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Root

Status: Stable

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

Description

Root[Function[t, p[t]], k]
    Represents the k-th root of the univariate polynomial p in the
    variable t. k is canonical: real roots first ascending, then
    complex roots ordered by Re ascending, |Im| ascending, with the
    negative-Im member of each conjugate pair first. N[Root[..]]
    and N[Root[..], prec] return a numerical approximation via a
    companion-matrix + Sturm + Newton pipeline.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= N[Root[Function[#^3 - 2 # - 5], 1], 30]
Out[1]= 2.094551481542326591482386540579

In[2]:= N[Root[Function[#^3 + # + 1], 1], 20]    (* real root first *)
Out[2]= -0.682327803828019327372

In[3]:= N[Root[Function[#^3 + # + 1], 2], 20]    (* conj pair: -Im first *)
Out[3]= 0.341163901914009663686 - 1.16154139999725193609*I

In[4]:= N[Root[Function[#^3 + # + 1], 3], 20]
Out[4]= 0.341163901914009663686 + 1.16154139999725193609*I

Implementation notes

Algorithm. Root is a held symbolic form. builtin_root unconditionally returns NULL — the symbol carries ATTR_HOLDALL | ATTR_PROTECTED, so by the time the builtin is reached the evaluator has already left the call verbatim, and there is nothing to compute. Root[Function[t, p[t]], k] denotes the k-th root of the univariate polynomial p in canonical ordering (real roots first ascending, then complex roots by ascending Re, ascending |Im|, negative-Im member of each conjugate pair first). The useful work happens in callers: N[Root[..]] / N[Root[..], prec] numericalise via a companion-matrix + Sturm + Newton pipeline (src/root_numeric.c); ToRadicals (src/radicals.c) converts a held Root of degree ≤ 4 (or a binomial of higher degree) into closed-form radicals, selecting the k-th root by matching against the numeric value; D[RootSum[...], x] threads through the body (src/deriv.c); and the rational integrator's NaiveLogPart fallback (src/intrat.c) constructs RootSum nodes when the logarithmic part has no closed-form real expression.

This file also implements the companion head RootSum. builtin_rootsum is not purely held: rootsum_try_lagrange recognises the post-differentiation shape Function[a(#)/(d'(#)(x-#))] and collapses it via the Hermite/Lagrange interpolation identity Σ_i a(α_i)/(d'(α_i)(x-α_i)) = a(x)/d(x) to the closed rational function a(x)/d(x), using find_x_minus_slot1, subst_slot1, D, and internal_together/internal_cancel. root_make_rootsum builds the Mathematica-canonical Slot[1] form, rewriting a bound variable to Slot[1] throughout.

Data structures. Expr* trees only. The held form is Root[Function[poly_in_t], k]; RootSum is RootSum[Function[poly], Function[body]] in either Slot[1] or 2-arg bound-variable form.

Attributes: HoldAll, 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]:= Root[#^2 - 2 &, 1]
Out[1]= Root[#1^2 - 2 &, 1]

In[2]:= N[Root[#^2 - 2 &, 2], 40]
Out[2]= -1.4142135623730950488016887242096980785697

In[1]:= N[Root[#^5 - # - 1 &, 1], 40]
Out[1]= 1.1673039782614186842560458998548421807206

In[1]:= {N[Root[#^3 - 2 &, 1], 30], N[Root[#^3 - 2 &, 2], 30], N[Root[#^3 - 2 &, 3], 30]}
Out[1]= {1.259921049894873164767210607278, -0.6299605249474365823836053036392 - 1.09112363597172140356007261419*I, -0.6299605249474365823836053036392 + 1.09112363597172140356007261419*I}

In[1]:= {N[Root[#^4 + # + 1 &, 1], 20], N[Root[#^4 + # + 1 &, 2], 20], N[Root[#^4 + # + 1 &, 3], 20], N[Root[#^4 + # + 1 &, 4], 20]}
Out[1]= {-0.72713608449119683998 - 0.430014288329715776416*I, -0.72713608449119683998 + 0.430014288329715776416*I, 0.72713608449119683998 - 0.934099289460529439642*I, 0.72713608449119683998 + 0.934099289460529439642*I}

Notes

Root[f, k] is the exact, held representation of the k-th root of a univariate polynomial — including roots like that of #^5 - # - 1 &, the classic example of a quintic with no radical solution (Abel–Ruffini), which Root nonetheless names and evaluates to arbitrary precision. The index k is canonical: real roots come first in ascending order, then complex roots ordered by real part, magnitude of imaginary part, and finally the negative-imaginary member of each conjugate pair. N[Root[..], prec] drives a companion-matrix + Sturm + Newton pipeline to the requested precision; the #^3 - 2 & and #^4 + # + 1 & examples recover the full set of real and complex roots in the canonical order.