7-Bedrock

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Cubic equation with complex coefficients

Forum|Forum|9 years ago
March 1, 2017
4 replies
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Hello everyone !

For the cubic equation with real coeffs there are Cardano relations to find out roots. Or the use of trigonometric substitution.

Do you know similar formulas for the case of complex coefficients ?

Google is pointing to a paper in Soviet (now : Russian) Journal of Numerical Analysis & Mathematical Modelling, vol. 4(1991)/uncertain year/ (No. 2), p.315, which is out of my reach for the time being.

Thank you & best regards, Liv.

Statistics_Analysis

Best answer by VladimirN

Please find this article in attachment.

L

LucMeekes

23-Emerald IV

Why not let Mathcad solve it:

the solution is a bit too wide to show here (like some 15 pages) The middle part is

It ends like:

Just a specific example:

The full symbolic solution is:

Luc

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Liv

Author

7-Bedrock

Oh, ok, it's an idea ! Thank you !

Forgot completely that there is symbolic calculation available. Should have thought about....

I'll try to make that work for my application.

(For the sake of completeness, let me recall polyroots function, which is able to provide the roots for both real or complex coefficients cases).

However, still curious about relations given in the mentioned Russian journal.

You got it in Prime or Mcd15 ? If in Mcd15 how u managed to display all that when solving w(x) symbolically for roots, because I'm getting in Mcd15 the message " The symbolic result returned is too large to display, but it can be used in subsequent calculations if assigned to a function or variable " .

Best, Liv.

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LucMeekes

23-Emerald IV

This was done in Mathcad 11... I can have my symbolic results virtually as large as I like. I believe that facility was not supported after Mathcad 11.

But in Mathcad 15 you should be able to use the results, even if you cannot see them.

In case you have a specific equation you want the full symbolic result of, I can do it and copy that to another expression. Then you can use it in Mathcad 15.

Success!
Luc

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LouP

12-Amethyst

The algebraic solutions/formulas for the cubic equation should remain valid for complex coefficients as well as for real coefficients. I'm not aware of anything in the process of finding these formulas that depends on the coefficients being real. Some assertions as to certain terms being real or imaginary may be depend on the coefficients being real, but not the formulas themselves.

Lou

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Liv

Author

7-Bedrock

Agree... & wondering the same initially : why & what step of Cardano procedure might be inadequate for complex coefficients in Mathcad. I'll try to look better later for that. I've tested for the moment Cardano formulas f real coefficients and such as they provide wrong roots in the case of complex coefficients (but one root my be correct if one of the coefficients is real).

Best, Liv

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LouP

12-Amethyst

Liv,

I tried a simple example (attached) in mathcad 11 using complex coefficients (almost same values as Luc), and found that symbolic and numeric solutions agreed.Can you post your example where you think the solutions for complex coefficients are incorrect?

For some reason, the symbolic evaluations in Luc's file didn't work in my Mathcad.

Lou

1_cubic_example_1(LP).pdf