Root versus Radicals

Q: Version 5 returns the eigenvalues of the following simple matrix in terms of Root objects.

What does this output mean? I prefer the explicit answer that Version 4 returned. How do I coerce Version 5 to return an explicit answer?

A: Daniel Lichtblau (danl@wolfram.com) answers: The Root form is a concise way of expressing algebraic numbers via the minimal polynomial they satisfy, along with a canonical ordering in the complex plane (specified by the second argument of Root). To get a radical form, you can use ToRadicals.

Clearly, this output is more complicated than the Root form.

In addition to size, there are other reasons to prefer the Root form:

It is typically faster to obtain.

It is numerically more stable to evaluate. In general, radical formulations are prone to numeric problems. Root objects do not have this liability.

When the roots of an irreducible cubic are all real but not rational, the so-called "casus irreducibilus" shows that they still must be expressed in terms of (mathworld.wolfram.com/CasusIrreducibilis.html). This means that numeric evaluation will give small imaginary parts unless, by happenstance, they exactly cancel. Small numeric error from round-off makes this unlikely.

For sufficiently complicated algebraics, it is often faster to evaluate the Root form numerically, at least at high precision.

Polynomial combinations of Root objects simplify using RootReduce.

Derivatives of Root objects with respect to a parameter are expressed in terms of Root objects. This is useful for eigenvalue sensitivity analysis.

So, one can avoid the Root form by using ToRadicals--but for all practical computation, you are better off working with Root objects.