Make SparseArrays an extension package

While this may seem a bit weird since the most normal case for Krylov subspace methods is on sparse arrays, there are multiple reasons for this.

1. SparseArrays pulls in SuiteSparse and thus GPL dependencies
2. This is by far the biggest part of the load. Normal Krylov.jl loads in about 8ms, but the SparseArrays part is ~150ms. So if you want to just work on operators, then you'd taking a decently sized hit.
3. This is required for downstream dependency reduction SciML/LinearSolve.jl#570

Project.toml

@@ -7,6 +7,12 @@ LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Printf = "de0858da-6303-5e67-8744-51eddeeeb8d7"
 SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 
+[weakdeps]
+SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
+
+[extensions]
+KrylovSparseArraysExt = "SparseArrays"
+
 [compat]
 julia = "1.6"
 LinearAlgebra = "1.6"
@@ -17,7 +23,8 @@ Test = "1.6"
 
 [extras]
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
+SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-test = ["Random", "Test"]
+test = ["Random", "SparseArrays", "Test"]

ext/KrylovSparseArraysExt.jl

@@ -0,0 +1,20 @@
+module KrylovSparseArraysExt
+
+using Krylov, SparseArrays
+using Krylov.LinearAlgebra
+using Krylov.LinearAlgebra: NoPivot, LowerTriangular
+using Krylov: FloatOrComplex, reduced_qr!, ktypeof, vector_to_matrix, knorm, kdot, kaxpy!, kdotr, kfill!
+
+function Krylov.ktypeof(v::S) where S <: SparseVector
+  T = eltype(S)
+  return Vector{T}
+end
+
+function Krylov.ktypeof(v::S) where S <: AbstractSparseVector
+  return S.types[2]  # return `CuVector` for a `CuSparseVector`
+end
+
+include("krylov_processes.jl")
+include("block_krylov_processes.jl")
+
+end