BUILT-IN MATHEMATICA SYMBOL

RootApproximant

RootApproximant[x]
converts the number x to one of the "simplest" algebraic numbers that approximates it well.

RootApproximant

finds an algebraic number of degree at most n that approximates x.

MORE INFORMATION

For degrees above 2, RootApproximant generates Root objects.

RootApproximant[x] effectively tests the total number of bits in the description of x by successively higher-degree algebraic numbers, and returns the first case for which the number of bits is small.

Results from RootApproximant may not be unique.

MinimalPolynomial yields the minimal polynomial for the result of RootApproximant.

The option Method specifies an additional cost p to be used for each successively higher power in determining the "simplest" approximation.

EXAMPLES

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Basic Examples (2)

Find a quadratic approximation to

:

Find algebraic approximants of any order:

Find a quadratic approximation to

:

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Find algebraic approximants of any order:

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Scope (4)

There is no simple quadratic approximation to N

:

At degree 4, there is a simple answer, which turns out to be equal to

:

Machine precision is not enough to recover the Root object equal to

:

30 digits of precision suffice to recover the exact value here:

When given exact input, RootApproximant uses machine-precision approximations:

RootApproximant works with complex numbers:

Options (1)

Assigning additional cost to higher powers can be used to lower the degree of result:

Applications (2)

Find successive algebraic approximations to

:

A complicated radical expression:

Use FindRoot to find the real root near 2.5:

Use RootApproximant to find an algebraic number close to the root:

Check whether the result is a root of the original expression:

Properties & Relations (3)

RootApproximant gives a Root object close to a given algebraic number:

The Root object found may not be exactly equal to the input:

Use RootReduce to find exact Root object representations of algebraic numbers:

Specifying a linear polynomial effectively finds a rational approximation to x:

Rationalize also gives a rational approximation, but it need not be the same:

Interestingly, the approximations can be found among continued fraction convergents:

Use LatticeReduce to recognize linear combinations of more general functions:

The final relationship

:

Possible Issues (1)

Recognizing an algebraic number may require using higher precision:

The result is not equal to a:

Providing the correct degree improves the chances of recognizing an algebraic number:

A penalty may be used to lower the degree; here it does not help in recognizing the number:

Using a higher-precision approximation allows the algebraic number to be recognized: