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ExampleSystems :: randomGeneralizedEigenvalueProblem

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

Synopsis

Description

Given n×n matrices A and B, a number λ is a generalized eigenvalue if there is a nonzero vecor v such that A x = λB 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.

i1 : (T,S,solsS) = 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 : solsT = track(S,T,solsS)

o2 = {{.418168-.854494*ii, 6.15821-7.29267*ii, -18.0216+.27293*ii,
     ------------------------------------------------------------------------
     13.8634+7.01974*ii}, {.823136+.501455*ii, 1.32303-.221144*ii,
     ------------------------------------------------------------------------
     .278182-.58351*ii, .39879+.804654*ii}, {2.08136-.158678*ii,
     ------------------------------------------------------------------------
     .963218+.256557*ii, -.322407+1.34374*ii, 1.35919-1.60029*ii}}

o2 : List
i3 : #solsT

o3 = 3

Ways to use randomGeneralizedEigenvalueProblem :