NRoots

NRoots[lhs==rhs,var]

yields a disjunction of equations which represent numerical approximations to the roots of a polynomial equation.

Details and Options

NRoots gives several identical equations when roots with multiplicity greater than one occur.
NRoots has the following options:

MaxIterations	Automatic	maximum number of iterations to use
Method	Automatic	method to use
PrecisionGoal	Automatic	the precision sought
StepMonitor	None	expression to evaluate at each step

Possible settings for the Method option include: "Aberth", "CompanionMatrix", and "JenkinsTraub".

Examples

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Basic Examples (1)

Numerically find the roots of a polynomial:

Change the result to rules:

Test the roots:

Scope (1)

Find the numerical roots of a polynomial with only real roots:

Find the numerical roots of a polynomial with real and complex roots:

Find the numerical roots of a polynomial with multiple roots:

Options (5)

Method (3)

Use the "Aberth" method, which simultaneously approximates all the roots of a univariate polynomial and converges cubically in general (but linearly at multiple zeros):

Use the "CompanionMatrix" method:

Use the "JenkinsTraub" method, which is a standard fast iterative globally convergent root-finding algorithm for polynomials:

PrecisionGoal (1)

Specifying PrecisionGoal can improve the precision of the roots returned:

Use default precision:

Specify larger precisions:

Compare with the digits from an exact computation:

StepMonitor (1)

Monitor the root-finding steps:

Using a different Method gives different convergence:

Applications (1)

Visualize the asymptotic rate of growth of the terms in the "look and say sequence," which is given by the positive real root of the following polynomial:

Find the roots numerically:

Visualize the roots in the complex plane, highlighting the unique positive one:

Properties & Relations (7)

Return numerical roots using NRoots:

Compare with the numericization of roots returned by Roots:

Compare with the numerical roots returned by NSolve:

Compare with the numericization of roots returned by Solve:

Compare with the numericization of roots returned by Reduce:

Compare with a single numerical root returned by FindRoot:

Compare with the numericization of a single root returned by FindInstance:

Possible Issues (3)

NRoots returns unevaluated when called on a non-polynomial equation:

NRoots can return small imaginary parts for polynomials with real roots:

Use Chop to remove them:

Increasing MaxIterations does not necessarily give more accurate results:

In such cases, increasing the PrecisionGoal can sometimes give more precise results:

Interactive Examples (1)

Interactively plot the real roots of a cubic polynomial:

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NRoots

Details and Options

Examples

Basic Examples (1)

Scope (1)