e-sign-changes

▸ function eSignChanges (p: number[][]): number

Defined in roots/descartes/expansion/e-sign-changes.ts:41

Returns the number of sign changes in the polynomial coefficents when ordered in descending order; zeros are ignored.

  • this function is often called Descartes in the literature

  • returns an upper bound of the number of positive real roots of the given polynomial

  • the upper bound returned is always a non-negative multiple of two (i.e. 0, 2, etc) higher than the actual number of real roots

  • the polynomial need not be square free

  • Descartes' rule of signs states (quoted from Wikipedia): "if the terms of a polynomial are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or is less than it by an even number. Multiple roots of the same value are counted separately."

  • see Descartes' rule of signs

example

eSignChanges([[1],[2],[-3],[0],[0],[3],[-1]]); //=> 3

Parameters:

NameTypeDescription
pnumber[][]a polynomial with coefficients given densely as an array of Shewchuk floating point expansions from highest to lowest power, e.g. [[5],[-3],[0]] represents the polynomial 5x^2 - 3x

Returns: number