Lowest Magnitude Eigenvalues of Large Sparse Matrices

Use the Arnoldi method with shift-inversion:

Eigenvalues[A, 3, Method -> {"Arnoldi", "Criteria" -> "Magnitude", "Shift" -> 0}]

gives you the three smallest eigenvalues by absolute value (by magnitude). See here: Efficiently find all values of parameter such that any of the eigenvalues of a matrix is equal to 1

After comments by @HenrikSchumacher it appears that the same can be achieved with

Eigenvalues[A, -3, Method -> {"Arnoldi", "Criteria" -> "Magnitude"}]

and there is no need for explicit shift-inversion.

And as @CarlWoll points out, this method is the default method for sparse matrices, so even

Eigenvalues[A, -3]

achieves the same effect. For non-sparse matrices, however, specifying the method can give a large speedup.

update

It now looks like you want the eigenvalues with smallest real part, not those with smallest magnitude. This you can also achieve with the Arnoldi method:

Eigenvalues[M, 3, Method -> {"Arnoldi", "Criteria" -> "RealPart"}]

gives the three eigenvalues with largest real part. To get the smallest ones, do

-Eigenvalues[-M, 3, Method -> {"Arnoldi", "Criteria" -> "RealPart"}]

The trick is that M and -M have the same eigenvalues (up to sign) and eigenvectors.

Of course this trick also works with the other methods presented above.

To calculate the lowest eigenvalues using Mathematica, I always introduce a "shift" in the following way:

mat1 = mat - IdentityMatrix[Length[mat]]*large

and then add large to the result of Eigenvalues[mat1]. This operation leaves the eigenvectors unchanged and I do not need to use Arnoldi specifically. Of course, Arnoldi does a similar shift inside, however I am not sure if that is used correctly in the Mathematica implementation.

Example:

mat={{1.,2.},{3.,4.}};

The eigenvalues are in the order given by Eigenvalues

{5.37228, -0.372281}.

The procedure above produces for any large enough large:

{-0.372281, 5.37228}

with the smallest first. Or Eigenvalues[mat1,1]+large just the lowest eigenvalue.