# 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, , 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 vecor $v$ 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 eignevectors 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 .