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Merged
merged 2 commits into from
Oct 20, 2025
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Include pivoting in LUL⁻¹ #75

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bc350a6
Include pivoting
MikaelSlevinsky Oct 20, 2025
7275b86
two more tests
MikaelSlevinsky Oct 20, 2025
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2 changes: 1 addition & 1 deletion Project.toml
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Original file line number Diff line number Diff line change
@@ -1,6 +1,6 @@
name = "MatrixFactorizations"
uuid = "a3b82374-2e81-5b9e-98ce-41277c0e4c87"
version = "3.1.1"
version = "3.1.2"

[deps]
ArrayLayouts = "4c555306-a7a7-4459-81d9-ec55ddd5c99a"
Expand Down
121 changes: 77 additions & 44 deletions src/lulinv.jl
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Expand Up @@ -10,6 +10,8 @@ The individual components of the factorization `F::LULinv` can be accessed via [
|:----------|:--------------------------------------------|
| `F.L` | `L` (unit lower triangular) part of `LUL⁻¹` |
| `F.U` | `U` (upper triangular) part of `LUL⁻¹` |
| `F.p` | (right) permutation `Vector` |
| `F.P` | (right) permutation `Matrix` |

Iterating the factorization produces the components `F.L` and `F.U`.

Expand All @@ -31,12 +33,12 @@ U factor:
-0.772002 3.0
0.0 7.772

julia> F.L * F.U / F.L ≈ A
julia> F.L * F.U / F.L ≈ A[F.p, F.p]
true

julia> l, u = lulinv(A); # destructuring via iteration
julia> l, u, p = lulinv(A); # destructuring via iteration

julia> l == F.L && u == F.U
julia> l == F.L && u == F.U && p == F.p
true

julia> A = [-150 334 778; -89 195 464; 5 -10 -27]
Expand All @@ -58,25 +60,27 @@ U factor:
0 -2 75
0 0 3

