear equations and minimizing functions. It is easy to implement and
has a provably fast rate of convergence under fairly mild assumptions. Because
of these and other nice properties, Newton's method is at the heart of many
solution techniques used to solve real-world problems. This article, gives a
short introduction to this method with a brief discussion of some of the main
issues in applying this algorithm for the solution of practical problems.
Newton's method is perhaps the best known method for finding the root of a non-
linear equation or for minimizing a general nonlinear function. This method, also
known as the Newton-Raphson method, can be traced back to Isaac Newton(c. 1669)
and Joseph Raphson (1690). Both Newton and Raphson appear to have derived
essentially the same method independently. Newton based his derivation on a lin-
earization of a higher-order polynomial and he showed how the method could be
used to solve a particular cubic equation. Raphson's scheme on the other hand more
closely resembles what we now know as Newton's method.
In its basic form, Newton's method is easy to implement and requires only the
ability to compute a function and its first and second derivatives. One of the main
advantages of Newton's method is the fast rate of convergence that it possesses and
a well-studied convergence theory that provides the underpinnings for many other
methods. In practice, however, Newton's method needs to be modified to make
it more robust and computationally efficient. With these modifications, Newton's
method (or one of its many variations) is arguably the method of choice for a wide
variety of problems in science and engineering.
*This work was supported by the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
**Lawrence Berkeley National Laboratory, Berkeley, CA 94720
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Meza, Juan C. Newton's Method, report, March 1, 2010; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc1012637/m1/1/: accessed January 20, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.