Solving Cubic Equations with Chebyshev Roots

In summary, the discriminant of a cubic polynomial can be used to determine the number of real and complex roots. The Chebyshev root method can be used to solve a general cubic equation, but it may result in imaginary values for the variable t. To avoid this, the function S_{1\over3}(t) = iC_{1\over3}(-it)-iC_{1\over3}(it) can be used, which is a real function of a real variable and has no singularities along the real axis.
  • #1
junglebeast
515
2
Say I have a monic polynomial,

x^3 + ax^2 + bx + c

with a=-2.372282, b=1.862273, c=-0.483023

The discriminant is given by

a^2 b^2 - 4 b^3 - 4 a^3 c - 27 c^2 + 18 ab c

which is < 0, indicating 1 real root and 2 complex conjugates.

A method for solving a general cubic using the Chebyshev root is explained here,
http://www.statemaster.com/encyclopedia/Cubic-equation

but (a^3 - 3b) is negative, which means that "t" will be imaginary. But t is then used in the Chebyshev cubic root, which is only defined for real numbers [-2, inf].
 
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  • #2
The site you give does NOT say that the Chebyshev cubic root is "only defined for real numbers [-2, inf]". It gives simple formulas for the root in the case that the argument is in that interval, and then tells how to expand it to complex arguments.
 
  • #3
EDIT - deleted previous response.

Let me clarify my remaining confusion from here:
http://www.exampleproblems.com/wiki/index.php/Cubic_equation

They define s = t^2, then say

If s < 0 then the reduction to Chebyshev polynomial form has given a t which is a pure imaginary number. In this case [tex]i C_{1\over3}(-it)-iC_{1\over3}(it)[/tex] is the sole real root. We are now evaluating a real root by means of a function of a purely imaginary argument; however we can avoid this by using the function

[tex]S_{1\over3}(t) = iC_{1\over3}(-it)-iC_{1\over3}(it) = 2 \operatorname{sinh}\left(\operatorname{arcsinh}\left({t\over2}\right)/3\right) [/tex],

which is a real function of a real variable with no singularities along the real axis.

Well, as they already pointed out t is imaginary, but then they say [tex]S_{1\over3}(t)[/tex] is a function of a real variable! But it's still defined in terms of t only! So how can I calculate [tex]S_{1\over3}(t)[/tex] given that I can't compute the imaginary t?
 
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1. What are Chebyshev cubic roots?

Chebyshev cubic roots, also known as Tschirnhaus transformations, are a set of equations used to solve cubic equations. They were discovered by Russian mathematician Pafnuty Chebyshev in the 19th century.

2. How are Chebyshev cubic roots derived?

Chebyshev cubic roots are derived through the use of a substitution method, where the original cubic equation is transformed into a simpler form. This involves replacing the x-term in the equation with a new variable, u, and solving for u.

3. What is the significance of Chebyshev cubic roots?

Chebyshev cubic roots have significant applications in mathematics and physics. They are used to find the roots of cubic equations, which can be used to solve various real-world problems, such as determining the trajectory of a projectile or finding the volume of a cube.

4. Are there any limitations to using Chebyshev cubic roots?

While Chebyshev cubic roots are a powerful tool for solving cubic equations, they do have some limitations. They cannot be used to solve equations with complex roots, and they may not provide all possible solutions to a given equation.

5. How can Chebyshev cubic roots be applied in engineering?

Chebyshev cubic roots have various applications in engineering, such as in control systems, signal processing, and circuit design. They can also be used to model and analyze physical systems, such as mechanical and electrical systems.

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