Photo by Martin Sanchez on Unsplash

Quartic Polynomial Roots-With Newton Plus a Little Bit

Calculate Quartic Roots Using Newton’s Approximation Plus a little Bit

This post presents a Quartic application of my recent Cubic related post; Polynomial Roots-Sector Arcs and Newton’s Approximation-Plus a Little Bit, which introduces alternative methods of approximating Cubic polynomial roots using Sector Arc Length in lieu of the Δx ‘Opposite’ side of a triangle of Newton’s Approximation. It found a good comparison of Sector Arc Length with Newton’s Approx-Plus a little Bit, thereby offering a choice of 2 methods giving good results in a single iteration.

Sector Arc Length

Uses the familiar Arc=R*theta where R is a sector radius and sector angle theta is expressed in radians of which there are 2 pi in 360 deg giving 1deg=0.01745 rad.

Newton Plus a little Bit

Adds a little bit (actually 11%) to the Δx value when applied from a node point of Inflection Ip(x, y). The result is similar to the single iteration Sector Arc Length method.

Newton’s approximation can become a bit busy when 2nd or even 3rd iterations are required, involving lengthy height and gradient calculations. This situation generally occurs where roots are close to turning points being housed in low gradient areas.