Newbie question about eigenvalues calculating

> A polynomial of degree 5 is in general not solvable by radicals, so 
you
> cannot expect maxima (or others) to give exact formulas.
> But you may find approximate values using allroots(charpoly(...));

I thought eigenvalues does the task automatically.

>
> For instance this gives me the eigenvalues for 50 instances of your
> matrices in less than one second of computation :
>
> define(B0(m), genmatrix (lambda ([i, j], diff(m^(j-1),m,i-1)/(i-1)!),
> 5,5))$
> define(B1(m), genmatrix (lambda ([i, j], diff(m^(j+4),m,i-1)/(i-1)!),
> 5,5))$
> A0(M):= genmatrix (lambda ([i, j], 1.0*sum(k^(i+j+3),k,0,M)), 5,5)$
> A1(M):= genmatrix (lambda ([i, j], 1.0*sum(k^(i+j+8),k,0,M)), 5,5)$
> U(M,m):=B0(m) - B1(m).(invert(A1(M))).A0(M)$
> eigen(M,m):=allroots(charpoly(U(M,m),x))$
> for M:5 thru 9 do for m:1 thru 10 do print(eigen(M,m));
>
> For larger values of m (try M=6, m=15), allroots cannot find all the
> roots (might be a problem of precision, some coefficients in the
> characteristic polynomial being quite large).
>
> Eric Reyssat
>
>
>
> Hiisi a ?crit :
> > Dear list! I recently installed Maxima for my university studying. I
> > study S-splines and have to calculate optimal parameters for them.
> > Here's description of the task.
> > A0:
> >
> 
matrix([S5,S6,S7,S8,S9],[S6,S7,S8,S9,S10],[S7,S8,S9,S10,S11],[S8,S9,S10,S11,S12],[S9,S10,S11,S12,S13]);
> > A1:
> >
> 
matrix([S10,S11,S12,S13,S14],[S11,S12,S13,S14,S15],[S12,S13,S14,S15,S16],[S13,S14,S15,S16,S17],[S14,S15,S16,S17,S18]);
> > where Si: sum(k^i,k,0,M); and k is a range from 0 to some K; M is a
> > number.
> > B0:
> >
> 
matrix([1,m,m^2,m^3,m^4],[0,1,2*m,3*m^2,4*m^3],[0,0,1,3*m,6*m^2],[0,0,0,1,4*m],[0,0,0,0,1]);
> > B1:
> >
> 
matrix([m^5,m^6,m^7,m^8,m^9],[5*m^4,6*m^5,7*m^6,8*m^7,9*m^8],[10*m^3,15*m^4,21*m^5,28*m^6,36*m^7],[10*m^2,20*m^3,35*m^4,56*m^5,84*m^6],[5*m,15*m^2,35*m^3,70*m^4,126*m^5]);
> > where m is a number.
> > Matrix U is B0 - B1.(invert(A1)).A0;
> > The task is to calculate the eigenvalues of U. I need to do it with
> > different m and M to find out the best couple. The problem is: I 
can't
> > do it. When I'm trying eigenvalues(U); Maxima is calculating but it
> > takes so long time and I didn't saw results yet even when it's been
> > running all night. I was trying to calculate eigenvalues of A0 for
> > example but received the following error: solve is unable to find 
the
> > roots of the characteristic polynomial. Is it too big or am I doing
> > something wrong?
> > Thanks for Re:!

Nice answer. Grate help. Now I need to present the results in a 
user-friendly form. I need to calculate maximum of all roots for each 
couple of m and M. Does allroots function returns list? Could I use 
maxi(eigenv(M, m))? I also need to calculate m/M ration. But it returns 
fraction like 3/5 when I need decimal fraction. How to do that?
Thanks for Re:!
--
Hiisi.
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