From 6e38694d14e69c7620a3045ff748cf3ffe6d8330 Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Thu, 26 Feb 2026 11:45:10 -0500
Subject: [PATCH 01/14] add Driver types

---
 src/algorithms.jl | 54 +++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 54 insertions(+)

diff --git a/src/algorithms.jl b/src/algorithms.jl
index e9e7b8e81..b6e1bff25 100644
--- a/src/algorithms.jl
+++ b/src/algorithms.jl
@@ -143,6 +143,60 @@ If this is not possible, for example when the output size is not known a priori
 this function may return `nothing`.
 """ initialize_output
 
+
+# Drivers
+# -------
+"""
+    abstract type Driver
+
+Supertype used for customizing various implementations of the same algorithm.
+"""
+abstract type Driver end
+
+"""
+    DefaultDriver <: Driver
+
+Select a default driver at runtime, based on the input matrix.
+See also [`default_householder_driver`](@ref).
+"""
+struct DefaultDriver <: Driver end
+
+"""
+    LAPACK <: Driver
+
+Driver to select LAPACK as the implementation strategy.
+"""
+struct LAPACK <: Driver end
+
+"""
+    CUSOLVER <: Driver
+
+Driver to select CUSOLVER as the implementation strategy.
+"""
+struct CUSOLVER <: Driver end
+
+"""
+    ROCSOLVER <: Driver
+
+Driver to select ROCSOLVER as the implementation strategy.
+"""
+struct ROCSOLVER <: Driver end
+
+"""
+    GLA <: Driver
+
+Driver to select GenericLinearAlgebra.jl as the implementation strategy.
+"""
+struct GLA <: Driver end
+
+"""
+    Native <: Driver
+
+Driver to select a native implementation in MatrixAlgebraKit as the implementation strategy.
+"""
+struct Native <: Driver end
+
+
 # Truncation strategy
 # -------------------
 """

From 858bf1c8a29596eddf9068659c1124ea39d09708 Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Thu, 26 Feb 2026 11:46:02 -0500
Subject: [PATCH 02/14] add `Householder` algorithm

---
 src/interface/decompositions.jl | 23 +++++++++++++++++++++++
 1 file changed, 23 insertions(+)

diff --git a/src/interface/decompositions.jl b/src/interface/decompositions.jl
index 19f20e8e5..5cdb87993 100644
--- a/src/interface/decompositions.jl
+++ b/src/interface/decompositions.jl
@@ -64,6 +64,29 @@ of `R` are non-negative.
 @algdef LAPACK_HouseholderQL
 @algdef LAPACK_HouseholderRQ
 
+"""
+    Householder(; [driver], kwargs...)
+
+Algorithm type to denote the algorithm for computing QR, RQ, QL or LQ decompositions of a matrix using Householder reflectors.
+The optional `driver` symbol can be used to choose between different implementations of this algorithm.
+
+### Keyword arguments
+
+- `positive::Bool = true` : Fix the gauge of the resulting factors by making the diagonal elements of `L` or `R` non-negative.
+- `pivoted::Bool = false` : Use column- or row-pivoting for low-rank input matrices.
+- `blocksize::Int` : Use a blocked version of the algorithm if `blocksize > 1`.
+
+Depending on the driver, various other keywords may be (un)available to customize the implementation.
+"""
+struct Householder{D <: Driver, KW} <: AbstractAlgorithm
+    driver::D
+    kwargs::KW
+end
+Householder(driver::Driver = DefaultDriver(); kwargs...) = Householder(driver, kwargs)
+
+default_householder_driver(A) = Native()
+default_householder_driver(::YALAPACK.MaybeBlasMat) = LAPACK()
+
 # General Eigenvalue Decomposition
 # -------------------------------
 """

From 07260b6873a28212e18057ddf4dcd9d7f3590d00 Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Thu, 26 Feb 2026 11:47:01 -0500
Subject: [PATCH 03/14] refactor `Housholder` QR implementations

---
 src/implementations/qr.jl | 85 +++++++++++++++++++++++++++++++--------
 1 file changed, 69 insertions(+), 16 deletions(-)

diff --git a/src/implementations/qr.jl b/src/implementations/qr.jl
index fd7ce01b6..583573995 100644
--- a/src/implementations/qr.jl
+++ b/src/implementations/qr.jl
@@ -89,6 +89,30 @@ end
 # Implementation
 # --------------
 # actual implementation
+function qr_full!(A::AbstractMatrix, QR, alg::Householder)
+    check_input(qr_full!, A, QR, alg)
+    Q, R = QR
+    driver = get(alg.kwargs, :driver, default_householder_driver(A))
+    alg_kwargs = Base.structdiff(alg.kwargs, NamedTuple{(:driver,)})
+    householder_qr!(driver, A, Q, R; alg_kwargs...)
+    return Q, R
+end
+function qr_compact!(A::AbstractMatrix, QR, alg::Householder)
+    check_input(qr_compact!, A, QR, alg)
+    Q, R = QR
+    driver = get(alg.kwargs, :driver, default_householder_driver(A))
+    alg_kwargs = Base.structdiff(alg.kwargs, NamedTuple{(:driver,)})
+    householder_qr!(driver, A, Q, R; alg_kwargs...)
+    return Q, R
+end
+function qr_null!(A::AbstractMatrix, N, alg::Householder)
+    check_input(qr_null!, A, N, alg)
+    driver = get(alg.kwargs, :driver, LAPACK())
+    alg_kwargs = Base.structdiff(alg.kwargs, NamedTuple{(:driver,)})
+    householder_qr_null!(A, N; alg_kwargs...)
+    return N
+end
+
 function qr_full!(A::AbstractMatrix, QR, alg::LAPACK_HouseholderQR)
     check_input(qr_full!, A, QR, alg)
     Q, R = QR
@@ -127,8 +151,12 @@ end
 
 # LAPACK logic
 # ------------
-function _lapack_qr!(
-        A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
+Base.@deprecate(
+    _lapack_qr!(A, Q, R; kwargs...),
+    householder_qr!(LAPACK(), A, Q, R; kwargs...)
+)
+function householder_qr!(
+        ::LAPACK, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
         positive = true, pivoted = false,
         blocksize = ((pivoted || A === Q) ? 1 : YALAPACK.default_qr_blocksize(A))
     )
@@ -193,8 +221,12 @@ function _lapack_qr!(
     return Q, R
 end
 
-function _lapack_qr_null!(
-        A::AbstractMatrix, N::AbstractMatrix;
+Base.@deprecate(
+    _lapack_qr_null!(A, N; kwargs...),
+    householder_qr_null!(LAPACK(), A, N, kwargs...)
+)
+function householder_qr_null!(
+        ::LAPACK, A::AbstractMatrix, N::AbstractMatrix;
         positive = true, pivoted = false, blocksize = YALAPACK.default_qr_blocksize(A)
     )
     m, n = size(A)
@@ -268,13 +300,19 @@ function _gpu_unmqr!(
     throw(MethodError(_gpu_unmqr!, (side, trans, A, τ, C)))
 end
 
-function _gpu_qr!(
-        A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix; positive = true, blocksize = 1, pivoted = false
+Base.@deprecate(
+    _gpu_qr!(A, Q, R; kwargs...),
+    householder_qr!(default_householder_driver(A), A, Q, R; kwargs...)
+)
+
+function householder_qr!(
+        driver::Union{CUSOLVER, ROCSOLVER}, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
+        positive = true, blocksize = 1, pivoted = false
     )
     blocksize > 1 &&
-        throw(ArgumentError("CUSOLVER/ROCSOLVER does not provide a blocked implementation for a QR decomposition"))
+        throw(ArgumentError(lazy"$driver does not provide a blocked implementation for a QR decomposition"))
     pivoted &&
-        throw(ArgumentError("CUSOLVER/ROCSOLVER does not provide a pivoted implementation for a QR decomposition"))
+        throw(ArgumentError(lazy"$driver does not provide a pivoted implementation for a QR decomposition"))
     m, n = size(A)
     minmn = min(m, n)
     computeR = length(R) > 0
@@ -309,16 +347,21 @@ function _gpu_qr!(
     return Q, R
 end
 
-function _gpu_qr_null!(
-        A::AbstractMatrix, N::AbstractMatrix; positive = true, blocksize = 1, pivoted = false
+Base.@deprecate(
+    _gpu_qr_null!(A, N; kwargs...),
+    householder_qr_null!(default_householder_driver(A), A, N; kwargs...)
+)
+function householder_qr_null!(
+        driver::Union{CUSOLVER, ROCSOLVER}, A::AbstractMatrix, N::AbstractMatrix;
+        positive = true, blocksize = 1, pivoted = false
     )
     blocksize > 1 &&
-        throw(ArgumentError("CUSOLVER/ROCSOLVER does not provide a blocked implementation for a QR decomposition"))
+        throw(ArgumentError(lazy"$driver does not provide a blocked implementation for a QR decomposition"))
     pivoted &&
-        throw(ArgumentError("CUSOLVER/ROCSOLVER does not provide a pivoted implementation for a QR decomposition"))
+        throw(ArgumentError(lazy"$driver does not provide a pivoted implementation for a QR decomposition"))
     m, n = size(A)
     minmn = min(m, n)
-    fill!(N, zero(eltype(N)))
+    zero!(N)
     one!(view(N, (minmn + 1):m, 1:(m - minmn)))
     A, τ = _gpu_geqrf!(A)
     N = _gpu_unmqr!('L', 'N', A, τ, N)
@@ -349,8 +392,12 @@ function qr_null!(A::AbstractMatrix, N, alg::Native_HouseholderQR)
     return N
 end
 
-function _native_qr!(
-        A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
+Base.@deprecate(
+    _native_qr!(A, Q, R; kwargs...),
+    householder_qr!(Native(), A, Q, R; kwargs...)
+)
+function householder_qr!(
+        ::Native, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
         positive::Bool = true # always true regardless of setting
     )
     m, n = size(A)
@@ -385,7 +432,13 @@ function _native_qr!(
     return Q, R
 end
 
