Re: Problems with NSolve

*To*: mathgroup at smc.vnet.net*Subject*: [mg88167] Re: [mg88144] Problems with NSolve*From*: Carl Woll <carlw at wolfram.com>*Date*: Sun, 27 Apr 2008 04:57:11 -0400 (EDT)*References*: <200804260743.DAA08637@smc.vnet.net>

kgwagh at gmail.com wrote: >Hi all > >I am trying to find the number of eigenvalue crossings for a matrix as >a function of the parameter 'u', on which the elements of the >(symmetric) matrix depend on linearly. The matrix elements also >involve randomly chosen constants. The plan is to find the >distribution of the crossings of these type of matrices as I scan over >the random numbers. > >So far I have been using the following : > >NSolve[{chpoly[u, dim, \[Alpha], \[Gamma], \[Epsilon], x] == 0, > D[chpoly[u, dim, \[Alpha], \[Gamma], \[Epsilon], x], x] == 0}, {x, > u}, WorkingPrecision -> prec] > >where chpoly is the characteristic polynomial of the matrix (with the >eigenvalue variable being x) and \alpha, \gamma and \epsilon are >constant parameter arrays of random numbers. For a double (or higher) >degeneracy of the eigenvalues both the characteristic equation and its >derivative should be zero. This approach has worked successfully only >upto 12*12 matrices (where one such computation takes 40 secs on my >laptop). For 13*13 my laptop takes 4000 sec. This seems to be somewhat >surprising, because these polynomials are of the order 'n' (where n >is dimension of the matrix) in both u and x - and n=13 does not sound >very computationally unreasonable. So I was wondering if there was a >faster approach I could take. > >Also, the problem essentially entails me to know the number (not the >values) of the real solutions to this system of polynomials. >CountRoots seemed ideal but it does not work for more than one >equation. So is there any alternative along this route? > >Any other alternatives are also welcome. >Thanks, >Kshitij. > > I would find it helpful if you specified what chpoly, \[Alpha] etc were to help diagnose you're problem. However, it seems you are coming across an issue described in the following thread: http://tinyurl.com/53cp5u I believe there are other threads that discuss this issue as well, but I couldn't find them in a quick search. Carl Woll Wolfram Research

**References**:**Problems with NSolve***From:*kgwagh@gmail.com