Which numerical methods are associated with Isaac Newton?

Isaac Newton is primarily associated with two significant numerical methods in mathematics:

• Newton’s Method: This is a powerful technique for finding successive approximations to the roots of a real-valued function, expressed as $f(x)=0$. Devised by Newton in 1669, the method uses the derivative of a function to iteratively improve an initial guess ($x_n$) toward a more accurate solution ($x_{n+1}$) using the recursive formula $x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}$.1

• Newton’s Interpolation Formula: In the field of interpolation, Newton developed a formula to construct a polynomial that passes through a set of given data points. When the input values ($x_i$) are equally spaced by a factor $h$, this method provides a polynomial function that fits the data, allowing for the estimation of values between known points.2

Additionally, Newton’s work with infinite series, such as the binomial series, laid the foundational algebraic groundwork for his broader contributions to calculus and numerical approximation.3