-function _native_qr_null!(A::AbstractMatrix, N::AbstractMatrix; positive::Bool = true)
+Base.@deprecate(
+    _native_qr_null!(A, N; kwargs...),
+    householder_qr_null!(Native(), A, N; kwargs...)
+)
+function householder_qr_null!(
+        ::Native, A::AbstractMatrix, N::AbstractMatrix; positive::Bool = true
+    )
     m, n = size(A)
     minmn = min(m, n)
     @inbounds for j in 1:minmn

From 043d1de4ace1e8673530d58716916dd6f379ef83 Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Thu, 26 Feb 2026 13:27:31 -0500
Subject: [PATCH 04/14] rename HouseholderReflection to avoid name clash

---
 src/common/householder.jl | 16 ++++++++--------
 1 file changed, 8 insertions(+), 8 deletions(-)

diff --git a/src/common/householder.jl b/src/common/householder.jl
index 0aac21bc9..015672328 100644
--- a/src/common/householder.jl
+++ b/src/common/householder.jl
@@ -1,32 +1,32 @@
 const IndexRange{T <: Integer} = Base.AbstractRange{T}
 
 # Elementary Householder reflection
-struct Householder{T, V <: AbstractVector, R <: IndexRange}
+struct HouseholderReflection{T, V <: AbstractVector, R <: IndexRange}
     β::T
     v::V
     r::R
 end
-Base.adjoint(H::Householder) = Householder(conj(H.β), H.v, H.r)
+Base.adjoint(H::HouseholderReflection) = HouseholderReflection(conj(H.β), H.v, H.r)
 
 function householder(x::AbstractVector, r::IndexRange = axes(x, 1), k = first(r))
     i = findfirst(==(k), r)
     i == nothing && error("k = $k should be in the range r = $r")
     β, v, ν = _householder!(x[r], i)
-    return Householder(β, v, r), ν
+    return HouseholderReflection(β, v, r), ν
 end
 # Householder reflector h that zeros the elements A[r,col] (except for A[k,col]) upon lmul!(h,A)
 function householder(A::AbstractMatrix, r::IndexRange, col::Int, k = first(r))
     i = findfirst(==(k), r)
     i == nothing && error("k = $k should be in the range r = $r")
     β, v, ν = _householder!(A[r, col], i)
-    return Householder(β, v, r), ν
+    return HouseholderReflection(β, v, r), ν
 end
 # Householder reflector that zeros the elements A[row,r] (except for A[row,k]) upon rmul!(A,h')
 function householder(A::AbstractMatrix, row::Int, r::IndexRange, k = first(r))
     i = findfirst(==(k), r)
     i == nothing && error("k = $k should be in the range r = $r")
     β, v, ν = _householder!(conj!(A[row, r]), i)
-    return Householder(β, v, r), ν
+    return HouseholderReflection(β, v, r), ν
 end
 
 # generate Householder vector based on vector v, such that applying the reflection
@@ -66,7 +66,7 @@ function _householder!(v::AbstractVector{T}, i::Int = 1) where {T}
     return β, v, ν
 end
 
-function LinearAlgebra.lmul!(H::Householder, x::AbstractVector)
+function LinearAlgebra.lmul!(H::HouseholderReflection, x::AbstractVector)
     v = H.v
     r = H.r
     β = H.β
@@ -87,7 +87,7 @@ function LinearAlgebra.lmul!(H::Householder, x::AbstractVector)
     end
     return x
 end
-function LinearAlgebra.lmul!(H::Householder, A::AbstractMatrix; cols = axes(A, 2))
+function LinearAlgebra.lmul!(H::HouseholderReflection, A::AbstractMatrix; cols = axes(A, 2))
     v = H.v
     r = H.r
     β = H.β
@@ -110,7 +110,7 @@ function LinearAlgebra.lmul!(H::Householder, A::AbstractMatrix; cols = axes(A, 2
     end
     return A
 end
-function LinearAlgebra.rmul!(A::AbstractMatrix, H::Householder; rows = axes(A, 1))
+function LinearAlgebra.rmul!(A::AbstractMatrix, H::HouseholderReflection; rows = axes(A, 1))
     v = H.v
     r = H.r
     β = H.β

From a2e0205fd280dbb106b69d20f52241ce7cd929ef Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Thu, 26 Feb 2026 13:27:38 -0500
Subject: [PATCH 05/14] reorganize QR

---
 src/implementations/qr.jl | 323 ++++++++++++++------------------------
 1 file changed, 120 insertions(+), 203 deletions(-)

diff --git a/src/implementations/qr.jl b/src/implementations/qr.jl
index 583573995..0c889ce13 100644
--- a/src/implementations/qr.jl
+++ b/src/implementations/qr.jl
@@ -86,75 +86,30 @@ for f! in (:qr_full!, :qr_compact!)
     end
 end
 
-# Implementation
-# --------------
-# actual implementation
-function qr_full!(A::AbstractMatrix, QR, alg::Householder)
-    check_input(qr_full!, A, QR, alg)
-    Q, R = QR
-    driver = get(alg.kwargs, :driver, default_householder_driver(A))
-    alg_kwargs = Base.structdiff(alg.kwargs, NamedTuple{(:driver,)})
-    householder_qr!(driver, A, Q, R; alg_kwargs...)
-    return Q, R
-end
-function qr_compact!(A::AbstractMatrix, QR, alg::Householder)
-    check_input(qr_compact!, A, QR, alg)
-    Q, R = QR
-    driver = get(alg.kwargs, :driver, default_householder_driver(A))
-    alg_kwargs = Base.structdiff(alg.kwargs, NamedTuple{(:driver,)})
-    householder_qr!(driver, A, Q, R; alg_kwargs...)
-    return Q, R
-end
-function qr_null!(A::AbstractMatrix, N, alg::Householder)
-    check_input(qr_null!, A, N, alg)
-    driver = get(alg.kwargs, :driver, LAPACK())
-    alg_kwargs = Base.structdiff(alg.kwargs, NamedTuple{(:driver,)})
-    householder_qr_null!(A, N; alg_kwargs...)
-    return N
-end
+# ==========================
+#      IMPLEMENTATIONS
+# ==========================
 
-function qr_full!(A::AbstractMatrix, QR, alg::LAPACK_HouseholderQR)
+# Householder
+# -----------
+function qr_full!(A, QR, alg::Householder)
     check_input(qr_full!, A, QR, alg)
-    Q, R = QR
-    _lapack_qr!(A, Q, R; alg.kwargs...)
-    return Q, R
+    return householder_qr!(alg.driver, A, QR...; alg.kwargs...)
 end
-function qr_compact!(A::AbstractMatrix, QR, alg::LAPACK_HouseholderQR)
+function qr_compact!(A, QR, alg::Householder)
     check_input(qr_compact!, A, QR, alg)
-    Q, R = QR
-    _lapack_qr!(A, Q, R; alg.kwargs...)
-    return Q, R
+    return householder_qr!(alg.driver, A, QR...; alg.kwargs...)
 end
-function qr_null!(A::AbstractMatrix, N, alg::LAPACK_HouseholderQR)
+function qr_null!(A, N, alg::Householder)
     check_input(qr_null!, A, N, alg)
-    _lapack_qr_null!(A, N; alg.kwargs...)
-    return N
+    return householder_qr_null!(alg.driver, A, N; alg.kwargs...)
 end
 
-function qr_full!(A::AbstractMatrix, QR, alg::DiagonalAlgorithm)
-    check_input(qr_full!, A, QR, alg)
-    Q, R = QR
-    _diagonal_qr!(A, Q, R; alg.kwargs...)
-    return Q, R
-end
-function qr_compact!(A::AbstractMatrix, QR, alg::DiagonalAlgorithm)
-    check_input(qr_compact!, A, QR, alg)
-    Q, R = QR
-    _diagonal_qr!(A, Q, R; alg.kwargs...)
-    return Q, R
-end
-function qr_null!(A::AbstractMatrix, N, alg::DiagonalAlgorithm)
-    check_input(qr_null!, A, N, alg)
-    _diagonal_qr_null!(A, N; alg.kwargs...)
-    return N
-end
+householder_qr!(::DefaultDriver, A, Q, R; kwargs...) =
+    householder_qr!(default_householder_driver(A), A, Q, R; kwargs...)
+householder_qr_null!(::DefaultDriver, A, N; kwargs...) =
+    householder_qr_null!(default_householder_driver(A), A, N; kwargs...)
 
