How do mathematica and maple solve cubics?

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Discussion Overview

The discussion revolves around how Mathematica and Maple solve cubic equations, focusing on whether these software tools utilize numerical methods, algebraic solutions, or both. Participants explore the complexities of symbolic computation and the underlying mathematical principles involved in solving cubics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants suggest that Mathematica can solve cubic equations both numerically and algebraically, depending on user instructions.
  • One participant notes the complexity of symbolic computation and mentions that advanced algebraic theorems are involved in the process.
  • Another participant expresses uncertainty about the specifics of how these programs handle the algebraic challenges, particularly in relation to the trisection of angles.
  • There is a discussion about the lack of specificity in the original question, with some participants encouraging the original poster to clarify their interest in numerical versus symbolic methods.

Areas of Agreement / Disagreement

Participants generally agree that both numerical and algebraic methods are employed by Mathematica and Maple, but there is no consensus on the specifics of how these methods are implemented or the challenges involved in solving cubics algebraically.

Contextual Notes

Some limitations include the original poster's vague question and the varying levels of understanding among participants regarding symbolic computation and its complexities.

Who May Find This Useful

This discussion may be useful for individuals interested in computational mathematics, particularly those exploring the capabilities of software like Mathematica and Maple in solving polynomial equations.

wumple
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Hi,

Does anyone know how mathematica and maple solve cubics? Do they do it numerically?
 
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Simon_Tyler said:
http://en.wikipedia.org/wiki/Cubic_function

Not to be rude, but that link by itself isn't terrifically helpful for him.


I don't know the answer to your question, but Mathematica can solve an equation numerically or algebraically, depending on the instructions you give it. Do you have a spefic problem in mind?
 
I once tried to learn the basic of symbolic computation but it's not easy, I quitted :) They use very advanced algebra theorems for convergences and so on. The numeric codes should be as complicated as the symbolic ones as well. They're all highly optimized.
 
dreiter said:
Not to be rude, but that link by itself isn't terrifically helpful for him.

I agree -- but so many times on this forum, you put in a lot of effort into an answer and never get a reply.

Notice that the OP has not replied to your request for more information about whether their interested in symbolics or numerics.

The OP posted a one line question which showed very little hint that they'd done any reading on the subject - so I assumed they were just wowed by the large algebraic mess that solving cubics gives. The wikipedia article gives a good discussion of the history and method.

If the OP then wanted to be more specific about numerics or say symbolic factorization over various rings, then they could have come back and clarified their interest.
 
Well my understanding is that solving a cubic algebraically comes down to trisecting an angle, which is of course not terribly doable...for example, I put in coefficients that should give me a real root of -1 for the depressed cubic

x^3 + ax + b = 0

but if I make the substitution x = u + v and go through the algebra, I come up with something of the form

x = ((1+z)^(1/3) + (1-z)^(1/3))

So how do these programs get around this issue?
 

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