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From: Rainer von Seggern on 28 Apr 2010 08:33
On 24 Apr., 12:27, matt <n...(a)none.com> wrote:
> Dear all,
>
>    I would like to be able to estimate _numerically_ the eigenvalues (ev)
> of a matrix M(x) as x changes, and display them in a plot. Let me use
> Matlab notation to further clarify what I'm trying to achieve.
>
> Let us suppose that I have M = [x-1, 3; x^2, x+4]. I would give some
> values to x, e.g x = -3:.01:3, and for each of these values I would
> update Mx_ev(:,i) = eig([x(i)-1, 3; x(i)^2, x(i)+4]), being Mx_ev a
> matrix of 2x601 storing the evolution of the ev of M(x) as x changes.
> Unfortunately, nothing guarantees that the "corresponding" ev always
> appear in the same position at different values of x: they may well
> exchange their positions for some x, showing crossing on the plot at
> those x.
>
> An obvious workaround to this is explicit sorting. But I would like to
> avoid it, because requires a significant amount of computation. And by
> the way, how am I supposed to check the occurrency of discontinuties in
> the ev evolutions, without knowing their rate of rise (or fall) in
> advance? Although checking for a variation of 2-3% from the previous
> value worked correctly for all the cases I tried, I suppose there must
> be a better (more accurated) way to checking for discontinuities.
>
> So the question is: do you know any algorithm that, starting from very
> good estimations, Mx_ev(:,i), computes the ev of M(x(i+1)), namely
> Mx_ev(:,i+1), with the same order in which the estimates are given?
>
> I tried the Shifted-QR algorithm with deflation, but without success. As
> an example, consider the following matrix (A) standing for M(x(i+1)).
>
> A =
>
>     0.0015 - 0.1645i  -0.0054 + 0.0012i
>     0.0032             0.0626 + 0.0122i
>
> We have the following estimate (s) of its second ev, i.e. we know that
> the second ev of M(x(i)), namely Mx_ev(2,i), is
>
> s =
>
>     0.0016 - 0.1684i
>
> What I was expecting is that, after some iteration, the Shifted-QR
> algorithm
>
>  >> n = size(A,1); I = eye(n,n);
>  >> [Q,R] = qr(A - s*I); A = R*Q + s*I, s = A(n,n),
>
> returned 0.0015 - 0.1646i, i.e. the continue evolution at x(i+1) of the
> second ev of M(x(i)), namely Mx_ev(2,i+1), in A(2,2). Instead, I got
>
> A =
>
>     0.0015 - 0.1646i  -0.0019 + 0.0024i
>          0             0.0626 + 0.0123i
>
> which shows clearly that the initial estimate 0.0016 - 0.1684i converged
> to 0.0626 + 0.0123i (the other ev) instead of 0.0015 - 0.1646i

Hi Matt
Perhaps it may be of interest for you that the matrix
M = [x-1 3; x**2 x+4]
has for real x only real eigenvalues since the characteristic
polynomial may be written in the form:
P(x,t) = (t - ((2*x+3)**2)/2)**2 - (3*x**2 + 25/4)
I hope this helps,
Rainer
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