rootfamily

Rootfamily is a term used in some mathematical and computational contexts to refer to a structured collection of roots associated with a polynomial, equation, or transformation. It is not a standard mathematical term; its meaning is defined by the workflow or software in which it appears.

Definition and scope: A rootfamily groups roots by a common object, such as the polynomial they come from, a parametric family, or a numerical approximation. Each entry may include the root value, multiplicity, algebraic or numeric representation, the field of definition, and any related transformation applied to track root movement under perturbation.

Construction and interpretation: Rootfamilies can be formed by direct computation of the polynomial's roots, by continuation methods as coefficients vary, or by symbolic-numeric methods that track roots through algebraic transformations. In root-tracking contexts, a base rootfamily might be extended as parameters change, producing a family of roots.

Examples: For p(x) = x^3 - 3x + 2, a rootfamily associated with p would include the roots x = 1 (multiplicity 2) and x = -2 (multiplicity 1) with their multiplicities and possibly their numerical approximations. In a parametric setting, the polynomial p_t(x) = x^3 - 3t x + 2t could have a rootfamily that tracks the roots as t varies from 1 to 2.

Applications: Rootfamilies are used in computer algebra systems, numerical analysis, and algebraic geometry to organize and compare roots across models, to facilitate root-tracking under perturbations, and to support symbolic-numeric computation.

Limitations and terminology: Because “rootfamily” is not standardized, definitions vary. When adopting this term, users should provide explicit definitions and data schemas in documentation or code.

See also: root, polynomial, root-finding, algebraic equation, root-locus, continuation method.