Support for eigen-solvers

[MasonProtter]

This currently doesn't work:

julia> using Enzyme

julia> let 
           autodiff(Forward, Duplicated(1.0, 1.0)) do x
               M = [x    1+x
                    1+x' x^2]
               
               sum(eigvecs(M))
           end
       end
ERROR: 
No forward mode derivative found for ejlstr$dsyevr_64_$libblastrampoline.so.5
 at context:   call void @"ejlstr$dsyevr_64_$libblastrampoline.so.5"(i8* noundef nonnull %5, i8* noundef nonnull %6, i8* noundef nonnull %7, i8* noundef nonnull %9, i64 %174, i8* noundef nonnull %11, i8* noundef nonnull %13, i8* noundef nonnull %15, i8* noundef nonnull %17, i8* noundef nonnull %19, i8* noundef nonnull %21, i64 noundef %149, i64 %157, i64 %175, i8* noundef nonnull %23, i64 %150, i64 %176, i8* noundef nonnull %25, i64 %177, i8* noundef nonnull %27, i8* noundef nonnull %4, i64 noundef 1, i64 noundef 1, i64 noundef 1) #141 [ "jl_roots"({} addrspace(10)* null, {} addrspace(10)* null, { i8*, {} addrspace(10)* } %173, {} addrspace(10)* null, { i8*, {} addrspace(10)* } %169, { i8*, {} addrspace(10)* } %107, {} addrspace(10)* null, { i8*, {} addrspace(10)* } %165, { i8*, {} addrspace(10)* } %156, {} addrspace(10)* null, {} addrspace(10)* null, {} addrspace(10)* null, {} addrspace(10)* null, {} addrspace(10)* null, {} addrspace(10)* null, {} addrspace(10)* null, { i8*, {} addrspace(10)* } %161, {} addrspace(10)* null, {} addrspace(10)* null, {} addrspace(10)* null, {} addrspace(10)* null) ], !dbg !262

Stacktrace:
 [1] syevr!
   @ ~/.julia/juliaup/julia-1.11.2+0.x64.linux.gnu/share/julia/stdlib/v1.11/LinearAlgebra/src/lapack.jl:5397


Stacktrace:
  [1] syevr!
    @ ~/.julia/juliaup/julia-1.11.2+0.x64.linux.gnu/share/julia/stdlib/v1.11/LinearAlgebra/src/lapack.jl:5397
  [2] eigen!
    @ ~/.julia/juliaup/julia-1.11.2+0.x64.linux.gnu/share/julia/stdlib/v1.11/LinearAlgebra/src/symmetriceigen.jl:8 [inlined]
  [3] #_eigen#96
    @ ~/.julia/juliaup/julia-1.11.2+0.x64.linux.gnu/share/julia/stdlib/v1.11/LinearAlgebra/src/eigen.jl:250 [inlined]
  [4] fwddiffejulia___eigen_96_12524wrap
    @ ~/.julia/juliaup/julia-1.11.2+0.x64.linux.gnu/share/julia/stdlib/v1.11/LinearAlgebra/src/eigen.jl:0
  [5] macro expansion
    @ ~/.julia/packages/Enzyme/R6sE8/src/compiler.jl:5340 [inlined]
  [6] enzyme_call
    @ ~/.julia/packages/Enzyme/R6sE8/src/compiler.jl:4878 [inlined]
  [7] ForwardModeThunk
    @ ~/.julia/packages/Enzyme/R6sE8/src/compiler.jl:4766 [inlined]
  [8] runtime_generic_fwd(activity::Type{…}, runtimeActivity::Val{…}, width::Val{…}, RT::Val{…}, f::LinearAlgebra.var"##_eigen#96", df::Nothing, primal_1::Bool, shadow_1_1::Nothing, primal_2::Bool, shadow_2_1::Nothing, primal_3::typeof(LinearAlgebra.eigsortby), shadow_3_1::Nothing, primal_4::typeof(LinearAlgebra._eigen), shadow_4_1::Nothing, primal_5::Matrix{…}, shadow_5_1::Matrix{…})
    @ Enzyme.Compiler ~/.julia/packages/Enzyme/R6sE8/src/rules/jitrules.jl:303
  [9] _eigen
    @ ~/.julia/juliaup/julia-1.11.2+0.x64.linux.gnu/share/julia/stdlib/v1.11/LinearAlgebra/src/eigen.jl:247 [inlined]
 [10] #eigen#94
    @ ~/.julia/juliaup/julia-1.11.2+0.x64.linux.gnu/share/julia/stdlib/v1.11/LinearAlgebra/src/eigen.jl:239 [inlined]
 [11] eigen
    @ ~/.julia/juliaup/julia-1.11.2+0.x64.linux.gnu/share/julia/stdlib/v1.11/LinearAlgebra/src/eigen.jl:238 [inlined]
 [12] eigvecs
    @ ~/.julia/juliaup/julia-1.11.2+0.x64.linux.gnu/share/julia/stdlib/v1.11/LinearAlgebra/src/eigen.jl:274 [inlined]
 [13] #1
    @ ./REPL[2]:6 [inlined]
 [14] fwddiffejulia__1_7197_inner_1wrap
    @ ./REPL[2]:0
 [15] macro expansion
    @ ~/.julia/packages/Enzyme/R6sE8/src/compiler.jl:5340 [inlined]
 [16] enzyme_call
    @ ~/.julia/packages/Enzyme/R6sE8/src/compiler.jl:4878 [inlined]
 [17] ForwardModeThunk
    @ ~/.julia/packages/Enzyme/R6sE8/src/compiler.jl:4766 [inlined]
 [18] autodiff
    @ ~/.julia/packages/Enzyme/R6sE8/src/Enzyme.jl:654 [inlined]
 [19] autodiff(mode::ForwardMode{false, FFIABI, true, false}, f::Const{var"#1#2"}, args::Duplicated{Float64})
    @ Enzyme ~/.julia/packages/Enzyme/R6sE8/src/Enzyme.jl:544
 [20] autodiff
    @ ~/.julia/packages/Enzyme/R6sE8/src/Enzyme.jl:516 [inlined]
 [21] autodiff(f::Function, m::ForwardMode{false, FFIABI, false, false}, args::Duplicated{Float64})
    @ Enzyme ~/.julia/packages/Enzyme/R6sE8/src/Enzyme.jl:1019
 [22] top-level scope
    @ REPL[2]:2
Some type information was truncated. Use `show(err)` to see complete types.

It'd be really nice to be able to differentiate through eigen-solver calls.

[wsmoses]

@michel2323 potentially another one for the blas rules?