julia> F.L * F.U / F.L == A
julia> F.L * F.U / F.L == A[F.p, F.p]
true
```
"""
struct LULinv{T, S <: AbstractMatrix{T}} <: Factorization{T}
struct LULinv{T, S <: AbstractMatrix{T}, IPIV <: AbstractVector{<:Integer}} <: Factorization{T}
factors::S
function LULinv{T, S}(factors) where {T, S <: AbstractMatrix{T}}
ipiv::IPIV
function LULinv{T, S, IPIV}(factors, ipiv) where {T, S <: AbstractMatrix{T}, IPIV <: AbstractVector{<:Integer}}
require_one_based_indexing(factors)
new{T, S}(factors)
new{T, S, IPIV}(factors, ipiv)
end
end


LULinv(factors::AbstractMatrix{T}) where T = LULinv{T, typeof(factors)}(factors)
LULinv{T}(factors::AbstractMatrix) where T = LULinv(convert(AbstractMatrix{T}, factors))
LULinv{T}(F::LULinv) where T = LULinv{T}(F.factors)
LULinv(factors::AbstractMatrix{T}, ipiv::AbstractVector{<:Integer}) where T = LULinv{T, typeof(factors), typeof(ipiv)}(factors, ipiv)
LULinv{T}(factors::AbstractMatrix, ipiv::AbstractVector{<:Integer}) where T = LULinv(convert(AbstractMatrix{T}, factors), ipiv)
LULinv{T}(F::LULinv) where T = LULinv{T}(F.factors, F.ipiv)

iterate(F::LULinv) = (F.L, Val(:U))
iterate(F::LULinv, ::Val{:U}) = (F.U, Val(:done))
iterate(F::LULinv, ::Val{:U}) = (F.U, Val(:p))
iterate(F::LULinv, ::Val{:p}) = (F.p, Val(:done))
iterate(F::LULinv, ::Val{:done}) = nothing


Expand Down Expand Up @@ -116,17 +120,22 @@ function getU(F::LULinv)
end

function getproperty(F::LULinv{T, <: AbstractMatrix}, d::Symbol) where T
n = size(F, 1)
if d === :L
return getL(F)
elseif d === :U
return getU(F)
elseif d === :p
return getfield(F, :ipiv)
elseif d === :P
return Matrix{T}(I, n, n)[:, F.p]
else
getfield(F, d)
end
end

propertynames(F::LULinv, private::Bool=false) =
(:L, :U, (private ? fieldnames(typeof(F)) : ())...)
(:L, :U, :p, :P, (private ? fieldnames(typeof(F)) : ())...)

function show(io::IO, mime::MIME{Symbol("text/plain")}, F::LULinv)
summary(io, F); println(io)
Expand All @@ -137,42 +146,60 @@ function show(io::IO, mime::MIME{Symbol("text/plain")}, F::LULinv)
end


lulinv(A::AbstractMatrix; kwds...) = lulinv(A, eigvals(A); kwds...)
function lulinv(A::AbstractMatrix{T}, λ::AbstractVector{T}; kwds...) where T
lulinv(A::AbstractMatrix, pivot::Union{Val{false}, Val{true}}=Val(true); kwds...) = lulinv(A, eigvals(A), pivot; kwds...)
function lulinv(A::AbstractMatrix{T}, λ::AbstractVector{T}, pivot::Union{Val{false}, Val{true}}=Val(true); kwds...) where T
S = lulinvtype(T)
lulinv!(copy_oftype(A, S), copy_oftype(λ, S); kwds...)
lulinv!(copy_oftype(A, S), copy_oftype(λ, S), pivot; kwds...)
end
function lulinv(A::AbstractMatrix{T1}, λ::AbstractVector{T2}; kwds...) where {T1, T2}
function lulinv(A::AbstractMatrix{T1}, λ::AbstractVector{T2}, pivot::Union{Val{false}, Val{true}}=Val(true); kwds...) where {T1, T2}
T = promote_type(T1, T2)
S = lulinvtype(T)
lulinv!(copy_oftype(A, S), copy_oftype(λ, S); kwds...)
lulinv!(copy_oftype(A, S), copy_oftype(λ, S), pivot; kwds...)
end

function lulinv!(A::Matrix{T}, λ::Vector{T}; rtol::Real = size(A, 1)*eps(real(float(oneunit(T))))) where T
function lulinv!(A::Matrix{T}, λ::Vector{T}, ::Val{Pivot} = Val(true); rtol::Real = size(A, 1)*eps(real(float(oneunit(T))))) where {T, Pivot}