-# LAPACK logic
-# ------------
-Base.@deprecate(
-    _lapack_qr!(A, Q, R; kwargs...),
-    householder_qr!(LAPACK(), A, Q, R; kwargs...)
-)
 function householder_qr!(
         ::LAPACK, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
         positive = true, pivoted = false,
@@ -220,91 +175,6 @@ function householder_qr!(
     end
     return Q, R
 end
-
-Base.@deprecate(
-    _lapack_qr_null!(A, N; kwargs...),
-    householder_qr_null!(LAPACK(), A, N, kwargs...)
-)
-function householder_qr_null!(
-        ::LAPACK, A::AbstractMatrix, N::AbstractMatrix;
-        positive = true, pivoted = false, blocksize = YALAPACK.default_qr_blocksize(A)
-    )
-    m, n = size(A)
-    minmn = min(m, n)
-    fill!(N, zero(eltype(N)))
-    one!(view(N, (minmn + 1):m, 1:(m - minmn)))
-    if blocksize > 1
-        nb = min(minmn, blocksize)
-        A, T = YALAPACK.geqrt!(A, similar(A, nb, minmn))
-        N = YALAPACK.gemqrt!('L', 'N', A, T, N)
-    else
-        A, τ = YALAPACK.geqrf!(A)
-        N = YALAPACK.unmqr!('L', 'N', A, τ, N)
-    end
-    return N
-end
-
-# Diagonal logic
-# --------------
-function _diagonal_qr!(
-        A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix; positive::Bool = true
-    )
-    # note: Ad and Qd might share memory here so order of operations is important
-    Ad = diagview(A)
-    Qd = diagview(Q)
-    Rd = diagview(R)
-    if positive
-        @. Rd = abs(Ad)
-        @. Qd = sign_safe(Ad)
-    else
-        Rd .= Ad
-        one!(Q)
-    end
-    return Q, R
-end
-
-_diagonal_qr_null!(A::AbstractMatrix, N; positive::Bool = true) = N
-
-# GPU logic
-# --------------
-# placed here to avoid code duplication since much of the logic is replicable across CUDA and AMDGPU
-function MatrixAlgebraKit.qr_full!(
-        A::AbstractMatrix, QR, alg::Union{CUSOLVER_HouseholderQR, ROCSOLVER_HouseholderQR}
-    )
-    check_input(qr_full!, A, QR, alg)
-    Q, R = QR
-    _gpu_qr!(A, Q, R; alg.kwargs...)
-    return Q, R
-end
-function MatrixAlgebraKit.qr_compact!(
-        A::AbstractMatrix, QR, alg::Union{CUSOLVER_HouseholderQR, ROCSOLVER_HouseholderQR}
-    )
-    check_input(qr_compact!, A, QR, alg)
-    Q, R = QR
-    _gpu_qr!(A, Q, R; alg.kwargs...)
-    return Q, R
-end
-function MatrixAlgebraKit.qr_null!(
-        A::AbstractMatrix, N, alg::Union{CUSOLVER_HouseholderQR, ROCSOLVER_HouseholderQR}
-    )
-    check_input(qr_null!, A, N, alg)
-    _gpu_qr_null!(A, N; alg.kwargs...)
-    return N
-end
-
-_gpu_geqrf!(A::AbstractMatrix) = throw(MethodError(_gpu_geqrf!, (A,)))
-_gpu_ungqr!(A::AbstractMatrix, τ::AbstractVector) = throw(MethodError(_gpu_ungqr!, (A, τ)))
-function _gpu_unmqr!(
-        side::AbstractChar, trans::AbstractChar, A::AbstractMatrix, τ::AbstractVector, C
-    )
-    throw(MethodError(_gpu_unmqr!, (side, trans, A, τ, C)))
-end
-
-Base.@deprecate(
-    _gpu_qr!(A, Q, R; kwargs...),
-    householder_qr!(default_householder_driver(A), A, Q, R; kwargs...)
-)
-
 function householder_qr!(
         driver::Union{CUSOLVER, ROCSOLVER}, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
         positive = true, blocksize = 1, pivoted = false
@@ -346,56 +216,6 @@ function householder_qr!(
     end
     return Q, R
 end
-
-Base.@deprecate(
-    _gpu_qr_null!(A, N; kwargs...),
-    householder_qr_null!(default_householder_driver(A), A, N; kwargs...)
-)
-function householder_qr_null!(
-        driver::Union{CUSOLVER, ROCSOLVER}, A::AbstractMatrix, N::AbstractMatrix;
-        positive = true, blocksize = 1, pivoted = false
-    )
-    blocksize > 1 &&
-        throw(ArgumentError(lazy"$driver does not provide a blocked implementation for a QR decomposition"))
-    pivoted &&
-        throw(ArgumentError(lazy"$driver does not provide a pivoted implementation for a QR decomposition"))
-    m, n = size(A)
-    minmn = min(m, n)
-    zero!(N)
-    one!(view(N, (minmn + 1):m, 1:(m - minmn)))
-    A, τ = _gpu_geqrf!(A)
-    N = _gpu_unmqr!('L', 'N', A, τ, N)
-    return N
-end
-
-# Native logic
-# --------------
-function qr_full!(A::AbstractMatrix, QR, alg::Native_HouseholderQR)
-    check_input(qr_full!, A, QR, alg)
-    Q, R = QR
-    A === Q &&
-        throw(ArgumentError("inplace Q not supported with native QR implementation"))
-    _native_qr!(A, Q, R; alg.kwargs...)
-    return Q, R
-end
-function qr_compact!(A::AbstractMatrix, QR, alg::Native_HouseholderQR)
-    check_input(qr_compact!, A, QR, alg)
-    Q, R = QR
-    A === Q &&
-        throw(ArgumentError("inplace Q not supported with native QR implementation"))
-    _native_qr!(A, Q, R; alg.kwargs...)
-    return Q, R
-end
-function qr_null!(A::AbstractMatrix, N, alg::Native_HouseholderQR)
-    check_input(qr_null!, A, N, alg)
-    _native_qr_null!(A, N; alg.kwargs...)
-    return N
-end
-
-Base.@deprecate(
-    _native_qr!(A, Q, R; kwargs...),
-    householder_qr!(Native(), A, Q, R; kwargs...)
-)
 function householder_qr!(
         ::Native, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
         positive::Bool = true # always true regardless of setting
@@ -410,7 +230,7 @@ function householder_qr!(
         for i in (j + 1):size(R, 1)
             R[i, j] = 0
         end
-        H = Householder(β, v, j:m)
+        H = HouseholderReflection(β, v, j:m)
         lmul!(H, A; cols = (j + 1):n)
         # A[j,j] == 1; store β instead
         A[j, j] = β
@@ -426,16 +246,46 @@ function householder_qr!(
     @inbounds for j in minmn:-1:1
         β = A[j, j]
         A[j, j] = 1
-        Hᴴ = Householder(conj(β), view(A, j:m, j), j:m)
+        Hᴴ = HouseholderReflection(conj(β), view(A, j:m, j), j:m)
         lmul!(Hᴴ, Q)
     end
     return Q, R
 end
 
-Base.@deprecate(
-    _native_qr_null!(A, N; kwargs...),
-    householder_qr_null!(Native(), A, N; kwargs...)
-)
+function householder_qr_null!(
+        ::LAPACK, A::AbstractMatrix, N::AbstractMatrix;
+        positive = true, pivoted = false, blocksize = YALAPACK.default_qr_blocksize(A)
+    )
+    m, n = size(A)
+    minmn = min(m, n)
+    fill!(N, zero(eltype(N)))
+    one!(view(N, (minmn + 1):m, 1:(m - minmn)))
+    if blocksize > 1
+        nb = min(minmn, blocksize)
+        A, T = YALAPACK.geqrt!(A, similar(A, nb, minmn))
+        N = YALAPACK.gemqrt!('L', 'N', A, T, N)
+    else
+        A, τ = YALAPACK.geqrf!(A)
+        N = YALAPACK.unmqr!('L', 'N', A, τ, N)
+    end
+    return N
+end
+function householder_qr_null!(
+        driver::Union{CUSOLVER, ROCSOLVER}, A::AbstractMatrix, N::AbstractMatrix;
+        positive = true, blocksize = 1, pivoted = false
+    )
+    blocksize > 1 &&
+        throw(ArgumentError(lazy"$driver does not provide a blocked implementation for a QR decomposition"))
+    pivoted &&
+        throw(ArgumentError(lazy"$driver does not provide a pivoted implementation for a QR decomposition"))
+    m, n = size(A)
+    minmn = min(m, n)
+    zero!(N)
+    one!(view(N, (minmn + 1):m, 1:(m - minmn)))
+    A, τ = _gpu_geqrf!(A)
+    N = _gpu_unmqr!('L', 'N', A, τ, N)
+    return N
+end
 function householder_qr_null!(
         ::Native, A::AbstractMatrix, N::AbstractMatrix; positive::Bool = true
     )
@@ -443,7 +293,7 @@ function householder_qr_null!(
     minmn = min(m, n)
     @inbounds for j in 1:minmn
         β, v, ν = _householder!(view(A, j:m, j), 1)
-        H = Householder(β, v, j:m)
+        H = HouseholderReflection(β, v, j:m)
         lmul!(H, A; cols = (j + 1):n)
         # A[j,j] == 1; store β instead
         A[j, j] = β
@@ -454,8 +304,75 @@ function householder_qr_null!(
     @inbounds for j in minmn:-1:1
         β = A[j, j]
         A[j, j] = 1
-        Hᴴ = Householder(conj(β), view(A, j:m, j), j:m)
+        Hᴴ = HouseholderReflection(conj(β), view(A, j:m, j), j:m)
         lmul!(Hᴴ, N)
     end
     return N
 end
+
+# Utility for sharing GPU implementations
+_gpu_geqrf!(A::AbstractMatrix) =
+    throw(MethodError(_gpu_geqrf!, (A,)))
+_gpu_ungqr!(A::AbstractMatrix, τ::AbstractVector) =
+    throw(MethodError(_gpu_ungqr!, (A, τ)))
+_gpu_unmqr!(side::AbstractChar, trans::AbstractChar, A::AbstractMatrix, τ::AbstractVector, C) =
+    throw(MethodError(_gpu_unmqr!, (side, trans, A, τ, C)))
+
+# Diagonal
+# --------
+function qr_full!(A, QR, alg::DiagonalAlgorithm)
+    check_input(qr_full!, A, QR, alg)
+    Q, R = QR
+    _diagonal_qr!(A, Q, R; alg.kwargs...)
+    return Q, R
+end
+function qr_compact!(A, QR, alg::DiagonalAlgorithm)
+    check_input(qr_compact!, A, QR, alg)
+    Q, R = QR
+    _diagonal_qr!(A, Q, R; alg.kwargs...)
+    return Q, R
+end
+function qr_null!(A, N, alg::DiagonalAlgorithm)
+    check_input(qr_null!, A, N, alg)
+    _diagonal_qr_null!(A, N; alg.kwargs...)
+    return N
+end
+
+function _diagonal_qr!(
+        A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix; positive::Bool = true
+    )
+    # note: Ad and Qd might share memory here so order of operations is important
+    Ad = diagview(A)
+    Qd = diagview(Q)
+    Rd = diagview(R)
+    if positive
+        @. Rd = abs(Ad)
+        @. Qd = sign_safe(Ad)
+    else
+        Rd .= Ad
+        one!(Q)
+    end
+    return Q, R
+end
+
+_diagonal_qr_null!(A::AbstractMatrix, N; positive::Bool = true) = N
+
+# Deprecations
+# ------------
+for drivertype in (:LAPACK, :CUSOLVER, :ROCSOLVER, :Native)
+    algtype = Symbol(drivertype, :_HouseholderQR)
+    @eval begin
+        Base.@deprecate(
+            qr_full!(A, QR, alg::$algtype),
+            qr_full!(A, QR, Householder($drivertype(), alg.kwargs))
+        )
+        Base.@deprecate(
+            qr_compact!(A, QR, alg::$algtype),
+            qr_compact!(A, QR, Householder($drivertype(), alg.kwargs))
+        )
+        Base.@deprecate(
+            qr_null!(A, N, alg::$algtype),
+            qr_null!(A, N, Householder($drivertype(), alg.kwargs))
+        )
+    end
+end

