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Newton's Method

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Throughout your course work in Mathematics, you've likely had to find the root, or zero, of a function. These functions have likely only been linear or quadratic. But how would we go about finding the roots of the equation of a higher-order polynomial? Or what about a cubic equation with a natural logarithm, such as

It is much more difficult to find the roots of higher-order functions like these algebraically. However, Calculus proposes a few methods for estimating the root of complex equations. This article will cover one method to help us solve the roots of functions like these nasty ones!

One method we can use to help us approximate the root(s) of a function is called Newton's Method (Yes, it was discovered by the same Newton you've studied in Physics)!

**Newton's Method **is a recursive approximation technique for finding the root of a differentiable function when other analytical methods fail.

The Newton's Method formula states that for a differentiable function *F(x**)* and an initial point *x _{0}*

where *n *= 0, 1, 2, ...

With multiple iterations of Newton's Method, the sequence of *x _{n}* will converge to a solution for

As the derivative of *F(x)* is in the fraction's denominator, if *F(x) *is a constant function with the first derivative of 0, Newton's Method will not work. Additionally, as we must compute the derivative analytically, functions with complex first derivatives may not work for Newton's Method.

With the Newton's Method formula in mind, see the graphical representation below.

Newton's Method aims to find an approximation for the root of a function. In terms of the graph, the zero of the function is the green point, *f(x)* = 0. Newton's Method uses an initial point (the pink *x _{0}* on the graph) and finds the tangent line at the point. The graph shows that the line tangent to

The new point, *x _{1}*, found via the tangent line at x

In cases where we cannot solve a function's root directly, Newton's Method is an appropriate method to use. However, there are certain cases where Newton's Method may fail:

The tangent line does not cross the

*x*-axisOccurs when

*f'(x)*is 0

Different approximations may approach different roots if there are multiple

This occurs when the initial

*x*isn't close enough to the root_{0}

Approximations don't approach the root at all

Approximation oscillates back and forth

Let's consider one such example where Newton's Method fails.

Suppose we have the function

This function has roots at and . However, let's say you wanted to use Newton's Method to find the roots of . With an initial guess of , Newton's Method will approach the root rather than the root even though is closer to . Try it for yourself and see!

Use three iterations of Newton's Method to approximate the root near of** .**

Since we already have an equation for , we can skip right to finding the derivative,

Using the Newton's Method formula with *x _{0}* = 3:

Rounding to the first six decimal places, we get

Let such that

Using the quadratic equation

Taking the square root of we get

Our approximation is pretty accurate!

It is also possible to use Newton's Method to approximate the square root of a number! The Newton's Method square root approximation formula is nearly identical to the Newton's Method formula.

To compute a square root for and with an initial guess for of

Let's apply the Newton's Method square root approximation equation to an example!

Use Newton's Method square root approximation equation to approximate by finding* x _{1}*, ...,

Our guess should be a positive number that is smaller than 2. So, let's start with .

Plugging our known values in

Rounding to the first six decimal places, we get

When we compute the exact value of rounding to the first six decimal places, we get a value of . Additionally, notice how the answer of every iteration of the Newton's Method square root approximation formula is the same as each iteration of Newton's Method.

However, the Newton's Method Square Root Approximation method is much faster and easier to compute.

**Newton's Method**is a recursive approximation technique for finding the root of a differentiable function when other analytical methods fail- The formula for Newton's Method states that for a differentiable function
*F(x**)*and an initial point*x*near the root_{0} - for
*n*= 0, 1, 2, ... - Newton's Method uses iterative tangent line approximations to estimate the root

- The formula for Newton's Method states that for a differentiable function
- Newton's Method may fail when:
- the first derivative of
*f(x)*is 0 *x*_{0}_{ }isn't close enough to the root- iterative approximations don't approach the root at all

- the first derivative of

_{n+1} = x_{n} - [f(x_{n})/f'(x_{n})] where n = 0, 1, 2, ...

More about Newton's Method

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