Unit 2.1 Roots of Nonlinear Equations and Taylor’s Polynomials
We want to find a real number that solves the equation
Such an is called a root of , which we will denote by .
A method used to find is called a rootfinding method.
Example: The roots of are and .
For many problems, the answer isn’t so obvious, so we turn to a numerical method.
We will look at methods that are iterative, i.e., we apply the same process over and over to get closer and closer to the root until we are “close enough”.
Big Idea: Replace with something easier
Finding such that is the same as finding where the graph of intersects the -axis. For some functions this is easy.
The easiest is a line (constant functions don’t cross the -axis).
Idea: Start with a guess at the root, or at least a place near the root.
We can get an approximation of where crosses the -axis by considering the tangent line through that point.
Linear Approximation
This is called a linear approximation, and we will use it in many other places in this course.
Example: For and starting at the point , we get the tangent line .
This suggests the next guess at the root should be .
Taylor’s Polynomials
A more general way to get the linear approximation is through the Taylor Polynomials.
Recall from calculus that a function can, under certain conditions, be written as an infinite series:
This is called the Taylor’s Series for , and the formula and its existence conditions are called Taylor’s Theorem. If , the Taylor Series is called a Maclaurin Series.
Example:
Practice Exercise
Find the Maclaurin Series of .
Taylor’s Polynomials and Error
You may recall from calculus that you proved Taylor’s Theorem by writing it as a finite sum, called the Taylor’s Polynomial of
order , and a remainder term.
Showing the remainder term went to zero as proved the result.
Here we want to use the Taylor’s Polynomial as an approximating polynomial, and think of the remainder term as the error in our approximation.
Mathematically the two ideas are the same, but thinking in terms of approximation plus error is better suited for our purposes:
For notational simplicity, we will call a Taylor’s Polynomial approximation of order to be . Then
Example
is a linear approximation to .
Newton’s Method derived from Taylor’s Polynomials.
We have
In the prior example with , , , we get
If is close to the root , then the root of should usually be closer to . The notion of “close” is critical and we will get to it more concretely later.
In general, solving for gives
A little algebra gives
Iterating this formula gives us Newton’s Method:
Practice Exercise
Apply the Newton’s Method formula to
and simplify to get an iterative equation.
Note: normally we won’t bother simplifying; and will be sent directly to the code as is. For certain functions that we use over and over (such as ), if we use Newton’s method we may wish to simplify.