From 337adbe90648dfc5a02fc02ecc999557c42d5252 Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Thu, 26 Feb 2026 13:27:46 -0500
Subject: [PATCH 06/14] make QR select Householder by default

---
 src/interface/qr.jl | 17 ++++++-----------
 1 file changed, 6 insertions(+), 11 deletions(-)

diff --git a/src/interface/qr.jl b/src/interface/qr.jl
index c881b5a6d..3d03eb863 100644
--- a/src/interface/qr.jl
+++ b/src/interface/qr.jl
@@ -69,18 +69,13 @@ See also [`lq_full(!)`](@ref lq_full) and [`lq_compact(!)`](@ref lq_compact).
 # Algorithm selection
 # -------------------
 default_qr_algorithm(A; kwargs...) = default_qr_algorithm(typeof(A); kwargs...)
-function default_qr_algorithm(T::Type; kwargs...)
+
+default_qr_algorithm(T::Type; kwargs...) =
     throw(MethodError(default_qr_algorithm, (T,)))
-end
-function default_qr_algorithm(::Type{T}; kwargs...) where {T <: AbstractMatrix}
-    return Native_HouseholderQR(; kwargs...)
-end
-function default_qr_algorithm(::Type{T}; kwargs...) where {T <: YALAPACK.MaybeBlasMat}
-    return LAPACK_HouseholderQR(; kwargs...)
-end
-function default_qr_algorithm(::Type{T}; kwargs...) where {T <: Diagonal}
-    return DiagonalAlgorithm(; kwargs...)
-end
+default_qr_algorithm(::Type{T}; kwargs...) where {T <: AbstractMatrix} =
+    Householder(; kwargs...)
+default_qr_algorithm(::Type{T}; kwargs...) where {T <: Diagonal} =
+    DiagonalAlgorithm(; kwargs...)
 
 for f in (:qr_full!, :qr_compact!, :qr_null!)
     @eval function default_algorithm(::typeof($f), ::Type{A}; kwargs...) where {A}

From b48ab9a5a74c08dada284f3cff08876cc6dd7107 Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Thu, 26 Feb 2026 14:31:00 -0500
Subject: [PATCH 07/14] merge QR implementations

---
 .../MatrixAlgebraKitAMDGPUExt.jl              |  12 +-
 .../MatrixAlgebraKitCUDAExt.jl                |  15 +-
 src/implementations/qr.jl                     | 160 ++++++++----------
 3 files changed, 79 insertions(+), 108 deletions(-)

diff --git a/ext/MatrixAlgebraKitAMDGPUExt/MatrixAlgebraKitAMDGPUExt.jl b/ext/MatrixAlgebraKitAMDGPUExt/MatrixAlgebraKitAMDGPUExt.jl
index 8a7c2ef29..c86622546 100644
--- a/ext/MatrixAlgebraKitAMDGPUExt/MatrixAlgebraKitAMDGPUExt.jl
+++ b/ext/MatrixAlgebraKitAMDGPUExt/MatrixAlgebraKitAMDGPUExt.jl
@@ -4,9 +4,9 @@ using MatrixAlgebraKit
 using MatrixAlgebraKit: @algdef, Algorithm, check_input
 using MatrixAlgebraKit: one!, zero!, uppertriangular!, lowertriangular!
 using MatrixAlgebraKit: diagview, sign_safe
-using MatrixAlgebraKit: LQViaTransposedQR, TruncationStrategy, NoTruncation, TruncationByValue, AbstractAlgorithm
+using MatrixAlgebraKit: ROCSOLVER, LQViaTransposedQR, TruncationStrategy, NoTruncation, TruncationByValue, AbstractAlgorithm
 using MatrixAlgebraKit: default_qr_algorithm, default_lq_algorithm, default_svd_algorithm, default_eigh_algorithm
-import MatrixAlgebraKit: _gpu_geqrf!, _gpu_ungqr!, _gpu_unmqr!, _gpu_gesvd!, _gpu_Xgesvdp!, _gpu_gesvdj!
+import MatrixAlgebraKit: geqrf!, ungqr!, unmqr!, _gpu_gesvd!, _gpu_Xgesvdp!, _gpu_gesvdj!
 import MatrixAlgebraKit: _gpu_heevj!, _gpu_heevd!, _gpu_heev!, _gpu_heevx!, _sylvester, svd_rank
 using AMDGPU
 using LinearAlgebra
@@ -28,10 +28,10 @@ function MatrixAlgebraKit.default_eigh_algorithm(::Type{T}; kwargs...) where {T
     return ROCSOLVER_DivideAndConquer(; kwargs...)
 end
 
-_gpu_geqrf!(A::StridedROCMatrix) = YArocSOLVER.geqrf!(A)
-_gpu_ungqr!(A::StridedROCMatrix, τ::StridedROCVector) = YArocSOLVER.ungqr!(A, τ)
-_gpu_unmqr!(side::AbstractChar, trans::AbstractChar, A::StridedROCMatrix, τ::StridedROCVector, C::StridedROCVecOrMat) =
-    YArocSOLVER.unmqr!(side, trans, A, τ, C)
+for f in (:geqrf!, :ungqr!, :unmqr!)
+    @eval $f(::ROCSOLVER, args...) = YArocSOLVER.$f(args...)
+end
+
 _gpu_gesvd!(A::StridedROCMatrix, S::StridedROCVector, U::StridedROCMatrix, Vᴴ::StridedROCMatrix) =
     YArocSOLVER.gesvd!(A, S, U, Vᴴ)
 # not yet supported
diff --git a/ext/MatrixAlgebraKitCUDAExt/MatrixAlgebraKitCUDAExt.jl b/ext/MatrixAlgebraKitCUDAExt/MatrixAlgebraKitCUDAExt.jl
index fb67149e3..5b3f91f8c 100644
--- a/ext/MatrixAlgebraKitCUDAExt/MatrixAlgebraKitCUDAExt.jl
+++ b/ext/MatrixAlgebraKitCUDAExt/MatrixAlgebraKitCUDAExt.jl
@@ -4,9 +4,9 @@ using MatrixAlgebraKit
 using MatrixAlgebraKit: @algdef, Algorithm, check_input
 using MatrixAlgebraKit: one!, zero!, uppertriangular!, lowertriangular!
 using MatrixAlgebraKit: diagview, sign_safe
-using MatrixAlgebraKit: LQViaTransposedQR, TruncationByValue, AbstractAlgorithm
+using MatrixAlgebraKit: CUSOLVER, LQViaTransposedQR, TruncationByValue, AbstractAlgorithm
 using MatrixAlgebraKit: default_qr_algorithm, default_lq_algorithm, default_svd_algorithm, default_eig_algorithm, default_eigh_algorithm
-import MatrixAlgebraKit: _gpu_geqrf!, _gpu_ungqr!, _gpu_unmqr!, _gpu_gesvd!, _gpu_Xgesvdp!, _gpu_Xgesvdr!, _gpu_gesvdj!, _gpu_geev!
+import MatrixAlgebraKit: geqrf!, ungqr!, unmqr!, _gpu_gesvd!, _gpu_Xgesvdp!, _gpu_Xgesvdr!, _gpu_gesvdj!, _gpu_geev!
 import MatrixAlgebraKit: _gpu_heevj!, _gpu_heevd!, _sylvester, svd_rank
 using CUDA, CUDA.CUBLAS
 using CUDA: i32
@@ -32,6 +32,7 @@ function MatrixAlgebraKit.default_eigh_algorithm(::Type{T}; kwargs...) where {TT
     return CUSOLVER_DivideAndConquer(; kwargs...)
 end
 
+
 # include for block sector support
 function MatrixAlgebraKit.default_qr_algorithm(::Type{Base.ReshapedArray{T, 2, SubArray{T, 1, A, Tuple{UnitRange{Int}}, true}, Tuple{}}}; kwargs...) where {T <: BlasFloat, A <: CuVecOrMat{T}}
     return CUSOLVER_HouseholderQR(; kwargs...)
@@ -50,14 +51,12 @@ function MatrixAlgebraKit.default_eigh_algorithm(::Type{Base.ReshapedArray{T, 2,
     return CUSOLVER_DivideAndConquer(; kwargs...)
 end
 
+for f in (:geqrf!, :ungqr!, :unmqr!)
+    @eval $f(::CUSOLVER, args...) = YACUSOLVER.$f(args...)
+end
+
 _gpu_geev!(A::StridedCuMatrix, D::StridedCuVector, V::StridedCuMatrix) =
     YACUSOLVER.Xgeev!(A, D, V)
-_gpu_geqrf!(A::StridedCuMatrix) =
-    YACUSOLVER.geqrf!(A)
-_gpu_ungqr!(A::StridedCuMatrix, τ::StridedCuVector) =
-    YACUSOLVER.ungqr!(A, τ)
-_gpu_unmqr!(side::AbstractChar, trans::AbstractChar, A::StridedCuMatrix, τ::StridedCuVector, C::StridedCuVecOrMat) =
-    YACUSOLVER.unmqr!(side, trans, A, τ, C)
 _gpu_gesvd!(A::StridedCuMatrix, S::StridedCuVector, U::StridedCuMatrix, Vᴴ::StridedCuMatrix) =
     YACUSOLVER.gesvd!(A, S, U, Vᴴ)
 _gpu_Xgesvdp!(A::StridedCuMatrix, S::StridedCuVector, U::StridedCuMatrix, Vᴴ::StridedCuMatrix; kwargs...) =
diff --git a/src/implementations/qr.jl b/src/implementations/qr.jl
index 0c889ce13..34b3689ed 100644
--- a/src/implementations/qr.jl
+++ b/src/implementations/qr.jl
@@ -110,60 +110,85 @@ householder_qr!(::DefaultDriver, A, Q, R; kwargs...) =
 householder_qr_null!(::DefaultDriver, A, N; kwargs...) =
     householder_qr_null!(default_householder_driver(A), A, N; kwargs...)
 