n = checksquare(A)
n == length(λ) || throw(ArgumentError("Eigenvalue count does not match matrix dimensions."))
v = zeros(T, n)
for i in 1:n-1
# We must find an eigenvector with nonzero "first" entry. A failed UL factorization of A-λI reveals this vector provided L₁₁ == 0.
for j in 1:length(λ)
F = ul!(view(A, i:n, i:n) - λ[j]*I; check=false)
nrm = norm(F.L)
if norm(F.L[1]) ≤ rtol*nrm # v[i] is a free parameter; we set it to 1.
fill!(v, zero(T))
v[i] = one(T)
# Next, we must scan the remaining free parameters, set them to 0, so that we find a nonsingular lower-triangular linear system for the nontrivial remaining part of the eigenvector.
idx = Int[]
for k in 2:n+1-i
if norm(F.L[k, k]) > rtol*nrm
push!(idx, k)
end
ipiv = collect(1:n)
for i in 1:n
F = ul!(view(A, i:n, i:n) - λ[i]*I; check = false)
nrm = norm(F.L)
if Pivot
ip = n+1-i
while (ip > 1) && (norm(F.L[ip, ip]) > rtol*nrm)
ip -= 1
end
fill!(v, zero(T))
v[i] = one(T)
idx = ip+1:n+1-i
if !isempty(idx)
v[idx.+(i-1)] .= -F.L[idx, ip]
ldiv!(LowerTriangular(view(F.L, idx, idx)), view(v, idx.+(i-1)))
end
ip = ip+i-1
if ip ≠ i
tmp = ipiv[i]
ipiv[i] = ipiv[ip]
ipiv[ip] = tmp
for j in 1:n
tmp = A[i, j]
A[i, j] = A[ip, j]
A[ip, j] = tmp
end
if !isempty(idx)
v[idx.+(i-1)] .= -F.L[idx, 1]
ldiv!(LowerTriangular(view(F.L, idx, idx)), view(v, idx.+(i-1)))
for j in 1:n
tmp = A[j, i]
A[j, i] = A[j, ip]
A[j, ip] = tmp
end
deleteat!(λ, j)
break
end
else
fill!(v, zero(T))
v[i] = one(T)
idx = 2:n+1-i
if !isempty(idx)
v[idx.+(i-1)] .= -F.L[idx, 1]
ldiv!(LowerTriangular(view(F.L, idx, idx)), view(v, idx.+(i-1)))
end
end
for k in 1:n
Expand All @@ -189,17 +216,23 @@ function lulinv!(A::Matrix{T}, λ::Vector{T}; rtol::Real = size(A, 1)*eps(real(f
A[k, i] = v[k]
end
end
return LULinv(A)
return LULinv(A, ipiv)
end

function ldiv!(F::LULinv, B::AbstractVecOrMat)
L = UnitLowerTriangular(F.factors)
return lmul!(L, ldiv!(UpperTriangular(F.factors), ldiv!(L, B)))
function ldiv!(A::LULinv, B::AbstractVecOrMat)
B .= B[invperm(A.p), :] # Todo: fix me
L = UnitLowerTriangular(A.factors)
lmul!(L, ldiv!(UpperTriangular(A.factors), ldiv!(L, B)))
B .= B[A.p, :] # Todo: fix me
return B
end

function rdiv!(B::AbstractVecOrMat, F::LULinv)
L = UnitLowerTriangular(F.factors)
return rdiv!(rdiv!(rmul!(B, L), UpperTriangular(F.factors)), L)
function rdiv!(A::AbstractVecOrMat, B::LULinv)
A .= A[:, B.p] # Todo: fix me
L = UnitLowerTriangular(B.factors)
rdiv!(rdiv!(rmul!(A, L), UpperTriangular(B.factors)), L)
A .= A[:, invperm(B.p)] # Todo: fix me
return A
end

det(F::LULinv) = det(UpperTriangular(F.factors))
Expand Down
82 changes: 53 additions & 29 deletions test/test_lulinv.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -6,18 +6,18 @@ using LinearAlgebra, MatrixFactorizations, Random, Test
λ = [17//1; -2; 3]
A = det(V)*V*Diagonal(λ)/V
@test A == Matrix{Int}(A)
L, U = lulinv(A, λ)
@test A ≈ L*U/L
@test A*L ≈ L*U
L, U, p = lulinv(A, λ)
@test A[p, p] ≈ L*U/L
@test A[p, p]*L ≈ L*U
end
@testset "A::Matrix{Rational{Int}}" begin
Random.seed!(0)
V = rand(-3:3//1, 5, 5)
λ = [-2; -1; 0; 1; 2//1]
A = V*Diagonal(λ)/V
