randomGeneralizedEigenvalueProblem -- an example of a 0-dimensional square polynomial system

Synopsis

Usage:

(T,S,solsS) = randomGeneralizedEigenvalueProblem(n,K)
Inputs:
- n, an integer, positive value representing the number of polynomials
Outputs:
- T'S'solsS, a sequence, target system T, start system S, and solutions solsS to the start system

Description

Given $n\times n$ matrices $A$ and $B$, a number $\lambda$ is a generalized eigenvalue if there is a nonzero vector $v$ such that $A x = \lambda B x$.

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 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.

i1 : randomGeneralizedEigenvalueProblem 3

o1 = ({(- .741046 - .108386*ii)lambda*x1 + (- .830833 - .538554*ii)lambda*x2
     ------------------------------------------------------------------------
     + (- .191734 - .403215*ii)lambda*x3 + (.892712 + .673395*ii)x1 + (.89189
     ------------------------------------------------------------------------
     + .231053*ii)x2 + (.0741835 + .808694*ii)x3, (- .348931 -
     ------------------------------------------------------------------------
     .562428*ii)lambda*x1 + (- .873665 - .415912*ii)lambda*x2 + (- .615911 -
     ------------------------------------------------------------------------
     .0147867*ii)lambda*x3 + (.29398 + .632944*ii)x1 + (.461944 +
     ------------------------------------------------------------------------
     .775187*ii)x2 + (.362835 + .706096*ii)x3, (- .246268 -
     ------------------------------------------------------------------------
     .153346*ii)lambda*x1 + (- .606588 - .848005*ii)lambda*x2 + (- .223028 -
     ------------------------------------------------------------------------
     .388829*ii)lambda*x3 + (.0258884 + .714827*ii)x1 + (.909047 +
     ------------------------------------------------------------------------
     .314897*ii)x2 + (.127435 + .254482*ii)x3, - x1 - x2 - x3 + 2},
     ------------------------------------------------------------------------
     {lambda*x1 - lambda - x1 + 1, lambda*x2 - lambda + (.5 - .866025*ii)x2 -
     ------------------------------------------------------------------------
     .5 + .866025*ii, lambda*x3 - lambda + (.5 + .866025*ii)x3 - .5 -
     ------------------------------------------------------------------------
     .866025*ii, - x1 - x2 - x3 + 2}, {{1, 0, 1, 1}, {-.5+.866025*ii, 1, 0,
     ------------------------------------------------------------------------
     1}, {-.5-.866025*ii, 1, 1, 0}})

o1 : Sequence

i2 : randomGeneralizedEigenvalueProblem 5

o2 = ({(- .977573 - .212436*ii)lambda*x1 + (- .205375 - .276652*ii)lambda*x2
     ------------------------------------------------------------------------
     + (- .130004 - .522285*ii)lambda*x3 + (- .971588 - .249992*ii)lambda*x4
     ------------------------------------------------------------------------
     + (- .581271 - .640172*ii)lambda*x5 + (.557119 + .873708*ii)x1 +
     ------------------------------------------------------------------------
     (.169813 + .965004*ii)x2 + (.350611 + .379495*ii)x3 + (.184779 +
     ------------------------------------------------------------------------
     .370833*ii)x4 + (.305423 + .732358*ii)x5, (- .592747 -
     ------------------------------------------------------------------------
     .831802*ii)lambda*x1 + (- .0958269 - .605398*ii)lambda*x2 + (- .171029 -
     ------------------------------------------------------------------------
     .340019*ii)lambda*x3 + (- .0328338 - .21113*ii)lambda*x4 + (- .919548 -
     ------------------------------------------------------------------------
     .560684*ii)lambda*x5 + (.7037 + .681869*ii)x1 + (.0647412 +
     ------------------------------------------------------------------------
     .877846*ii)x2 + (.237252 + .116721*ii)x3 + (.339222 + .062212*ii)x4 +
     ------------------------------------------------------------------------
     (.562839 + .629991*ii)x5, (- .501243 - .154289*ii)lambda*x1 + (- .883549
     ------------------------------------------------------------------------
     - .942865*ii)lambda*x2 + (- .818142 - .781168*ii)lambda*x3 + (- .611415
     ------------------------------------------------------------------------
     - .638389*ii)lambda*x4 + (- .847715 - .763255*ii)lambda*x5 + (.276259 +
     ------------------------------------------------------------------------
     .605659*ii)x1 + (.0340514 + .507989*ii)x2 + (.444183 + .644366*ii)x3 +
     ------------------------------------------------------------------------
     (.465736 + .40273*ii)x4 + (.479826 + .815167*ii)x5, (- .467203 -
     ------------------------------------------------------------------------
     .765564*ii)lambda*x1 + (- .0821679 - .10394*ii)lambda*x2 + (- .767433 -
     ------------------------------------------------------------------------
     .155346*ii)lambda*x3 + (- .0594513 - .0625324*ii)lambda*x4 + (- .477291
     ------------------------------------------------------------------------
     - .0048212*ii)lambda*x5 + (.96518 + .681683*ii)x1 + (.150294 +
     ------------------------------------------------------------------------
     .656391*ii)x2 + (.194945 + .518585*ii)x3 + (.164647 + .713493*ii)x4 +
     ------------------------------------------------------------------------
     (.97723 + .0595849*ii)x5, (- .305946 - .53632*ii)lambda*x1 + (- .280679
     ------------------------------------------------------------------------
     - .475179*ii)lambda*x2 + (- .0215389 - .283851*ii)lambda*x3 + (- .270725
     ------------------------------------------------------------------------
     - .075503*ii)lambda*x4 + (- .46452 - .16645*ii)lambda*x5 + (.914199 +
     ------------------------------------------------------------------------
     .887381*ii)x1 + (.174853 + .626892*ii)x2 + (.987173 + .568273*ii)x3 +
     ------------------------------------------------------------------------
     (.909537 + .566034*ii)x4 + (.0645275 + .283709*ii)x5, - x1 - x2 - x3 -
     ------------------------------------------------------------------------
     x4 - x5 + 4}, {lambda*x1 - lambda - x1 + 1, lambda*x2 - lambda + (-
     ------------------------------------------------------------------------
     .309017 - .951057*ii)x2 + .309017 + .951057*ii, lambda*x3 - lambda +
     ------------------------------------------------------------------------
     (.809017 - .587785*ii)x3 - .809017 + .587785*ii, lambda*x4 - lambda +
     ------------------------------------------------------------------------
     (.809017 + .587785*ii)x4 - .809017 - .587785*ii, lambda*x5 - lambda + (-
     ------------------------------------------------------------------------
     .309017 + .951057*ii)x5 + .309017 - .951057*ii, - x1 - x2 - x3 - x4 - x5
     ------------------------------------------------------------------------
     + 4}, {{1, 0, 1, 1, 1, 1}, {.309017+.951057*ii, 1, 0, 1, 1, 1},
     ------------------------------------------------------------------------
     {-.809017+.587785*ii, 1, 1, 0, 1, 1}, {-.809017-.587785*ii, 1, 1, 1, 0,
     ------------------------------------------------------------------------
     1}, {.309017-.951057*ii, 1, 1, 1, 1, 0}})

o2 : Sequence

Ways to use randomGeneralizedEigenvalueProblem :

randomGeneralizedEigenvalueProblem(ZZ)

For the programmer

The object randomGeneralizedEigenvalueProblem is a method function.