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export { "randomGeneralizedEigenvalueProblem" }
randomGeneralizedEigenvalueProblem = method()
randomGeneralizedEigenvalueProblem ZZ := n -> (
lambda := symbol lambda;
R := CC[lambda, vars(53..n+52)];
A := sub(random(CC^n,CC^n), R);
B := sub(random(CC^n,CC^n), R);
x := transpose matrix{drop(gens R,1)};
T := flatten entries (A*x-R_0*B*x) | {n - 1 - sum flatten entries x};
S := apply(n,i->(R_0-exp(ii*2*pi*(i/n)))*(x_(i,0)-1)) | {n - 1 - sum flatten entries x};
solsS := apply(n,i->point{{exp(ii*2*pi*(i/n))} | toList(i:1) | {0} | toList(n-i-1:1)});
(T,S,solsS)
)
beginDocumentation()
doc ///
Key
randomGeneralizedEigenvalueProblem
(randomGeneralizedEigenvalueProblem,ZZ)
Headline
an example of a 0-dimensional square polynomial system
Usage
(T,S,solsS) = randomGeneralizedEigenvalueProblem(n,K)
Inputs
n:ZZ
positive value representing the number of polynomials
Outputs
T'S'solsS:Sequence
target system T, start system S, and solutions solsS to the start system
Description
Text
Given $n\times n$ matrices $A$ and $B$,
a number $\lambda$ is a {\bf generalized eigenvalue}
if there is a nonzero vector $v$ such that $A x = \lambda B x$.
Text
This function creates a square target system representing the problem
for random $A$ and $B$ and a start system representing
the problem with eigenvalues that
are $n$-th roots of unity and the corresponding eigenvectors
form the standard basis.
This system was solved in May 2020, using @TO track@ in Macaulay2 v1.15
with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.
For a system of 3 polynomials there were 3 solutions found in 0.0764 seconds.
For a system of 5 polynomials there were 5 solutions found in 0.0478 seconds.
Example
randomGeneralizedEigenvalueProblem 3
randomGeneralizedEigenvalueProblem 5
///
-* TEST ///
F = first randomGeneralizedEigenvalueProblem 5
sols = solveSystem F
assert(#sols==5)
/// *-
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