Cubic and Quartic Formulae

[Manolo Sobrino]

There are a few reasons. Check again the closed-form solutions. You want 3 or 4 several line expressions with nested radicals as answers on a calculator screen. That's not very useful, besides it would only look nice (not really) for integer or rational coefficients. That's the same reason why closed form eigenvalues are either trivial or you don't really want to see them.

And then, in the numerical world people don't use (or shouldn't be using) those expressions as they are numerically unstable. There are quite a few much better methods.

The interesting part about them is that they do exist and they are algebraic expressions, yet the quintic doesn't have one of this kind (Abel-Ruffini theorem). I guess you won't be using calculators to introduce Galois theory.


TI have actually given details on the nifty algorithm they used in their numerical polynomial root finder, which appeared first on the TI 85:

ftp://ftp.ti.com/pub/graph-ti/calc-apps/...lyroot.txt

Taking into account the constraints of the machine it was designed for and its intended use, all of this makes sense to me.