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ToRadicals

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

ToRadicals[expr]
    attempts to express all Root objects in expr in terms of radicals.

ToRadicals can always give expressions in terms of radicals when the
    highest degree of the polynomial that appears in any Root object is
    four.  Binomial Root objects of the form Root[Function[a #^n + b], k]
    are also reduced to radicals for any degree n.  Other Root objects
    of degree five or higher are returned unchanged.
If Root objects in expr contain parameters, ToRadicals[expr] may yield
    a result that is not equal to expr for all values of the parameters.
ToRadicals automatically threads over lists, equations, inequalities,
    and logic functions.

Examples

No verified examples yet for this function.

Implementation notes

Algorithm. builtin_to_radicals (builtin_to_radicals) converts held Root[Function[poly], k] objects into closed-form radical expressions. The top-level walker is a structural recurrence that rebuilds every EXPR_FUNCTION node, so a Root buried inside List/Equal/Less/And/Or/... is handled identically.

Per Root node: (1) extract the polynomial body, accepting both the Slot[1] form Function[expr] and the bound-variable form Function[t, expr]; (2) substitute the slot/variable with a fresh symbol x$ so the standard get_coeff/get_degree_poly univariate machinery applies; (3) dispatch on degree d — d=1 linear (-c0/c1), d=2 quadratic formula, d=3 Cardano, d=4 Ferrari (depressed quartic + resolvent cubic), d≥5 only the binomial fast-path a·x^n + b, otherwise the Root is left untouched — each path producing all d radical roots as a fresh Expr**; (4) select the k-th root in Mathilda's canonical Root ordering by computing N[Root[poly, k]] at machine precision (root_numericalize) and picking the radical root closest in the complex plane, falling back to the natural per-formula order (index k-1) when coefficients are parametric and numeric evaluation is unavailable.

Data structures. Expr* trees; degree dispatch reuses the polynomial coefficient extractors from src/poly/poly.c. Intermediate radical-expression bookkeeping rides on eval_and_free and the Plus/Times/Power normalisation. Inputs are borrowed and deep-copied into the output.

Complexity / limits. Closed forms exist only up to degree 4 (Abel–Ruffini); degree ≥ 5 is supported solely for binomials. Root selection costs one numeric Root evaluation per node.

  • Protected.
  • Closed-form radicals are always returned when the polynomial has degree

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= ToRadicals[Root[#^2 - 2 &, 1]]
Out[1]= -Sqrt[2]

The smaller root of x^2 + x - 1 is the negative reciprocal of the golden ratio, recovered exactly in radicals:

In[1]:= ToRadicals[Root[#^2 + # - 1 &, 1]]
Out[1]= 1/2 (-1 - Sqrt[5])

Cardano's formula appears automatically for the real root of a depressed cubic:

In[1]:= ToRadicals[Root[1 + #1 + #1^3 &, 1]]
Out[1]= -1/3 ((1/2 (27 + 3 Sqrt[93]))^(1/3) - 3/(1/2 (27 + 3 Sqrt[93]))^(1/3))

It threads through the implicit Root objects produced by Solve, here giving the three cube roots of two:

In[1]:= ToRadicals[Solve[x^3 - 2 == 0, x]]
Out[1]= {{x -> 2^(1/3)}, {x -> -(-1)^(1/3) 2^(1/3)}, {x -> (-1)^(2/3) 2^(1/3)}}

Notes

ToRadicals[expr] rewrites Root objects in expr using radicals. It always succeeds when the underlying polynomial has degree at most four (and for binomial Root[a #^n + b &, k] of any degree); degree-five-and-higher Root objects are returned unchanged. It threads automatically over lists, equations, and the results of Solve.