Determinant of Characterisitic Polynomial

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Discussion Overview

The discussion revolves around the challenges of calculating the discriminant of the characteristic polynomial of a matrix with complex entries, specifically focusing on computational efficiency when dealing with larger matrices. Participants explore the performance of different software tools, including Mathematica and MATLAB, in this context.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes a lengthy computation process for evaluating the discriminant of a characteristic polynomial for a 10x10 matrix, suggesting that their code may be basic and seeking more efficient methods.
  • Another participant expresses skepticism about the time required for such calculations, sharing their experience of solving a characteristic polynomial for a smaller matrix quickly and questioning the complexity of the entries involved.
  • A third participant clarifies that while finding the characteristic polynomial is quick in Mathematica, the evaluation of the discriminant is what takes considerable time, especially with non-numeric entries.
  • One participant notes the requirement to use Mathematica for their research group, indicating a limitation in exploring alternative software solutions.
  • Another participant acknowledges the complexity of the discriminant of a polynomial and suggests that non-numeric entries may contribute to the lengthy computation time.

Areas of Agreement / Disagreement

Participants express differing views on the efficiency of software tools for calculating the discriminant, with some questioning the time required for computations and others affirming the challenges posed by complex entries. No consensus is reached regarding the best approach or solution.

Contextual Notes

Participants mention the potential impact of using non-numeric entries on computation time, as well as the limitations imposed by the requirement to use specific software. There are unresolved questions about the efficiency of different programming approaches and the nature of the matrix entries.

azuresky101
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I'm doing a calculation which finds the characteristic polynomial of a matrix, HH, with rather complex entries and then determines the discriminant of that polynomial. For smaller matrices up to around 7x7 it finishes evaluating the Discriminant command within a few hours, but at a 10x10, which is what I need, it takes days to evaluate. Running in parallel won't work for the built in Discriminant command. I was wondering if there was a more efficient way to evaluate the discriminant of high-order polynomial, as I know my code is basic.

CPHH = Collect[CharacteristicPolynomial[HH, x],x]];
DD = Discriminant[CPHH, x, Method -> SylvesterMatrix];
Simplify[DD, TimeConstraint -> Infinity]

If the rest of the code would be useful in determining a solution I'd be happy to post it

***Meant Discriminant in Title***
 
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Well, I have not had to do Matlab nor linear algebra since grad school (a couple of decades ago), so, I had to look up what Characteristic Polynomial is...I just saw a video in YouTube where the prof solved the Char. Poly for a 3x3 matrix in a few minutes...does it really take a few hours to solve a 7x7 in matlab?...I feel like I could solve it by hand in less than one!

What do you mean by "rather complex entries"?

Back then, MATLAB was not faster than a C or Fortran program; lately, I have seen postings in forums that seem to indicate that that continues to be the case...maybe you should write your own program to solve this.
 
Finding the characteristic polynomial takes seconds in mathematica. Its evaluating the discriminant of that polynomial that takes so long.

A typical entry in the 21x21 looks like this 1/2 h (61 h k1 - 159 h k2 - 14 g Hz uB). I forgot the particle is spin 10 so the hamiltonian is 21x21, not 10x10 like I said earlier.

Unfortunately it's required of me to use mathematica, as that is what everyone in the research group uses.
 
Oh, so it is mathematica and not matlab...same problem, I don't use either.

I looked up Discriminant of a Polynomial, now...I see that they can get pretty lengthy and you seem to be using non-numeric entries which I presume makes things more time consuming.

Anyway, good luck, I am afraid I can't help.
 

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