+# dispatch helpers
+for f in (:geqrt!, :gemqrt!, :geqp3!, :geqrf!, :ungqr!, :unmqr!)
+    @eval begin
+        $f(::LAPACK, args...) = YALAPACK.$f(args...)
+    end
+end
+
 function householder_qr!(
-        ::LAPACK, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
+        driver::Union{LAPACK, CUSOLVER, ROCSOLVER}, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
         positive = true, pivoted = false,
-        blocksize = ((pivoted || A === Q) ? 1 : YALAPACK.default_qr_blocksize(A))
+        blocksize = ((driver !== LAPACK() || pivoted || A === Q) ? 1 : YALAPACK.default_qr_blocksize(A))
     )
+    # error messages for disallowing driver - setting combinations
+    (blocksize == 1 || driver === LAPACK()) ||
+        throw(ArgumentError(lazy"$driver does not provide a blocked QR decomposition"))
+    (!pivoted || driver === LAPACK()) ||
+        throw(ArgumentError(lazy"$driver does not provide a pivoted QR decomposition"))
+    pivoted && (blocksize > 1) &&
+        throw(ArgumentError(lazy"$driver does not provide a blocked pivoted QR decomposition"))
+
     m, n = size(A)
     minmn = min(m, n)
     computeR = length(R) > 0
     inplaceQ = Q === A
 
-    if pivoted && (blocksize > 1)
-        throw(ArgumentError("LAPACK does not provide a blocked implementation for a pivoted QR decomposition"))
-    end
-    if inplaceQ && (computeR || positive || blocksize > 1 || m < n)
-        throw(ArgumentError("inplace Q only supported if matrix is tall (`m >= n`), R is not required, and using the unblocked algorithm (`blocksize=1`) with `positive=false`"))
-    end
+    (inplaceQ && (computeR || positive || blocksize > 1 || m < n)) &&
+        throw(ArgumentError("inplace Q only supported if matrix is tall (`m >= n`), R is not required, and using the unblocked algorithm (`blocksize = 1`) with `positive = false`"))
 
+    # Compute QR in packed form
     if blocksize > 1
         nb = min(minmn, blocksize)
         if computeR # first use R as space for T
-            A, T = YALAPACK.geqrt!(A, view(R, 1:nb, 1:minmn))
+            A, T = geqrt!(driver, A, view(R, 1:nb, 1:minmn))
         else
-            A, T = YALAPACK.geqrt!(A, similar(A, nb, minmn))
+            A, T = geqrt!(driver, A, similar(A, nb, minmn))
         end
-        Q = YALAPACK.gemqrt!('L', 'N', A, T, one!(Q))
+        Q = gemqrt!(driver, 'L', 'N', A, T, one!(Q))
     else
         if pivoted
-            A, τ, jpvt = YALAPACK.geqp3!(A)
+            A, τ, jpvt = geqp3!(driver, A)
         else
-            A, τ = YALAPACK.geqrf!(A)
+            A, τ = geqrf!(driver, A)
         end
         if inplaceQ
-            Q = YALAPACK.ungqr!(A, τ)
+            Q = ungqr!(driver, A, τ)
         else
-            Q = YALAPACK.unmqr!('L', 'N', A, τ, one!(Q))
+            Q = unmqr!(driver, 'L', 'N', A, τ, one!(Q))
         end
     end
 
+
     if positive # already fix Q even if we do not need R
-        @inbounds for j in 1:minmn
-            s = sign_safe(A[j, j])
-            @simd for i in 1:m
-                Q[i, j] *= s
+        if driver === LAPACK()
+            @inbounds for j in 1:minmn
+                s = sign_safe(A[j, j])
+                @simd for i in 1:m
+                    Q[i, j] *= s
+                end
             end
+        else
+            # guaranteed τ exists and no longer needed
+            τ .= sign_safe.(diagview(A))
+            Qf = view(Q, 1:m, 1:minmn) # first minmn columns of Q
+            Qf .= Qf .* transpose(τ)
         end
     end
 
     if computeR
         R̃ = uppertriangular!(view(A, axes(R)...))
         if positive
-            @inbounds for j in n:-1:1
-                @simd for i in 1:min(minmn, j)
-                    R̃[i, j] = R̃[i, j] * conj(sign_safe(R̃[i, i]))
+            if driver === LAPACK()
+                @inbounds for j in n:-1:1
+                    @simd for i in 1:min(minmn, j)
+                        R̃[i, j] = R̃[i, j] * conj(sign_safe(R̃[i, i]))
+                    end
                 end
+            else
+                R̃f = view(R̃, 1:minmn, 1:n) # first minmn rows of R
+                R̃f .= conj.(τ) .* R̃f
             end
         end
         if !pivoted
@@ -175,47 +200,6 @@ function householder_qr!(
     end
     return Q, R
 end
-function householder_qr!(
-        driver::Union{CUSOLVER, ROCSOLVER}, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
-        positive = true, blocksize = 1, pivoted = false
-    )
-    blocksize > 1 &&
-        throw(ArgumentError(lazy"$driver does not provide a blocked implementation for a QR decomposition"))
-    pivoted &&
-        throw(ArgumentError(lazy"$driver does not provide a pivoted implementation for a QR decomposition"))
-    m, n = size(A)
-    minmn = min(m, n)
-    computeR = length(R) > 0
-    inplaceQ = Q === A
-    if inplaceQ && (computeR || positive || m < n)
-        throw(ArgumentError("inplace Q only supported if matrix is tall (`m >= n`), R is not required and using `positive=false`"))
-    end
-
-    A, τ = _gpu_geqrf!(A)
-    if inplaceQ
-        Q = _gpu_ungqr!(A, τ)
-    else
-        Q = _gpu_unmqr!('L', 'N', A, τ, one!(Q))
-    end
-    # henceforth, τ is no longer needed and can be reused
-
-    if positive # already fix Q even if we do not need R
-        # TODO: report that `lmul!` and `rmul!` with `Diagonal` don't work with CUDA
-        τ .= sign_safe.(diagview(A))
-        Qf = view(Q, 1:m, 1:minmn) # first minmn columns of Q
-        Qf .= Qf .* transpose(τ)
-    end
-
-    if computeR
-        R̃ = uppertriangular!(view(A, axes(R)...))
-        if positive
-            R̃f = view(R̃, 1:minmn, 1:n) # first minmn rows of R
-            R̃f .= conj.(τ) .* R̃f
-        end
-        copyto!(R, R̃)
-    end
-    return Q, R
-end
 function householder_qr!(
         ::Native, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
         positive::Bool = true # always true regardless of setting
@@ -253,41 +237,36 @@ function householder_qr!(
 end
 
 function householder_qr_null!(
-        ::LAPACK, A::AbstractMatrix, N::AbstractMatrix;
-        positive = true, pivoted = false, blocksize = YALAPACK.default_qr_blocksize(A)
+        driver::Union{LAPACK, CUSOLVER, ROCSOLVER}, A::AbstractMatrix, N::AbstractMatrix;
+        positive = true, pivoted = false,
+        blocksize = ((driver !== LAPACK() || pivoted) ? 1 : YALAPACK.default_qr_blocksize(A))
     )
+    # error messages for disallowing driver - setting combinations
+    (blocksize == 1 || driver === LAPACK()) ||
+        throw(ArgumentError(lazy"$driver does not provide a blocked QR decomposition"))
+    (!pivoted || driver === LAPACK()) ||
+        throw(ArgumentError(lazy"$driver does not provide a pivoted QR decomposition"))
+    pivoted && (blocksize > 1) &&
+        throw(ArgumentError(lazy"$driver does not provide a blocked pivoted QR decomposition"))
+
     m, n = size(A)
     minmn = min(m, n)
-    fill!(N, zero(eltype(N)))
+    zero!(N)
     one!(view(N, (minmn + 1):m, 1:(m - minmn)))
+
     if blocksize > 1
         nb = min(minmn, blocksize)
-        A, T = YALAPACK.geqrt!(A, similar(A, nb, minmn))
-        N = YALAPACK.gemqrt!('L', 'N', A, T, N)
+        A, T = geqrt!(driver, A, similar(A, nb, minmn))
+        N = gemqrt!(driver, 'L', 'N', A, T, N)
     else
-        A, τ = YALAPACK.geqrf!(A)
-        N = YALAPACK.unmqr!('L', 'N', A, τ, N)
+        A, τ = geqrf!(driver, A)
+        N = unmqr!(driver, 'L', 'N', A, τ, N)
     end
     return N
 end
 function householder_qr_null!(
-        driver::Union{CUSOLVER, ROCSOLVER}, A::AbstractMatrix, N::AbstractMatrix;
-        positive = true, blocksize = 1, pivoted = false
-    )
-    blocksize > 1 &&
-        throw(ArgumentError(lazy"$driver does not provide a blocked implementation for a QR decomposition"))
-    pivoted &&
-        throw(ArgumentError(lazy"$driver does not provide a pivoted implementation for a QR decomposition"))
-    m, n = size(A)
-    minmn = min(m, n)
-    zero!(N)
-    one!(view(N, (minmn + 1):m, 1:(m - minmn)))
-    A, τ = _gpu_geqrf!(A)
-    N = _gpu_unmqr!('L', 'N', A, τ, N)
-    return N
-end
-function householder_qr_null!(
-        ::Native, A::AbstractMatrix, N::AbstractMatrix; positive::Bool = true
+        ::Native, A::AbstractMatrix, N::AbstractMatrix;
+        positive::Bool = true
     )
     m, n = size(A)
     minmn = min(m, n)
@@ -310,13 +289,6 @@ function householder_qr_null!(
     return N
 end
 
