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In this blog, we look at the first ever example of the use of the Newton-Raphson method...

Given an equation \(y = f(x)\), the Newton-Raphson method uses tangents to try to find roots of the equation, where \(f(x) = 0\). The method uses the following recurrence relation which, often, converges to a specific root.

\(x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}\)

The method is named after two mathematicians born in the 17th century, Isaac Newton and Joseph Raphson. Newton is very famous both in mathematics and in science, but who was Joseph Raphson? Not much information is available on his early life, but we know he graduated from Cambridge University in 1692. During his time at university, he published the book Analysis Aequationum Universalis, where the technique for the Newton-Raphson method appeared in print for the first time. (Newton, independently in 1671, had developed a very similar method, but his work was not published until 1736.)

The first ever example of the Newton-Raphson method appears on page 9 of the book:

The example uses the Newton-Raphson method to find the positive root of the equation \(a^{2}=2\), which is written as \(aa=2\) in the book. Remembering that the modern recurrence relation for the Newton-Raphson method is

\(x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})},\)

in the example we have \(f(x_{n}) = x_n^{2} - 2\), so that \(f'(x_{n}) = 2x_n\). Starting at \(x_0 = 1\), we find that

\(x_1 = x_0 - \frac{x_0^{2} - 2}{2 \times x_0}\)

\(x_1 = 1 - \frac{1^2 - 2}{2 \times 1}\)

\(x_1 = 1.5\)

Then,

\(x_2 = x_1 - \frac{x_1^{2} - 2}{2 \times x_1}\)

\(x_2 = 1.5 - \frac{1.5^2 - 2}{2 \times 1.5}\)

\(x_2 = 1.41\dot{6}\)

which is written as

\(g = 1.417\)

in the book.

Then,

\(x_3 = x_2 - \frac{x_2^{2} - 2}{2 \times x_2}\)

\(x_3 = 1.41\dot{6} - \frac{1.41\dot{6}^2 - 2}{2 \times 1.41\dot{6}}\)

\(x_3 = 1.414215686\)

to 9 decimal places, which is written as

\(g = 1.414217\)

in the book.

Finally,

\(x_4 = x_3 - \frac{x_3^{2} - 2}{2 \times x_3}\)

\(x_4 = 1.414215686 - \frac{1.414215686^2 - 2}{2 \times 1.414215686}\)

\(x_4 = 1.414213562\)

to 9 decimal places, which is written as

\(a = 1.414213562378\)

in the book.

We can use the following graph to explain what is happening as we converge towards the root \(\sqrt{2}\):

• We are attempting to find the positive root of the equation \(x^2-2=0\).
• We choose to start at \(x_0=1\), which is close to where the red curve  \(y=x^2-2\) intersects the \(x\)-axis.
• We draw a green tangent to the point \((x_0, f(x_0))\) [or \((1, -1)\)] on the red curve.
• The green tangent cuts through the \(x\)-axis at the point \(x_1=1.5\).
• We draw a blue tangent to the point \((x_1, f(x_1))\) [or \((1.5, 0.25)\)] on the red curve.
• The blue tangent cuts through the \(x\)-axis at the point \(x_2=1.41\dot{6}\).
• And so on...