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ExampleSystems :: randomGeneralizedEigenvalueProblem

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

Synopsis

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 :

For the programmer

The object randomGeneralizedEigenvalueProblem is a method function.