-# Utility for sharing GPU implementations
-_gpu_geqrf!(A::AbstractMatrix) =
-    throw(MethodError(_gpu_geqrf!, (A,)))
-_gpu_ungqr!(A::AbstractMatrix, τ::AbstractVector) =
-    throw(MethodError(_gpu_ungqr!, (A, τ)))
-_gpu_unmqr!(side::AbstractChar, trans::AbstractChar, A::AbstractMatrix, τ::AbstractVector, C) =
-    throw(MethodError(_gpu_unmqr!, (side, trans, A, τ, C)))
 
 # Diagonal
 # --------

From c45935fc08e5bfb4c6c0755ada5832011af6ffc8 Mon Sep 17 00:00:00 2001
From: lkdvos <ldevos98@gmail.com>
Date: Thu, 26 Feb 2026 16:57:23 -0500
Subject: [PATCH 08/14] also implement LQ

---
 src/implementations/lq.jl | 309 ++++++++++++++++++++------------------
 src/interface/lq.jl       |  17 +--
 2 files changed, 166 insertions(+), 160 deletions(-)

diff --git a/src/implementations/lq.jl b/src/implementations/lq.jl
index b11533a8b..4e2c11507 100644
--- a/src/implementations/lq.jl
+++ b/src/implementations/lq.jl
@@ -86,65 +86,48 @@ for f! in (:lq_full!, :lq_compact!)
     end
 end
 
-# Implementation
-# --------------
-function lq_full!(A::AbstractMatrix, LQ, alg::LAPACK_HouseholderLQ)
-    check_input(lq_full!, A, LQ, alg)
-    L, Q = LQ
-    _lapack_lq!(A, L, Q; alg.kwargs...)
-    return L, Q
-end
-function lq_compact!(A::AbstractMatrix, LQ, alg::LAPACK_HouseholderLQ)
-    check_input(lq_compact!, A, LQ, alg)
-    L, Q = LQ
-    _lapack_lq!(A, L, Q; alg.kwargs...)
-    return L, Q
-end
-function lq_null!(A::AbstractMatrix, Nᴴ, alg::LAPACK_HouseholderLQ)
-    check_input(lq_null!, A, Nᴴ, alg)
-    _lapack_lq_null!(A, Nᴴ; alg.kwargs...)
-    return Nᴴ
-end
+# ==========================
+#      IMPLEMENTATIONS
+# ==========================
 
-function lq_full!(A::AbstractMatrix, LQ, alg::LQViaTransposedQR)
+# Householder
+# -----------
+function lq_full!(A, LQ, alg::Householder)
     check_input(lq_full!, A, LQ, alg)
-    L, Q = LQ
-    lq_via_qr!(A, L, Q, alg.qr_alg)
-    return L, Q
+    return householder_lq!(alg.driver, A, LQ...; alg.kwargs...)
 end
-function lq_compact!(A::AbstractMatrix, LQ, alg::LQViaTransposedQR)
+function lq_compact!(A, LQ, alg::Householder)
     check_input(lq_compact!, A, LQ, alg)
-    L, Q = LQ
-    lq_via_qr!(A, L, Q, alg.qr_alg)
-    return L, Q
+    return householder_lq!(alg.driver, A, LQ...; alg.kwargs...)
 end
-function lq_null!(A::AbstractMatrix, Nᴴ, alg::LQViaTransposedQR)
+function lq_null!(A, Nᴴ, alg::Householder)
     check_input(lq_null!, A, Nᴴ, alg)
-    lq_null_via_qr!(A, Nᴴ, alg.qr_alg)
-    return Nᴴ
+    return householder_lq_null!(alg.driver, A, Nᴴ; alg.kwargs...)
 end
 
-function lq_full!(A::AbstractMatrix, LQ, alg::DiagonalAlgorithm)
-    check_input(lq_full!, A, LQ, alg)
-    L, Q = LQ
-    _diagonal_lq!(A, L, Q; alg.kwargs...)
-    return L, Q
+householder_lq!(::DefaultDriver, A, L, Q; kwargs...) =
+    householder_lq!(default_householder_driver(A), A, L, Q; kwargs...)
+householder_lq_null!(::DefaultDriver, A, Nᴴ; kwargs...) =
+    householder_lq_null!(default_householder_driver(A), A, Nᴴ; kwargs...)
+
+# dispatch helpers
+for f in (:gelqt!, :gemlqt!, :gelqf!, :unglq!, :unmlq!)
+    @eval begin
+        $f(::LAPACK, args...) = YALAPACK.$f(args...)
+    end
 end
-function lq_compact!(A::AbstractMatrix, LQ, alg::DiagonalAlgorithm)
-    check_input(lq_compact!, A, LQ, alg)
-    L, Q = LQ
-    _diagonal_lq!(A, L, Q; alg.kwargs...)
-    return L, Q
+
+function householder_lq!(driver::Union{CUSOLVER, ROCSOLVER, GLA}, A, L, Q; kwargs...)
+    qr_alg = driver === GLA() ? GLA_HouseholderQR(; kwargs...) : Householder(driver; kwargs...)
+    return lq_via_qr!(A, L, Q, qr_alg)
 end
-function lq_null!(A::AbstractMatrix, N, alg::DiagonalAlgorithm)
-    check_input(lq_null!, A, N, alg)
-    return _diagonal_lq_null!(A, N; alg.kwargs...)
+function householder_lq_null!(driver::Union{CUSOLVER, ROCSOLVER, GLA}, A, Nᴴ; kwargs...)
+    qr_alg = driver === GLA() ? GLA_HouseholderQR(; kwargs...) : Householder(driver; kwargs...)
+    return lq_null_via_qr!(A, Nᴴ, qr_alg)
 end
 
-# LAPACK logic
-# ------------
-function _lapack_lq!(
-        A::AbstractMatrix, L::AbstractMatrix, Q::AbstractMatrix;
+function householder_lq!(
+        driver::LAPACK, A::AbstractMatrix, L::AbstractMatrix, Q::AbstractMatrix;
         positive = true, pivoted = false,
         blocksize = ((pivoted || A === Q) ? 1 : YALAPACK.default_qr_blocksize(A))
     )
@@ -153,31 +136,29 @@ function _lapack_lq!(
     computeL = length(L) > 0
     inplaceQ = Q === A
 
-    if pivoted && (blocksize > 1)
-        throw(ArgumentError("LAPACK does not provide a blocked implementation for a pivoted LQ decomposition"))
-    end
-    if inplaceQ && (computeL || positive || blocksize > 1 || n < m)
-        throw(ArgumentError("inplace Q only supported if matrix is wide (`m <= n`), L is not required, and using the unblocked algorithm (`blocksize=1`) with `positive=false`"))
-    end
+    pivoted && (blocksize > 1) &&
+        throw(ArgumentError(lazy"$driver does not provide a blocked pivoted LQ decomposition"))
+    (inplaceQ && (computeL || positive || blocksize > 1 || n < m)) &&
+        throw(ArgumentError("inplace Q only supported if matrix is wide (`m <= n`), L is not required, and using the unblocked algorithm (`blocksize = 1`) with `positive = false`"))
 
     if blocksize > 1
         mb = min(minmn, blocksize)
         if computeL # first use L as space for T
-            A, T = YALAPACK.gelqt!(A, view(L, 1:mb, 1:minmn))
+            A, T = gelqt!(driver, A, view(L, 1:mb, 1:minmn))
         else
-            A, T = YALAPACK.gelqt!(A, similar(A, mb, minmn))
+            A, T = gelqt!(driver, A, similar(A, mb, minmn))
         end
-        Q = YALAPACK.gemlqt!('R', 'N', A, T, one!(Q))
+        Q = gemlqt!(driver, 'R', 'N', A, T, one!(Q))
     else
-        A, τ = YALAPACK.gelqf!(A)
+        A, τ = gelqf!(driver, A)
         if inplaceQ
-            Q = YALAPACK.unglq!(A, τ)
+            Q = unglq!(driver, A, τ)
         else
-            Q = YALAPACK.unmlq!('R', 'N', A, τ, one!(Q))
+            Q = unmlq!(driver, 'R', 'N', A, τ, one!(Q))
         end
     end
 
