“Arnoldi” method for Eigenvalues inside FindRoot

Question

I'm trying to implement a function which, given a matrix with one free parameter, would return the value of the parameter at which the lowest eigenvalue of the matrix is equal to a certain number.

Importantly, I'm planning to run this algorithm for extremely large sparse matrices, so I would like to use Arnoldi method.

Here's my attempt:

FitMat[matrix_, lowest_, param_, starting_, howmany_] := 
  Module[{mat, fu},
   mat[x_] := matrix /. {param -> x};
   fu[x_] := Min[Eigenvalues[mat[x], howmany
      , Method -> {"Arnoldi", "Criteria" -> "RealPart"}
      ]];
   Return[
    x /. FindRoot[fu[x] == lowest, {x, starting}
      ]
    ];
   ];

FitMat[( {
   {1, 2, 1},
   {3, 4, 1},
   {x, 4, 9}
  } ), -3, x, 50, 1]

This, however, results in the following error:

Eigenvalues::arm: Method -> Arnoldi can only be used for matrices of machine- or arbitrary-precision real numbers.

Please note that replacing mat[x_] and/or fu[x_] with mat[x_?NumericQ] and/or fu[x_?NumericQ] totally ruins the code, even if the Method specification is not used.

Could anyone please fix my solution or come up with a better one?

(Of course, the problem I'm trying to solve is highly non-linear; however, I typically do have a pretty good estimate for the value of starting. So, for small matrices the same code without specifying the Method works well.)

Answer 1

I think there are a few things going on.

First, as the error message says, the Arnoldi method requires machine- or arbitrary-precision real numbers. Your matrix is made of integers, hence the message.

Second, you do need fu[x_?NumericQ] to avoid FindRoot from prematurely evaluating fu without a number.

Finally, for your example, there appears to be no such root. If you insert

Print[Plot[fu[x], {x, -100, 100}, PlotRange -> All]];

into FitMat, you'll see

Clearly, this never equals -3, so FindRoot fails.

If you try

FitMat[({{1., 2., 1.}, {3., 4., 1.}, {x, 4., 9.}}), 15, x, 50, 1]

instead, you get the answer 63.0769 with no errors or messages.

EDIT:

To find where the smallest (most negative) eigenvalue equals -3 you can use the -1 option for the Arnoldi method together with a Shift. No Min needed!

FitMat[matrix_, lowest_, param_, starting_] := 
  Module[{mat, fu}, mat[x_] := matrix /. {param -> x};
   fu[x_?NumericQ] := 
    Eigenvalues[mat[x], -1, 
     Method -> {"Arnoldi", "Criteria" -> "RealPart", 
       "Shift" -> -1000}];
   Print[Plot[{fu[x], lowest}, {x, -100, 100}, PlotRange -> All]];
   Return[x /. FindRoot[fu[x] == lowest, {x, starting}]];];

FitMat[({{1., 2., 1.}, {3., 4., 1.}, {x, 4., 9.}}), -3, x, 50]

(* 52. *)