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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
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Team Newton's Method Teachers
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 x0 near the root
where n = 0, 1, 2, ...
With multiple iterations of Newton's Method, the sequence of xn will converge to a solution for F(x) = 0.
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 finds a line tangent to the initial point to find an approximation for the root of f(x) - Vaia Original
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 x0 on the graph) and finds the tangent line at the point. The graph shows that the line tangent to touches the x-axis near the root.
On the second iteration, Newton's Method constructs a new tangent line based on the last approximation found by the tangent line - Vaia Original
The new point, x1, found via the tangent line at x0, is translated onto the graph of the function, and a new tangent line is found. This process is repeated until a plausible estimation is found for f(x) = 0.
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-axis
Occurs when f'(x) is 0
Different approximations may approach different roots if there are multiple
This occurs when the initial x0 isn't close enough to the root
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 x0 = 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 x1, ..., x5.
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.
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What is Newton's method?
Newton's method is a recursive approximation technique for finding the root of a differentiable function when other analytical methods fail.
What is Newton's method formula?
The formula for Newton's Method says xn+1 = xn - [f(xn)/f'(xn)] where n = 0, 1, 2, ...
How to use newton's method?
To use Newton's Method, you need a differentiable function and an initial starting point. From there, plug in points iteratively until a plausible approximation is achieved.
What is the assumption of newton's method?
Newton's Method assumes that a line tangent to the function crosses the x-axis near the root of the function.
What is the advantages and theory of newton's method?
Newton's Method allows one to approximate the root of the function the algebra and analytical methods fail. It has a rather fast rate of convergence, meaning a limited number of iterations are required for an accurate approximation.
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