The Newton-Raphson Method

The Newton-Raphson method is a very efficient algorithm to search for the zero of a real function. Most of the time it converges more quickly than a dichotomy approach. Nevertheless, there are some traps to avoid by studying the function and/or its curve.

It is attributed to the mathematicians Newton and Raphson for their respective contributions in 1685 and 1690. However, it is also the result of a slow maturation during the ages, before and after the Newton era. Its definitive principle as known today is from the mathematician Thomas Simpson in 1740. Before, it is derived from the works of mathematicians as the French François Viète in the 16th centuries, Iranians Sharaf al-Din al-Tusi and Jamshīd al-Kāshī in the 13th and 14th centuries. Finally it is a generalization of the Hero of Alexandria method in the 1st century BC also known as the Babylonian method.

Hero method

This technique extracts the square root from a positive number a. In other words, solving the equation x²=a for x or search for the zero of the function f(x)=x²-a . The method is iterative and geometric. The idea is to search for rectangles getting closer to a square, all of whose surface areas are equal to a.

Newton-Raphson method

From a geometric point of view this method substitutes a sequence of straight lines to the curve of a function we are searching for a zero. These lines are tangents, the best local linear approximation to the curve. The first tangent line is an important choice.

Read the complete article.

Thanks to Chao-Kuei Hung and Richard O'Keefe for the reviews of the article below.