Use pivoting as the default in LU regardless of the element type. (#32989)

For types that weren't subtypes of AbstractFloat, we used to try
to LU factorize without pivoting and only use pivoting when it failed.
This caused large numerical errors when computing the LU for element
types which promoted to float like numbers such as most integers.
The behavior was never documented and is error prone. Hence, this
PR removes the behavior.

Author: andreasnoack

stdlib/LinearAlgebra/src/lu.jl

@@ -179,13 +179,6 @@ function generic_lufact!(A::StridedMatrix{T}, ::Val{Pivot} = Val(true);
     return LU{T,typeof(A)}(A, ipiv, convert(BlasInt, info))
 end
 
-# floating point types doesn't have to be promoted for LU, but should default to pivoting
-function lu(A::Union{AbstractMatrix{T}, AbstractMatrix{Complex{T}}},
-            pivot::Union{Val{false}, Val{true}} = Val(true);
-            check::Bool = true) where {T<:AbstractFloat}
-    lu!(copy(A), pivot; check = check)
-end
-
 function lutype(T::Type)
     # In generic_lufact!, the elements of the lower part of the matrix are
     # obtained using the division of two matrix elements. Hence their type can
@@ -274,26 +267,10 @@ julia> l == F.L && u == F.U && p == F.p
 true
 ```
 """
-function lu(A::AbstractMatrix{T}, pivot::Union{Val{false}, Val{true}};
+function lu(A::AbstractMatrix{T}, pivot::Union{Val{false}, Val{true}}=Val(true);
             check::Bool = true) where T
     S = lutype(T)
-    AA = similar(A, S)
-    copyto!(AA, A)
-    lu!(AA, pivot; check = check)
-end
-# We can't assume an ordered field so we first try without pivoting
-function lu(A::AbstractMatrix{T}; check::Bool = true) where T
-    S = lutype(T)
-    AA = similar(A, S)
-    copyto!(AA, A)
-    F = lu!(AA, Val(false); check = false)
-    if issuccess(F)
-        return F
-    else
-        AA = similar(A, S)
-        copyto!(AA, A)
-        return lu!(AA, Val(true); check = check)
-    end
+    lu!(copy_oftype(A, S), pivot; check = check)
 end
 
 lu(S::LU) = S

stdlib/LinearAlgebra/test/lu.jl

@@ -301,10 +301,19 @@ include("trickyarithmetic.jl")
 @testset "lu with type whose sum is another type" begin
     A = TrickyArithmetic.A[1 2; 3 4]
     ElT = TrickyArithmetic.D{TrickyArithmetic.C,TrickyArithmetic.C}
-    B = lu(A)
+    B = lu(A, Val(false))
     @test B isa LinearAlgebra.LU{ElT,Matrix{ElT}}
-    C = lu(A, Val(false))
-    @test C isa LinearAlgebra.LU{ElT,Matrix{ElT}}
+end
+
+@testset "Issue #30917. Determinant of integer matrix" begin
+    @test det([1 1 0 0 1 0 0 0
+               1 0 1 0 0 1 0 0
+               1 0 0 1 0 0 1 0
+               0 1 1 1 0 0 0 0
+               0 1 0 0 0 0 1 1
+               0 0 1 0 1 0 0 1
+               0 0 0 1 1 1 0 0
+               0 0 0 0 1 1 0 1]) ≈ 6
 end
 
 end # module TestLU