Add column-pivoted QR factorization fallback on failed LU factorization

In action:

```julia
using LinearSolve

A = [1.0 0 0 0
     0  1 0 0
     0  0 1 0
    0 0 0 0]
b = rand(4)
prob = LinearProblem(A, b)
sol = solve(prob, LinearSolve.DefaultLinearSolver(LinearSolve.DefaultAlgorithmChoice.LUFactorization; safetyfallback=false))
@test sol.retcode === ReturnCode.Failure
@test sol.u == zeros(4)

sol = solve(prob)
@test sol.u ≈ svd(A)\b
```

Previously it would just fail, now it falls back to the column-pivoted QR in order to successfully factorize when necessary.

Author: ChrisRackauckas

docs/src/solvers/solvers.md

@@ -82,6 +82,12 @@ use `Krylov_GMRES()`.
 
 ## Full List of Methods
 
+### Polyalgorithms
+
+```@docs
+LinearSolve.DefaultLinearSolver
+```
+
 ### RecursiveFactorization.jl
 
 !!! note

src/LinearSolve.jl

@@ -119,8 +119,23 @@ EnumX.@enumx DefaultAlgorithmChoice begin
     KrylovJL_LSMR
 end
 
+"""
+    DefaultLinearSolver(;safetyfallback=true)
+
+The default linear solver. This is the algorithm chosen when `solve(prob)`
+is called. It's a polyalgorithm that detects the optimal method for a given
+`A, b` and hardware (Intel, AMD, GPU, etc.).
+
+## Keyword Arguments
+
+* `safetyfallback`: determines whether to fallback to a column-pivoted QR factorization
+  when an LU factorization fails. This can be required if `A` is rank-deficient. Defaults
+  to true.
+"""
 struct DefaultLinearSolver <: SciMLLinearSolveAlgorithm
     alg::DefaultAlgorithmChoice.T
+    safetyfallback::Bool
+    DefaultLinearSolver(alg; safetyfallback=true) = new(alg,safetyfallback)
 end
 
 const BLASELTYPES = Union{Float32, Float64, ComplexF32, ComplexF64}

src/default.jl

@@ -41,6 +41,11 @@ end
     ex = Expr(:if, ex.args...)
 end
 
+# Handle special case of Column-pivoted QR fallback for LU
+function __setfield!(cache::DefaultLinearSolverInit, alg::DefaultLinearSolver, v::QRFactorization{ColumnNorm})
+
+end
+
 # Legacy fallback
 # For SciML algorithms already using `defaultalg`, all assume square matrix.
 defaultalg(A, b) = defaultalg(A, b, OperatorAssumptions(true))
@@ -352,11 +357,32 @@ end
         kwargs...)
     ex = :()
     for alg in first.(EnumX.symbol_map(DefaultAlgorithmChoice.T))
-        newex = quote
-            sol = SciMLBase.solve!(cache, $(algchoice_to_alg(alg)), args...; kwargs...)
-            SciMLBase.build_linear_solution(alg, sol.u, sol.resid, sol.cache;
-                retcode = sol.retcode,
-                iters = sol.iters, stats = sol.stats)
+        if alg in Symbol.((DefaultAlgorithmChoice.LUFactorization,
+            DefaultAlgorithmChoice.RFLUFactorization,
+            DefaultAlgorithmChoice.MKLLUFactorization,
+            DefaultAlgorithmChoice.AppleAccelerateLUFactorization,
+            DefaultAlgorithmChoice.GenericLUFactorization))
+            newex = quote
+                sol = SciMLBase.solve!(cache, $(algchoice_to_alg(alg)), args...; kwargs...)
+                if sol.retcode === ReturnCode.Failure && alg.safetyfallback
+                    ## TODO: Add verbosity logging here about using the fallback
+                    sol = SciMLBase.solve!(cache, QRFactorization(ColumnNorm()), args...; kwargs...)
+                    SciMLBase.build_linear_solution(alg, sol.u, sol.resid, sol.cache;
+                        retcode = sol.retcode,
+                        iters = sol.iters, stats = sol.stats)
+                else
+                    SciMLBase.build_linear_solution(alg, sol.u, sol.resid, sol.cache;
+                        retcode = sol.retcode,
+                        iters = sol.iters, stats = sol.stats)
+                end
+            end
+        else
+            newex = quote
+                sol = SciMLBase.solve!(cache, $(algchoice_to_alg(alg)), args...; kwargs...)
+                SciMLBase.build_linear_solution(alg, sol.u, sol.resid, sol.cache;
+                    retcode = sol.retcode,
+                    iters = sol.iters, stats = sol.stats)
+            end
         end
         alg_enum = getproperty(LinearSolve.DefaultAlgorithmChoice, alg)
         ex = if ex == :()

test/default_algs.jl

@@ -158,4 +158,18 @@ prob = LinearProblem(A, b)
 @test_broken SciMLBase.successful_retcode(solve(prob))
 
 prob2 = LinearProblem(A2, b)
-@test SciMLBase.successful_retcode(solve(prob2))
+@test SciMLBase.successful_retcode(solve(prob2))
+
+# Column-Pivoted QR fallback on failed LU
+A = [1.0 0 0 0
+     0  1 0 0
+     0  0 1 0
+    0 0 0 0]
+b = rand(4)
+prob = LinearProblem(A, b)
+sol = solve(prob, LinearSolve.DefaultLinearSolver(LinearSolve.DefaultAlgorithmChoice.LUFactorization; safetyfallback=false))
+@test sol.retcode === ReturnCode.Failure
+@test sol.u == zeros(4)
+
+sol = solve(prob)
+@test sol.u ≈ svd(A)\b