-    if positive # already fix Q even if we do not need R
+    if positive # already fix Q even if we do not need L
         @inbounds for j in 1:n
             @simd for i in 1:minmn
                 s = sign_safe(A[i, i])
@@ -200,28 +181,102 @@ function _lapack_lq!(
     end
     return L, Q
 end
-
-function _lapack_lq_null!(
-        A::AbstractMatrix, Nᴴ::AbstractMatrix;
+function householder_lq_null!(
+        driver::LAPACK, A::AbstractMatrix, Nᴴ::AbstractMatrix;
         positive = true, pivoted = false, blocksize = YALAPACK.default_qr_blocksize(A)
     )
     m, n = size(A)
     minmn = min(m, n)
-    fill!(Nᴴ, zero(eltype(Nᴴ)))
+    zero!(Nᴴ)
     one!(view(Nᴴ, 1:(n - minmn), (minmn + 1):n))
     if blocksize > 1
         mb = min(minmn, blocksize)
-        A, T = YALAPACK.gelqt!(A, similar(A, mb, minmn))
-        Nᴴ = YALAPACK.gemlqt!('R', 'N', A, T, Nᴴ)
+        A, T = gelqt!(driver, A, similar(A, mb, minmn))
+        Nᴴ = gemlqt!(driver, 'R', 'N', A, T, Nᴴ)
     else
-        A, τ = YALAPACK.gelqf!(A)
-        Nᴴ = YALAPACK.unmlq!('R', 'N', A, τ, Nᴴ)
+        A, τ = gelqf!(driver, A)
+        Nᴴ = unmlq!(driver, 'R', 'N', A, τ, Nᴴ)
+    end
+    return Nᴴ
+end
+function householder_lq!(
+        ::Native, A::AbstractMatrix, L::AbstractMatrix, Q::AbstractMatrix;
+        positive::Bool = true # always true regardless of setting
+    )
+    m, n = size(A)
+    minmn = min(m, n)
+    @inbounds for i in 1:minmn
+        for j in 1:(i - 1)
+            L[i, j] = A[i, j]
+        end
+        β, v, L[i, i] = _householder!(conj!(view(A, i, i:n)), 1)
+        for j in (i + 1):size(L, 2)
+            L[i, j] = 0
+        end
+        H = HouseholderReflection(conj(β), v, i:n)
+        rmul!(A, H; rows = (i + 1):m)
+        # A[i, i] == 1; store β instead
+        A[i, i] = β
+    end
+    # copy remaining rows for m > n
+    @inbounds for j in 1:size(L, 2)
+        for i in (minmn + 1):m
+            L[i, j] = A[i, j]
+        end
+    end
+    # build Q
+    one!(Q)
+    @inbounds for i in minmn:-1:1
+        β = A[i, i]
+        A[i, i] = 1
+        Hᴴ = HouseholderReflection(β, view(A, i, i:n), i:n)
+        rmul!(Q, Hᴴ)
+    end
+    return L, Q
+end
+function householder_lq_null!(::Native, A::AbstractMatrix, Nᴴ::AbstractMatrix; positive::Bool = true)
+    m, n = size(A)
+    minmn = min(m, n)
+    @inbounds for i in 1:minmn
+        β, v, ν = _householder!(conj!(view(A, i, i:n)), 1)
+        H = HouseholderReflection(conj(β), v, i:n)
+        rmul!(A, H; rows = (i + 1):m)
+        # A[i, i] == 1; store β instead
+        A[i, i] = β
+    end
+    # build Nᴴ
+    zero!(Nᴴ)
+    one!(view(Nᴴ, 1:(n - minmn), (minmn + 1):n))
+    @inbounds for i in minmn:-1:1
+        β = A[i, i]
+        A[i, i] = 1
+        Hᴴ = HouseholderReflection(β, view(A, i, i:n), i:n)
+        rmul!(Nᴴ, Hᴴ)
     end
     return Nᴴ
 end
 
+
 # LQ via transposition and QR
 # ---------------------------
+function lq_full!(A::AbstractMatrix, LQ, alg::LQViaTransposedQR)
+    check_input(lq_full!, A, LQ, alg)
+    L, Q = LQ
+    lq_via_qr!(A, L, Q, alg.qr_alg)
+    return L, Q
+end
+function lq_compact!(A::AbstractMatrix, LQ, alg::LQViaTransposedQR)
+    check_input(lq_compact!, A, LQ, alg)
+    L, Q = LQ
+    lq_via_qr!(A, L, Q, alg.qr_alg)
+    return L, Q
+end
+function lq_null!(A::AbstractMatrix, Nᴴ, alg::LQViaTransposedQR)
+    check_input(lq_null!, A, Nᴴ, alg)
+    lq_null_via_qr!(A, Nᴴ, alg.qr_alg)
+    return Nᴴ
+end
+
 function lq_via_qr!(
         A::AbstractMatrix, L::AbstractMatrix, Q::AbstractMatrix, qr_alg::AbstractAlgorithm
     )
@@ -250,8 +305,26 @@ function lq_null_via_qr!(A::AbstractMatrix, N::AbstractMatrix, qr_alg::AbstractA
     return N
 end
 
-# Diagonal logic
-# --------------
+
+# Diagonal
+# --------
+function lq_full!(A::AbstractMatrix, LQ, alg::DiagonalAlgorithm)
+    check_input(lq_full!, A, LQ, alg)
+    L, Q = LQ
+    _diagonal_lq!(A, L, Q; alg.kwargs...)
+    return L, Q
+end
+function lq_compact!(A::AbstractMatrix, LQ, alg::DiagonalAlgorithm)
+    check_input(lq_compact!, A, LQ, alg)
+    L, Q = LQ
+    _diagonal_lq!(A, L, Q; alg.kwargs...)
+    return L, Q
+end
+function lq_null!(A::AbstractMatrix, N, alg::DiagonalAlgorithm)
+    check_input(lq_null!, A, N, alg)
+    return _diagonal_lq_null!(A, N; alg.kwargs...)
+end
+
 function _diagonal_lq!(
         A::AbstractMatrix, L::AbstractMatrix, Q::AbstractMatrix; positive::Bool = true
     )
@@ -271,84 +344,22 @@ end
 
 _diagonal_lq_null!(A::AbstractMatrix, N; positive::Bool = true) = N
 
-# Native logic
-# -------------
-function lq_full!(A::AbstractMatrix, LQ, alg::Native_HouseholderLQ)
-    check_input(lq_full!, A, LQ, alg)
-    L, Q = LQ
-    A === Q &&
-        throw(ArgumentError("inplace Q not supported with native LQ implementation"))
-    _native_lq!(A, L, Q; alg.kwargs...)
-    return L, Q
-end
-function lq_compact!(A::AbstractMatrix, LQ, alg::Native_HouseholderLQ)
-    check_input(lq_compact!, A, LQ, alg)
-    L, Q = LQ
-    A === Q &&
-        throw(ArgumentError("inplace Q not supported with native LQ implementation"))
-    _native_lq!(A, L, Q; alg.kwargs...)
-    return L, Q
-end
-function lq_null!(A::AbstractMatrix, N, alg::Native_HouseholderLQ)
-    check_input(lq_null!, A, N, alg)
-    _native_lq_null!(A, N; alg.kwargs...)
-    return N
-end
-
-function _native_lq!(
-        A::AbstractMatrix, L::AbstractMatrix, Q::AbstractMatrix;
-        positive::Bool = true # always true regardless of setting
-    )
-    m, n = size(A)
-    minmn = min(m, n)
-    @inbounds for i in 1:minmn
-        for j in 1:(i - 1)
-            L[i, j] = A[i, j]
-        end
-        β, v, L[i, i] = _householder!(conj!(view(A, i, i:n)), 1)
-        for j in (i + 1):size(L, 2)
-            L[i, j] = 0
-        end
-        H = Householder(conj(β), v, i:n)
-        rmul!(A, H; rows = (i + 1):m)
-        # A[i, i] == 1; store β instead
-        A[i, i] = β
-    end
-    # copy remaining rows for m > n
-    @inbounds for j in 1:size(L, 2)
-        for i in (minmn + 1):m
-            L[i, j] = A[i, j]
-        end
-    end
-    # build Q
-    one!(Q)
-    @inbounds for i in minmn:-1:1
-        β = A[i, i]
-        A[i, i] = 1
-        Hᴴ = Householder(β, view(A, i, i:n), i:n)
-        rmul!(Q, Hᴴ)
-    end
-    return L, Q
-end
-
-function _native_lq_null!(A::AbstractMatrix, Nᴴ::AbstractMatrix; positive::Bool = true)
-    m, n = size(A)
-    minmn = min(m, n)
-    @inbounds for i in 1:minmn
-        β, v, ν = _householder!(conj!(view(A, i, i:n)), 1)
-        H = Householder(conj(β), v, i:n)
-        rmul!(A, H; rows = (i + 1):m)
-        # A[i, i] == 1; store β instead
-        A[i, i] = β
-    end
-    # build Nᴴ
-    fill!(Nᴴ, zero(eltype(Nᴴ)))
-    one!(view(Nᴴ, 1:(n - minmn), (minmn + 1):n))
-    @inbounds for i in minmn:-1:1
-        β = A[i, i]
-        A[i, i] = 1
-        Hᴴ = Householder(β, view(A, i, i:n), i:n)
-        rmul!(Nᴴ, Hᴴ)
+# Deprecations
+# ------------
+for drivertype in (:LAPACK, :Native)
+    algtype = Symbol(drivertype, :_HouseholderLQ)
+    @eval begin
+        Base.@deprecate(
+            lq_full!(A, LQ, alg::$algtype),
+            lq_full!(A, LQ, Householder($drivertype(), alg.kwargs))
+        )
+        Base.@deprecate(
+            lq_compact!(A, LQ, alg::$algtype),
+            lq_compact!(A, LQ, Householder($drivertype(), alg.kwargs))
+        )
+        Base.@deprecate(
+            lq_null!(A, Nᴴ, alg::$algtype),
+            lq_null!(A, Nᴴ, Householder($drivertype(), alg.kwargs))
+        )
     end
-    return Nᴴ
 end
diff --git a/src/interface/lq.jl b/src/interface/lq.jl
index 8254c826f..12b6d8144 100644
--- a/src/interface/lq.jl
+++ b/src/interface/lq.jl
@@ -69,18 +69,13 @@ See also [`qr_full(!)`](@ref lq_full) and [`qr_compact(!)`](@ref lq_compact).
 # Algorithm selection
 # -------------------
 default_lq_algorithm(A; kwargs...) = default_lq_algorithm(typeof(A); kwargs...)
-function default_lq_algorithm(T::Type; kwargs...)
+
+default_lq_algorithm(T::Type; kwargs...) =
     throw(MethodError(default_lq_algorithm, (T,)))
-end
-function default_lq_algorithm(::Type{T}; kwargs...) where {T <: AbstractMatrix}
-    return Native_HouseholderLQ(; kwargs...)
-end
-function default_lq_algorithm(::Type{T}; kwargs...) where {T <: YALAPACK.MaybeBlasMat}
-    return LAPACK_HouseholderLQ(; kwargs...)
-end
-function default_lq_algorithm(::Type{T}; kwargs...) where {T <: Diagonal}
-    return DiagonalAlgorithm(; kwargs...)
-end
+default_lq_algorithm(::Type{T}; kwargs...) where {T <: AbstractMatrix} =
+    Householder(; kwargs...)
+default_lq_algorithm(::Type{T}; kwargs...) where {T <: Diagonal} =
+    DiagonalAlgorithm(; kwargs...)
 
