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Re: How can I calc eigenvalues to fourier series?
Posted:
Jun 2, 2011 3:15 PM
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On Jun 2, 9:37 am, Marco Esteves <marcosilvaeste...@gmail.com> wrote:
> Sometimes you've got to calculate eigenvalues to solve fourier series.
> I don't know how to calc in matlab...
> Example of eigenvalues calc problem:
>
> a*d/lambda*sin(lambda/d)+b*cos(lambda/d)=0
>
> my inputs are the constants a,b and d with a*b>0.
> my output is lambda,the eigenvalue. But lambda is an array, indexed at
> discrete time,so can be called lambda_n with n=1:infinity
>
> Thank you
The way you have written the problem, you need to solve for 1 root
(eigenvalue) at a time using a routine like "fzero". The problem of
finding all the eigenvalues this way is that you need a good initial
guess for each of them. For example, if you can bracket the
eigenvalues, meaning that you can define intervals in which each of
the eigenvalues must lie, then you can make progress because you can
choose your initial guesses intelligently. For this problem you can
do that. Divide through by COS and you'll see that you have
TAN(g*lambda) = g*lambda/h, where g and h are some constants. Plot
the LHS and the RHS versus the variable g*lamba. All intersections
correspond to eigenvalues you seek. You can use some basic asymptotic
analysis of the tangent function to find brackets for each of the
roots.
However, if this eigenvalue equation comes from some underlying
boundary value problem (i.e. an ODE), then a potentially better
procedure is to solve the ODE numerically. For example, do a finite
differencing of the ODE. If the ODE involving the eigenvalue is
linear and homogeneous then you will ultimately get a linear problem
of the form Ax = lambda*x (or perhaps a generalized eigenvalue
problem like A*x = lambda*B*x). The eigenvalues of the
discretization matrix A will then be the eigenvalues you seek, and the
eigenvectors of A the associated eigenfunctions of the boundary value
problem. Note that the boundary conditions will be incorporated into
your discretization matrix A.
Jim
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