fix equality of QRCompactWY (#41363)

Equality for `QRCompactWY` did not ignore the subdiagonal entries of
`T` leading to nondeterministic behavior.

This is pulled out from #41228, since this change should be less
controversial than the other changes there and this particular bug just
came up in ChainRules again.

(cherry picked from commit 74fab49)

stdlib/LinearAlgebra/src/LinearAlgebra.jl

@@ -16,7 +16,8 @@ import Base: USE_BLAS64, abs, acos, acosh, acot, acoth, acsc, acsch, adjoint, as
     setindex!, show, similar, sin, sincos, sinh, size, sqrt,
     strides, stride, tan, tanh, transpose, trunc, typed_hcat, vec
 using Base: IndexLinear, promote_op, promote_typeof,
-    @propagate_inbounds, @pure, reduce, typed_vcat, require_one_based_indexing
+    @propagate_inbounds, @pure, reduce, typed_vcat, require_one_based_indexing,
+    splat
 using Base.Broadcast: Broadcasted, broadcasted
 import Libdl
 

stdlib/LinearAlgebra/src/qr.jl

@@ -127,6 +127,40 @@ Base.iterate(S::QRCompactWY) = (S.Q, Val(:R))
 Base.iterate(S::QRCompactWY, ::Val{:R}) = (S.R, Val(:done))
 Base.iterate(S::QRCompactWY, ::Val{:done}) = nothing
 
+# returns upper triangular views of all non-undef values of `qr(A).T`:
+#
+# julia> sparse(qr(A).T .== qr(A).T)
+# 36×100 SparseMatrixCSC{Bool, Int64} with 1767 stored entries:
+# ⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
+# ⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
+# ⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿
+# ⠀⠀⠀⠀⠀⠂⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
+# ⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⠐⠙⢿⣿⣿⣿⣿
+# ⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⢙⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠙⢿⣿⣿
+# ⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠙⢿
+# ⠀⡀⠀⠀⠀⠀⠀⠀⠂⠒⠒⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀
+# ⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢀⠀
+#
+function _triuppers_qr(T)
+    blocksize, cols = size(T)
+    return Iterators.map(0:div(cols - 1, blocksize)) do i
+        n = min(blocksize, cols - i * blocksize)
+        return UpperTriangular(view(T, 1:n, (1:n) .+ i * blocksize))
+    end
+end
+
+function Base.hash(F::QRCompactWY, h::UInt)
+    return hash(F.factors, foldr(hash, _triuppers_qr(F.T); init=hash(QRCompactWY, h)))
+end
+function Base.:(==)(A::QRCompactWY, B::QRCompactWY)
+    return A.factors == B.factors && all(splat(==), zip(_triuppers_qr.((A.T, B.T))...))
+end
+function Base.isequal(A::QRCompactWY, B::QRCompactWY)
+    return isequal(A.factors, B.factors) && all(zip(_triuppers_qr.((A.T, B.T))...)) do (a, b)
+        isequal(a, b)::Bool
+    end
+end
+
 """
     QRPivoted <: Factorization
 

stdlib/LinearAlgebra/test/factorization.jl

@@ -0,0 +1,60 @@
+# This file is a part of Julia. License is MIT: https://julialang.org/license
+
+module TestFactorization
+using Test, LinearAlgebra
+
+@testset "equality for factorizations - $f" for f in Any[
+    bunchkaufman,
+    cholesky,
+    x -> cholesky(x, Val(true)),
+    eigen,
+    hessenberg,
+    lq,
+    lu,
+    qr,
+    x -> qr(x, ColumnNorm()),
+    svd,
+    schur,
+]
+    A = randn(3, 3)
+    A = A * A' # ensure A is pos. def. and symmetric
+    F, G = f(A), f(A)
+
+    @test F == G
+    @test isequal(F, G)
+    @test hash(F) == hash(G)
+
+    f === hessenberg && continue
+
+    # change all arrays in F to have eltype Float32
+    F = typeof(F).name.wrapper(Base.mapany(1:nfields(F)) do i
+        x = getfield(F, i)
+        return x isa AbstractArray{Float64} ? Float32.(x) : x
+    end...)
+    # round all arrays in G to the nearest Float64 representable as Float32
+    G = typeof(G).name.wrapper(Base.mapany(1:nfields(G)) do i
+        x = getfield(G, i)
+        return x isa AbstractArray{Float64} ? Float64.(Float32.(x)) : x
+    end...)
+
+    @test F == G broken=!(f === eigen || f === qr)
+    @test isequal(F, G) broken=!(f === eigen || f === qr)
+    @test hash(F) == hash(G)
+end
+
+@testset "equality of QRCompactWY" begin
+    A = rand(100, 100)
+    F, G = qr(A), qr(A)
+
+    @test F == G
+    @test isequal(F, G)
+    @test hash(F) == hash(G)
+
+    G.T[28, 100] = 42
+
+    @test F != G
+    @test !isequal(F, G)
+    @test hash(F) != hash(G)
+end
+
+end

stdlib/LinearAlgebra/test/testgroups

@@ -25,3 +25,4 @@ givens
 structuredbroadcast
 addmul
 ldlt
+factorization