Cubic and Quartic Formulae

[LCieParagon]

(08-06-2015 09:01 AM)Manolo Sobrino Wrote:  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.

Actually, I do introduce calculators when speaking about Galois Theory. I teach differential equations, Calculus 3, Calculus 2, and Pre-Calculus. I explain that there are certain forms of quintics (and higher degrees) that can be solved numerically. I have a chart in my classroom regarding this.

Additionally, parisse noted that it's more practical to use approximations. I disagree with this. It's in fact a portion of the Ti-nSpire CAS (and the TI-89 for that matter) I prefer to the HP Prime. For instance (if we are talking about degrees), one can find sin(36), sin (18), sin(54), and sin(72) exactly. This angles are incredibly useful for architecture or engineering. One can apply half-angle, addition/subtraction, and double angle formulae to receive every whole number degree divisible by 3.

But it gets better. One can find the numerical value of sin(1) by applying the cubic formula. One can solve equations like sin(3x) = 3sin(x) - 4sin^3(x) by letting x = 1 and find the exact answers.

Additionally, if decide to take logarithms of these numbers (like sin(1) because logarithms of numbers close to 0+ diverge to -inf), we will see that our error will be huge if we use approximations. If we use exact answers (by simply using the copy/paste commands), our answers will be significantly more accurate, if not exact.