Change default symmetriceigen algorithm back to RobustRepresentations (#1363)

See #1313

A brief summary: RobustRepresentations (`LAPACK.syevr!`) was the only
algorithm implemented until Julia v1.11. Then,
JuliaLang/julia#49355 implemented interface to
other algorithms, and changed the default to DivideAndConquer
(`LAPACK.syevd!`) based on its better numerical accuracy and
performance. But, it turned out that in some LAPACK implementation, the
DivideAndConquer fails more frequently than RobustRepresentations
(#1313). Based on the discussion in #1313 , this PR reverts the default
algorithm to RobustRepresentations.

Once the problem with the RobustRepresentations in the problematic
LAPACK implementation has been sorted out, the default could be changed
to DivideAndConquer.

(cherry picked from commit b6ad8f9)

Author: jaemolihm

src/symmetriceigen.jl

@@ -10,9 +10,9 @@ eigencopy_oftype(A::Symmetric{<:Complex}, S) = copyto!(similar(parent(A), S), A)
     default_eigen_alg(A)
 
 Return the default algorithm used to solve the eigensystem `A v = λ v` for a symmetric matrix `A`.
-Defaults to `LinearAlegbra.DivideAndConquer()`, which corresponds to the LAPACK function `LAPACK.syevd!`.
+Defaults to `LinearAlegbra.RobustRepresentations()`, which corresponds to the LAPACK function `LAPACK.syevr!`.
 """
-default_eigen_alg(@nospecialize(A)) = DivideAndConquer()
+default_eigen_alg(@nospecialize(A)) = RobustRepresentations()
 
 # Eigensolvers for symmetric and Hermitian matrices
 function eigen!(A::RealHermSymComplexHerm{<:BlasReal,<:StridedMatrix}; alg::Algorithm = default_eigen_alg(A), sortby::Union{Function,Nothing}=nothing)
@@ -37,9 +37,9 @@ matrix `F.vectors`. (The `k`th eigenvector can be obtained from the slice `F.vec
 Iterating the decomposition produces the components `F.values` and `F.vectors`.
 
 `alg` specifies which algorithm and LAPACK method to use for eigenvalue decomposition:
-- `alg = DivideAndConquer()` (default): Calls `LAPACK.syevd!`.
+- `alg = DivideAndConquer()`: Calls `LAPACK.syevd!`.
 - `alg = QRIteration()`: Calls `LAPACK.syev!`.
-- `alg = RobustRepresentations()`: Multiple relatively robust representations method, Calls `LAPACK.syevr!`.
+- `alg = RobustRepresentations()` (default): Multiple relatively robust representations method, Calls `LAPACK.syevr!`.
 
 See James W. Demmel et al, SIAM J. Sci. Comput. 30, 3, 1508 (2008) for
 a comparison of the accuracy and performance of different algorithms.
@@ -140,14 +140,18 @@ end
 Return the eigenvalues of `A`.
 
 `alg` specifies which algorithm and LAPACK method to use for eigenvalue decomposition:
-- `alg = DivideAndConquer()` (default): Calls `LAPACK.syevd!`.
+- `alg = DivideAndConquer()`: Calls `LAPACK.syevd!`.
 - `alg = QRIteration()`: Calls `LAPACK.syev!`.
-- `alg = RobustRepresentations()`: Multiple relatively robust representations method, Calls `LAPACK.syevr!`.
+- `alg = RobustRepresentations()` (default): Multiple relatively robust representations method, Calls `LAPACK.syevr!`.
 
 See James W. Demmel et al, SIAM J. Sci. Comput. 30, 3, 1508 (2008) for
 a comparison of the accuracy and performance of different methods.
 
 The default `alg` used may change in the future.
+
+!!! compat "Julia 1.12"
+    The `alg` keyword argument requires Julia 1.12 or later.
+
 """
 function eigvals(A::RealHermSymComplexHerm; alg::Algorithm = default_eigen_alg(A), sortby::Union{Function,Nothing}=nothing)
     S = eigtype(eltype(A))