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The following code crashes the Kernel with no error message.

Solve[a (2 Sqrt[2] Sqrt[
     1 + b^2 + Sqrt[1 + b^2]] (m12 - Sqrt[1 + b^2] m12 + b m13) + 
    a b (d1 + Sqrt[1 + b^2] d1 + 2 d2 + 2 b m23)) == 
 2 Sqrt[1 + b^2] d1 b, 
b]

The equation is reproduced below. equation

Notice Solve is only solving for a single variable b, in a single equation, with no options or domain specification. The equation isn't really huge either.

Do you know why? More importantly, do you know a way around this? A solution where all variables are assumed to be real is sufficient.

Notes

  • It took between 2 and 3 hours for my Kernel to crash, but my computer is only a Mid 2014 MacBook Pro
  • I don't think it is a memory issue, as Activity Monitor didn't seem to record a red zone.
  • Mathematica version 13.0.1.0
  • macOS Big Sur, Version 11.7.9
  • MacBook Pro (Retina, 15-inch, Mid 2014)
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    $\begingroup$ I'm not getting a quick solution in V13.3. In general, if Mathematica returns a solution after running for a long time it will be very large. Do you think such a solution would be helpful to you? Perhaps you need to consider some other way of analysing your problem? (It looks like an eigenvalue type problem - perhaps you might look at what happens in a special case, e.g. a diagonal matrix) $\endgroup$ Commented Oct 20, 2023 at 16:56
  • $\begingroup$ It is easy... If all variables other than b are numeric then substitute them before solving for b. The other option is to get rid of roots to get polynomial equation in b. Then you get output in a second. $\endgroup$ Commented Oct 20, 2023 at 17:40

1 Answer 1

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Getting rid of roots I got this equation which is polynomial in b of degree 8 (which Solve has no problems to solve for b):

(1 + b^2) (4 a^2 b^2 d1^2 - 2 a^4 b^2 d1^2 + 8 a^2 b^2 d1 d2 - 
     4 a^4 b^2 d1 d2 - 8 a^2 b^2 m12^2 - 16 a^2 b^3 m12 m13 + 
     8 a^2 b^2 m13^2 + 8 a^2 b^3 d1 m23 - 
     4 a^4 b^3 d1 m23)^2 - (-4 b^2 d1^2 + 4 a^2 b^2 d1^2 - 
    2 a^4 b^2 d1^2 - 4 b^4 d1^2 + 4 a^2 b^4 d1^2 - a^4 b^4 d1^2 - 
    4 a^4 b^2 d1 d2 - 4 a^4 b^2 d2^2 + 8 a^2 b^2 m12^2 + 
    8 a^2 b^4 m12^2 + 8 a^2 b^2 m13^2 + 8 a^2 b^4 m13^2 - 
    4 a^4 b^3 d1 m23 - 8 a^4 b^3 d2 m23 - 4 a^4 b^4 m23^2)^2 == 0

It has quadruple root b=0, you can verify that also original equation has root b=0.

Then there are other four roots and you need to verify each of them whether they are also roots of original equation.

If you substitute this Thread[{a, d1, d2, m12, m13, m23} -> {4, 7, 2, 1, 6, 9}] to your original equation and solve for b you get solutions:

{b->0,b->-2.16191}

If you substitute the same to my equation you get solutions:

{b->0,b->0,b->0,b->0,b->-2.16191,b->-0.245123,b->-0.616311,b->-0.27024}

You see that two of them are the solution of the original equation.

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