BUILT-IN MATHEMATICA SYMBOL

Root

represents the exact k

root of the polynomial equation

.

Root

represents the exact root of the general equation

near

.

Root

represents n roots of the equation

near

.

MORE INFORMATION

f must be a Function object such as .

Root is automatically reduced so that f has the smallest possible degree and smallest integer coefficients.

The ordering used by Root takes real roots to come before complex ones, and takes complex conjugate pairs of roots to be adjacent.

The coefficients in the polynomial can involve symbolic parameters.

For linear and quadratic polynomials , Root is automatically reduced to explicit rational or radical form.

For other polynomials, ToRadicals can be used to convert to explicit radicals.

Root represents an exact root of the general equation , which can be transcendental.

In Root, must be an approximate real or complex number such that exactly one root of lies within the numerical region defined by its precision.

Root represents n roots, counting multiplicity, that lie within the numerical region defined by the precision of .

N finds the approximate numerical value of a Root object.

Operations such as Abs, Re, Round, and Less can be used on Root objects.

Root is treated as a numeric quantity if f contains no symbolic parameters.

Root by default isolates the complex roots of a polynomial using validated numerical methods. SetOptions[Root, ExactRootIsolation->True] will make Root use symbolic methods that are usually much slower.

EXAMPLES

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Basic Examples (2)

Solution to a quintic:

Numerical values:

Real solutions to an exp-log equation:

Solution to a quintic:

In[1]:=

Out[1]=

Numerical values:

In[2]:=

Out[2]=

Real solutions to an exp-log equation:

In[1]:=

Out[1]=

Scope (11)

Some exact values are generated automatically:

Evaluate to high precision:

Roots of a polynomial:

Real roots of an exp-log function:

Roots of an analytic function in a bounded region:

A triple root:

Roots of a polynomial with symbolic coefficients:

Roots of a quadratic with symbolic coefficients:

When a, b, c and the roots are real, the roots are always ordered by their values:

The "standard" formulas for the roots of a quadratic do not guarantee the ordering of roots:

Find the series with respect to a parameter:

Complex components of roots:

Exact comparisons:

Generalizations & Extensions (2)

Algebraic number coefficients are automatically lifted to integers:

Find Puiseux series at branch points:

Options (1)

The setting of

is reflected in the third argument of a Root object:

Root isolation is performed the first time the numerical value of the root is needed:

The symbolic complex root isolation method is usually slower than the validated numeric one:

The root isolation method may affect the ordering of nonreal roots:

Applications (16)

Solve polynomial equations of any degree in closed form in terms of Root:

Solve the characteristic equation of a Hilbert matrix:

Using Eigenvalues:

Find the minimum of a parameterized polynomial:

Solve a constant coefficient differential equation of any degree:

Solve a constant coefficient difference equation of any degree:

Resolve a piecewise function:

Solve univariate exp-log equations and inequalities over the reals:

Solve univariate elementary function equations over bounded intervals and regions:

Solve univariate analytic equations over bounded intervals and regions:

Find real roots of high-degree sparse polynomials and algebraic functions:

Solve univariate transcendental optimization problems:

Integrate a piecewise function with an exp-log inequality condition:

Evaluate the hard hexagon entropy constant:

Solve Kepler's equation:

Compute the Laplace limit constant:

Plot a root as a function of a parameter:

Properties & Relations (7)

Extract the polynomial from a Root object:

Series expansions of implicit solutions to equations:

Use RootReduce to canonicalize algebraic numbers:

Simplify combinations of Root objects:

Reduce an equation for a parameter in a Root object:

Use RootApproximant to generate Root objects from numbers:

Roots are numeric expressions:

Possible Issues (4)

Series at branch points may not be valid in all directions:

Canonicalization is only possible for parameter-free roots:

Parameterized roots can have complicated branch cuts in the complex parameter plane:

A non-polynomial Root object may represent a cluster of distinct roots:

Numerical computation with a higher precision yields an approximation of one of the roots:

The choice of root stays the same for subsequent computations:

Neat Examples (1)

A high power of a Pisot number that is nearly an integer: