Sorta Shifts

The Arnoldi process does not allow for shifts.  More accurately you can put them in all you want but they do not alter the computation at all in exact arithmetic.

The question is:  “Are there any things like shifts that I can put in to the Arnoldi process and alter (hopefully improve in some way) the iteration.  I have a dumb idea of using low rank matrices to perturb A.

n=12;
A=RandomReal[{-1,1},{n,n}];
C2R[z_]:={Re[z],Im[z]}
ones=ConstantArray[1,n];
onepm=Table[(((-1))^i),{i,1,n}];

Data1= Transpose[
Table[
ϵ=(0.8^n);
Map[C2R,Eigenvalues[A+ϵ Outer[Times,ones,ones]]],
{n,3,12}]];
Data2= Transpose[
Table[
ϵ=(0.8^n);
Map[C2R,Eigenvalues[A+ϵ Outer[Times,onepm,onepm]]],
{n,3,12}]];
Show[
ListPlot[Map[C2R,Eigenvalues[A]],
PlotStyle->{Red, PointSize[0.02]}],
ListPlot[Data1],
ListPlot[Data2],
PlotRange->All,
AspectRatio->Automatic
]

n=12;
A=RandomReal[{-1,1},{n,n}];
C2R[z_]:={Re[z],Im[z]}
ones=ConstantArray[1,n];
onepm=Table[(((-1))^i),{i,1,n}];

Data3= Flatten[
Table[
Map[C2R,Eigenvalues[A+ϵ  Outer[Times,ones,onepm]+δ Outer[Times,onepm,ones]]],
{ϵ,-0.1,0.1,0.01},
{δ,-0.1,0.1,0.01}],1];

Dimensions[Data3]

Show[ListPlot[Data3],
ListPlot[Map[C2R,Eigenvalues[A]],
PlotStyle->{Red, PointSize[0.02]}],
PlotRange->All,
AspectRatio->Automatic
]

(0.8^4)

n=12;
A=RandomReal[{-1,1},{n,n}];
C2R[z_]:={Re[z],Im[z]}
onepm=Table[(((-1))^i),{i,1,n}];

Data= Transpose[
Table[
Map[C2R,Eigenvalues[A+ϵ Outer[Times,ones,ones]]],
{ϵ,0,1,0.01}]];
ListPlot[
Data
]

C2R[Eigenvalues[A+ϵ Outer[Times,ones,ones]]]