L, U = lulinv(A, λ)
@test A ≈ L*U/L
@test A*L ≈ L*U
L, U, p = lulinv(A, λ)
@test A[p, p] ≈ L*U/L
@test A[p, p]*L ≈ L*U
end
@testset "Full Jordan blocks" begin
Random.seed!(0)
Expand All @@ -28,9 +28,10 @@ using LinearAlgebra, MatrixFactorizations, Random, Test
J[3, 4] = 1
J[4, 5] = 1
A = V*J/V
L, U = lulinv(A, λ)
@test A ≈ L*U/L
@test A*L ≈ L*U
L, U, p = lulinv(A, λ)
@test A[p, p] ≈ L*U/L
@test A[p, p]*L ≈ L*U

end
@testset "Alg ≠ geo" begin
Random.seed!(0)
Expand All @@ -40,17 +41,17 @@ using LinearAlgebra, MatrixFactorizations, Random, Test
J[1, 2] = 1
J[3, 4] = 1
A = V*J/V
L, U = lulinv(A, λ)
@test A ≈ L*U/L
@test A*L ≈ L*U
L, U, p = lulinv(A, λ)
@test A[p, p] ≈ L*U/L
@test A[p, p]*L ≈ L*U
λ = [1; -2; -2; 1; 1//1]
L, U = lulinv(A, λ)
@test A ≈ L*U/L
@test A*L ≈ L*U
L, U, p = lulinv(A, λ)
@test A[p, p] ≈ L*U/L
@test A[p, p]*L ≈ L*U
λ = [1; -2; 1; -2; 1//1]
L, U = lulinv(A, λ)
@test A ≈ L*U/L
@test A*L ≈ L*U
L, U, p = lulinv(A, λ)
@test A[p, p] ≈ L*U/L
@test A[p, p]*L ≈ L*U
end
@testset "Index scan is correct" begin
Random.seed!(0)
Expand All @@ -60,15 +61,15 @@ using LinearAlgebra, MatrixFactorizations, Random, Test
J = [J1 zeros(Rational{Int}, size(J1)); zeros(Rational{Int}, size(J2)) J2]
λ = diag(J)
A = V*J/V
L, U = lulinv(A, λ)
@test A ≈ L*U/L
@test A*L ≈ L*U
L, U, p = lulinv(A, λ)
@test A[p, p] ≈ L*U/L
@test A[p, p]*L ≈ L*U
end
@testset "A::Matrix{Float64}" begin
A = [4.0 3; 6 3]
L, U = lulinv(A)
@test A ≈ L*U/L
@test A*L ≈ L*U
L, U, p = lulinv(A)
@test A[p, p] ≈ L*U/L
@test A[p, p]*L ≈ L*U
end
@testset "size, det, logdet, logabsdet" begin
A = [4.0 3; 6 3]
Expand All @@ -90,7 +91,7 @@ using LinearAlgebra, MatrixFactorizations, Random, Test
seekstart(bf)
@test String(take!(bf)) ==
"""
LULinv{Float64, Matrix{Float64}}
LULinv{Float64, Matrix{Float64}, Vector{Int64}}
L factor:
2×2 Matrix{Float64}:
1.0 0.0
Expand All @@ -102,15 +103,38 @@ U factor:
end
@testset "propertynames" begin
names = sort!(collect(string.(Base.propertynames(lulinv([2 1; 1 2])))))
@test names == ["L", "U"]
@test names == ["L", "P", "U", "p"]
allnames = sort!(collect(string.(Base.propertynames(lulinv([2 1; 1 2]), true))))
@test allnames == ["L", "U", "factors"]
@test allnames == ["L", "P", "U", "factors", "ipiv", "p"]
end
@testset "A::Matrix{Int64}, λ::Vector{Rational{Int64}}" begin
A = [-150 334 778; -89 195 464; 5 -10 -27]
F = lulinv(A, [17, -2, 3//1])
@test A * F.L == F.L * F.U
@test A[F.p, F.p] * F.L == F.L * F.U
G = LULinv{Float64}(F)
@test A * G.L ≈ G.L * G.U
@test A[G.p, G.p] * G.L ≈ G.L * G.U
end
@testset "A is lower triangular" begin
Random.seed!(0)
for A in (tril!(rand(-5:5, 8, 8)), tril!(randn(8, 8)), tril!(randn(Float32, 8, 8)))
F = lulinv(A)
@test A[F.p, F.p] * F.L ≈ F.L * F.U
L, U, P = F.L, F.U, F.P
@test P'A*P*L ≈ L*U
@test P'A*P ≈ L*U/L
@test A ≈ P*L*U/(P*L)
@test A ≈ P*L*U/L*P'
end
end
@testset "Pivoting is necessary" begin
A = [1 0; 0 2//1]
λ = [2; 1//1]
@test_throws SingularException lulinv(A, λ, Val(false))
L, U, p = lulinv(A, λ)
@test p == [2, 1]
@test A[p, p] == L*U/L
F = lulinv(A, λ)
@test F\A ≈ I
@test A/F ≈ I
end
end
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