How to define the number of roots of the transcendental equations?

Question

To obtain the natural frequency of the clamped beam, I need to solve the following equation:

Cos[x]Cosh[x]==1

I use the Solve function to do this. Since this transcendental equation has infinite roots, I prescribe the range of the roots. The code I write is

Solve[Cos[x] Cosh[x] == 1 && 0 < x < 500, x, Reals]

I obtain the roots belonging to [0,500]. Now I want to prescribe the number of the roots rather than the range of the roots. How can I do this?

You can find it with CountRoots[Cos[x] Cosh[x] - 1, {x, 0, 500}], however since it counts also multiple roots it might be resonable to use Solve, DeletDuplicates and finally Length. — Artes, Nov 07 '17 at 09:39
@Artes Thanks for your comments. I mean I want to obtain the roots with prescribed numbers. For example, I want to obtain the first 100 roots. — Ice0cean, Nov 07 '17 at 09:42
I do not think that most root finding methods can find real roots in increasing order, thus it wouldn't really make sense to have such a function. You can keep extending the interval until you have the desired number of roots. If you just want to find one root, whichever that may be, use FindRoot. — Szabolcs, Nov 07 '17 at 09:56
You can play with IsolatingInterval for polynomials, however there is no specific functionality for transcendental equations. Nevertheless you can do it simply with Solve and Take[...,100]. There might be a bit more efficient method, but in general, it strongly depends on functions you deal with, so you shouldn't expect a universal approach. — Artes, Nov 07 '17 at 10:02

score 3 · Accepted Answer · answered Nov 07 '17 at 10:19

Modifying Daniel's solution here, we have (where I only take the second to twelfth roots, since we already know $x=0$ is a root):

k = 0;
Reap[NDSolve[{y'[x] == D[Exp[-x] (1 - Cos[x] Cosh[x]), x], y[0] == 0, 
              WhenEvent[y[x] == 0, Sow[x]; k++; If[k >= 11, "StopIntegration"]]},
             {}, {x, 0, ∞}, WorkingPrecision -> 20]][[-1, 1]]
{4.7300407456851316296, 7.8532046220416923611, 10.995607838630982039,
 14.137165489317331787, 17.278759656830400174, 20.420352243844582474,
 23.561944901469892955, 26.703537553710273995, 29.845130209052511595,
 32.986722860941260188, 36.128315516208914460}

Note the introduction of the damping $\exp(-x)$ factor for stability. The roots so obtained can be polished further with FindRoot[] if need be.

How to define the number of roots of the transcendental equations?

1 Answers1