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Newton's Method

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Newton's Method

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!

Newton's Method of Approximation

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.

Newton's Method Formula

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.

The Calculus behind Newton's Method

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

Newton's Method tangent line approximation StudySmarterNewton's Method finds a line tangent to the initial point to find an approximation for the root of f(x) - StudySmarter 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.

Newton's Method tangent line approximation StudySmarterOn the second iteration, Newton's Method constructs a new tangent line based on the last approximation found by the tangent line - StudySmarter 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.

When Newton's Method fails

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!

Newton's Method Examples

Example 1

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

Step 1: Find the derivative of f(x)

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

Step 2: Use x0 = 3 to complete the first Newton's Method iteration

Using the Newton's Method formula with x0 = 3:

Step 3: Continue iterations until finding x3

Rounding to the first six decimal places, we get

Step 4: Compare to the actual value

Let such that

Using the quadratic equation

Taking the square root of we get

Our approximation is pretty accurate!

Newton's Method of Approximating Square Roots

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

Square root approximation using Newton's Method Example

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.

Step 1: Establish an initial guess for x0

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

Step 2: Use x0 = 1 and plug into equation

Plugging our known values in

Step 3: Continue iterations until finding x5

Rounding to the first six decimal places, we get

Step 4: Compare to actual value and Newton's Method approximation

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 - Key takeaways

  • 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 x0 near the root
    • for n = 0, 1, 2, ...
    • Newton's Method uses iterative tangent line approximations to estimate the root
  • Newton's Method may fail when:
    • the first derivative of f(x) is 0
    • x0 isn't close enough to the root
    • iterative approximations don't approach the root at all

Frequently Asked Questions about Newton's Method

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 says xn+1 = xn - [f(xn)/f'(xn)] where n = 0, 1, 2, ...

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.

Newton's Method assumes that a line tangent to the function crosses the x-axis near the root of the function.

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.

Final Newton's Method Quiz

Question

What is Newton's Method?

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Answer

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

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Question

Newton's Method uses...

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Answer

tangent lines

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Question

What is an example of when Newton's Method fails?

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Answer

x0 is not close enough to the root

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Question

What is the difference between Newton's Method and the Newton's Method square root approximation formula?

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Answer

The Newton's Method square root formula is easier to compute and work with than Newton's Method.

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Question

Who was Newton's Method discovered by?

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Answer

Isaac Newton

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Question

Describe the geometry behind Newton's Method.

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Answer

Newton's Method aims to find an approximation for the root of a function. Newton's Method uses an initial point and finds the tangent line at the point. Wherever the tangent line touches the x-axis is where the new guessing point is. Each iteration should produce a more accurate guess as the tangent line moves along the graph of the function.

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Question

What kind of equations is Newton's Method good for solving?

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Answer

higher-order polynomials or complex equations including logarithms

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Question

When might Newton's Method not be the right choice for solving for roots?

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Answer

When the first derivative is too complex.

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