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Concept

Root finding

The computation of where a function is zero, as opposed to the argument that a zero exists.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

root 1root 2root 3each point coloured by which of the three roots Newton's method takes it tothe boundary is not three sectors: every point of it has all three basins arbitrarily close by

Where Newton's method goes instead

An algorithm designed to find roots, run from every starting point at once. Three roots, three basins, and a boundary at which all three are arbitrarily close — so a rule with no randomness in it has starting points whose answer cannot be predicted.

dynamics · newton basins average 2.0001.0003.0002.000the 3 roots add to 6.000 and multiply to 6.000, both of which are coefficients divided bythe leading onetheir average is at 2.000, marked — moving the constant term moves the roots about andcannot move that point at all

What the coefficients already know

Finding the roots of a polynomial is hard and often impossible in closed form. Reading off their sum, their product and how many of them are real is none of those things — those numbers are sitting in the coefficients, and no root-finding is required to get at them.

algebra · polynomial roots

Named alongside it

The objects these essays reach for when they reach for this one.

Complex numbersBasin of attractionCoefficientConvergenceDegreeDerivativeDiscriminantFactorisationFractalIterationNewtons methodPolynomial

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