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Volume 9, Issue 3


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In and Out
Edited by Paul Abbott

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 ( 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 ( 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.

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