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Quintic Polynomials — Finding Roots From Primary and Secondary Nodes; a Double Shot!
Using twin function proximal nodes for application of Newton’s Approximation to find roots
While Newton’s approximation is universal, its accuracy largely depends on having application nodes proximal to roots to eliminate the need for iterations. It is also impacted by the steepness of the gradient at the root and the variability of the function’s root gradient with the node tangent. To improve options for node selection I recently posted, Quartic Polynomials-Calculate Roots Using Twin Function Nodes, which uses the architecture and symmetry of polynomials to find roots using ‘SOSO’; a confected twin function with Same Opposite Same Opposite +/- coefficients. SOSO, is a process of finding the intercepts of twin functions by simultaneous equations, to produce the nodes.
This post furthers the application of SOSO by creating secondary nodes by lowering or raising primary nodes closer to roots to improve accuracy of Newton’s and other root approximation methods. Because nodes are very simple to calculate, roots normally requiring repeat approximations can be easily targeted.
I will find three roots by this method and the remaining two roots by a method I outlined in: Cubic Polynomials — A Simpler Approach.
An alternative is to calculate the four maximum with SOSO Quintic, and calculate the fifth by deduction from the coefficient of B!
This post assumes knowledge of algebra and introductory calculus (differentiating polynomials) at the high school level.
Secondary Nodes
To demonstrate the concept in the scope of this post, I have used simple node adjustments by adding or subtracting the constant term to get closer to a root in one application.
Generally, a node with height Yn ≤ 1.5 will give a sufficiently accurate result, unless nodes are proximal to turning points where gradients are small. In these cases a further shift may be required.
Overshooting the x axis with a raising or lowering is acceptable, as Newton’s Approximation can of course be applied from anywhere in the x-y grid.
