NRoots

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`NRoots`

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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)Summary of the most common use cases

Numerically find the roots of a polynomial:

In[1]:=1

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https://wolfram.com/xid/0y8ouw-esw8ih

Out[1]=1

Change the result to rules:

In[2]:=2

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https://wolfram.com/xid/0y8ouw-k41nig

Out[2]=2

Test the roots:

In[3]:=3

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https://wolfram.com/xid/0y8ouw-o69fvw

Out[3]=3

Scope (1)Survey of the scope of standard use cases

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

In[1]:=1

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https://wolfram.com/xid/0y8ouw-onbzkb

Out[1]=1

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

In[2]:=2

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https://wolfram.com/xid/0y8ouw-q5dai3

Out[2]=2

Find the numerical roots of a polynomial with multiple roots:

In[3]:=3

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https://wolfram.com/xid/0y8ouw-jpyfa0

Out[3]=3

Options (5)Common values & functionality for each option

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):

In[3]:=3

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https://wolfram.com/xid/0y8ouw-t4j3sf

Out[3]=3

Use the "CompanionMatrix" method:

In[1]:=1

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https://wolfram.com/xid/0y8ouw-pmry9v

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0y8ouw-l4x1ty

Out[1]=1

PrecisionGoal (1)

Specifying PrecisionGoal can improve the precision of the roots returned:

In[1]:=1

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https://wolfram.com/xid/0y8ouw-eml8tk

Use default precision:

In[2]:=2

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https://wolfram.com/xid/0y8ouw-g1zd03

Out[2]=2

Specify larger precisions:

In[3]:=3

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https://wolfram.com/xid/0y8ouw-bgdz9x

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0y8ouw-lvcnjl

Out[4]=4

Compare with the digits from an exact computation:

In[5]:=5

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https://wolfram.com/xid/0y8ouw-gujuiw

Out[5]=5

StepMonitor (1)

Monitor the root-finding steps:

In[1]:=1

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https://wolfram.com/xid/0y8ouw-1v9578

Out[1]=1

Using a different Method gives different convergence:

In[2]:=2

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https://wolfram.com/xid/0y8ouw-yjjrcw

Out[2]=2

Applications (1)Sample problems that can be solved with this function

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:

In[2]:=2