 for f in (:lq_full!, :lq_compact!, :lq_null!)
     @eval function default_algorithm(::typeof($f), ::Type{A}; kwargs...) where {A}

From 5b1bc265f50838a9f4b6e95234cb9d5f9ed400b1 Mon Sep 17 00:00:00 2001
From: lkdvos <ldevos98@gmail.com>
Date: Thu, 26 Feb 2026 16:57:32 -0500
Subject: [PATCH 09/14] update algorithm list

---
 src/interface/decompositions.jl | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/interface/decompositions.jl b/src/interface/decompositions.jl
index 5cdb87993..f7026a09e 100644
--- a/src/interface/decompositions.jl
+++ b/src/interface/decompositions.jl
@@ -411,8 +411,8 @@ const GPU_SVDAlgorithm = Union{CUSOLVER_SVDAlgorithm, ROCSOLVER_SVDAlgorithm}
 const GPU_SVDPolar = Union{CUSOLVER_SVDPolar}
 const GPU_Randomized = Union{CUSOLVER_Randomized}
 
-const QRAlgorithms = Union{LAPACK_HouseholderQR, CUSOLVER_HouseholderQR, ROCSOLVER_HouseholderQR}
-const LQAlgorithms = Union{LAPACK_HouseholderLQ, LQViaTransposedQR}
+const QRAlgorithms = Union{Householder, LAPACK_HouseholderQR, Native_HouseholderQR, CUSOLVER_HouseholderQR, ROCSOLVER_HouseholderQR}
+const LQAlgorithms = Union{Householder, LAPACK_HouseholderLQ, Native_HouseholderLQ, LQViaTransposedQR}
 const SVDAlgorithms = Union{LAPACK_SVDAlgorithm, GPU_SVDAlgorithm}
 const PolarAlgorithms = Union{PolarViaSVD, PolarNewton}
 

From 06f3211bac92422390d3bc4ac5d647b16e41c389 Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Fri, 27 Feb 2026 09:07:24 -0500
Subject: [PATCH 10/14] update GLA QR implementation

---
 ...MatrixAlgebraKitGenericLinearAlgebraExt.jl | 73 +++++++------------
 src/implementations/qr.jl                     |  2 +-
 2 files changed, 29 insertions(+), 46 deletions(-)

diff --git a/ext/MatrixAlgebraKitGenericLinearAlgebraExt.jl b/ext/MatrixAlgebraKitGenericLinearAlgebraExt.jl
index fd4cec351..e9e1e3959 100644
--- a/ext/MatrixAlgebraKitGenericLinearAlgebraExt.jl
+++ b/ext/MatrixAlgebraKitGenericLinearAlgebraExt.jl
@@ -1,8 +1,7 @@
 module MatrixAlgebraKitGenericLinearAlgebraExt
 
 using MatrixAlgebraKit
-using MatrixAlgebraKit: sign_safe, check_input, diagview, gaugefix!, one!, default_fixgauge
-using MatrixAlgebraKit: left_orth_alg
+using MatrixAlgebraKit: sign_safe, check_input, diagview, gaugefix!, one!, zero!, default_fixgauge
 using GenericLinearAlgebra: svd!, svdvals!, eigen!, eigvals!, Hermitian, qr!
 using LinearAlgebra: I, Diagonal, lmul!
 
@@ -57,38 +56,25 @@ function MatrixAlgebraKit.eigh_vals!(A::AbstractMatrix, D, ::GLA_QRIteration)
     return eigvals!(Hermitian(A); sortby = real)
 end
 
-function MatrixAlgebraKit.default_qr_algorithm(::Type{T}; kwargs...) where {T <: StridedMatrix{<:Union{Float16, ComplexF16, BigFloat, Complex{BigFloat}}}}
-    return GLA_HouseholderQR(; kwargs...)
-end
-
-function MatrixAlgebraKit.qr_full!(A::AbstractMatrix, QR, alg::GLA_HouseholderQR)
-    check_input(qr_full!, A, QR, alg)
-    Q, R = QR
-    return _gla_householder_qr!(A, Q, R; alg.kwargs...)
-end
-
-function MatrixAlgebraKit.qr_compact!(A::AbstractMatrix, QR, alg::GLA_HouseholderQR)
-    check_input(qr_compact!, A, QR, alg)
-    Q, R = QR
-    return _gla_householder_qr!(A, Q, R; alg.kwargs...)
-end
-
-function MatrixAlgebraKit.qr_null!(A::AbstractMatrix, N, alg::GLA_HouseholderQR)
-    check_input(qr_null!, A, N, alg)
-    return _gla_householder_qr_null!(A, N; alg.kwargs...)
-end
-
-function _gla_householder_qr!(A::AbstractMatrix, Q, R; positive = true, blocksize = 1, pivoted = false)
-    pivoted && throw(ArgumentError("Only pivoted = false implemented for GLA_HouseholderQR."))
-    (blocksize == 1) || throw(ArgumentError("Only blocksize = 1 implemented for GLA_HouseholderQR."))
+function MatrixAlgebraKit.householder_qr!(
+        driver::MatrixAlgebraKit.GLA, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
+        positive::Bool = true, pivoted::Bool = false, blocksize::Int = 1
+    )
+    blocksize == 1 ||
+        throw(ArgumentError(lazy"$driver does not provide a blocked QR decomposition"))
+    pivoted &&
+        throw(ArgumentError(lazy"$driver does not provide a pivoted QR decomposition"))
 
     m, n = size(A)
-    k = min(m, n)
+    minmn = min(m, n)
+    computeR = length(R) > 0
+
+    # compute QR
     Q̃, R̃ = qr!(A)
     lmul!(Q̃, MatrixAlgebraKit.one!(Q))
 
     if positive
-        @inbounds for j in 1:k
+        @inbounds for j in 1:minmn
             s = sign_safe(R̃[j, j])
             @simd for i in 1:m
                 Q[i, j] *= s
@@ -96,42 +82,39 @@ function _gla_householder_qr!(A::AbstractMatrix, Q, R; positive = true, blocksiz
         end
     end
 
-    computeR = length(R) > 0
     if computeR
         if positive
             @inbounds for j in n:-1:1
-                @simd for i in 1:min(k, j)
+                @simd for i in 1:min(minmn, j)
                     R[i, j] = R̃[i, j] * conj(sign_safe(R̃[i, i]))
                 end
-                @simd for i in (min(k, j) + 1):size(R, 1)
+                @simd for i in (min(minmn, j) + 1):size(R, 1)
                     R[i, j] = zero(eltype(R))
                 end
             end
         else
-            R[1:k, :] .= R̃
-            MatrixAlgebraKit.zero!(@view(R[(k + 1):end, :]))
+            R[1:minmn, :] .= R̃
+            MatrixAlgebraKit.zero!(@view(R[(minmn + 1):end, :]))
         end
     end
     return Q, R
 end
 
-function _gla_householder_qr_null!(
-        A::AbstractMatrix, N::AbstractMatrix;
-        positive = true, blocksize = 1, pivoted = false
+function MatrixAlgebraKit.householder_qr_null!(
+        driver::MatrixAlgebraKit.GLA, A::AbstractMatrix, N::AbstractMatrix;
+        positive::Bool = true, pivoted::Bool = false, blocksize::Int = 1
     )
-    pivoted && throw(ArgumentError("Only pivoted = false implemented for GLA_HouseholderQR."))
-    (blocksize == 1) || throw(ArgumentError("Only blocksize = 1 implemented for GLA_HouseholderQR."))
+    blocksize == 1 ||
+        throw(ArgumentError(lazy"$driver does not provide a blocked QR decomposition"))
+    pivoted &&
+        throw(ArgumentError(lazy"$driver does not provide a pivoted QR decomposition"))
+
     m, n = size(A)
     minmn = min(m, n)
-    fill!(N, zero(eltype(N)))
+    zero!(N)
     one!(view(N, (minmn + 1):m, 1:(m - minmn)))
     Q̃, = qr!(A)
-    lmul!(Q̃, N)
-    return N
-end
-
-function MatrixAlgebraKit.default_lq_algorithm(::Type{T}; kwargs...) where {T <: StridedMatrix{<:Union{Float16, ComplexF16, BigFloat, Complex{BigFloat}}}}
-    return MatrixAlgebraKit.LQViaTransposedQR(GLA_HouseholderQR(; kwargs...))
+    return lmul!(Q̃, N)
 end
 
 MatrixAlgebraKit.left_orth_alg(alg::GLA_HouseholderQR) = MatrixAlgebraKit.LeftOrthViaQR(alg)
diff --git a/src/implementations/qr.jl b/src/implementations/qr.jl
index 34b3689ed..4cac2e003 100644
--- a/src/implementations/qr.jl
+++ b/src/implementations/qr.jl
@@ -331,7 +331,7 @@ _diagonal_qr_null!(A::AbstractMatrix, N; positive::Bool = true) = N
 
 # Deprecations
 # ------------
-for drivertype in (:LAPACK, :CUSOLVER, :ROCSOLVER, :Native)
+for drivertype in (:LAPACK, :CUSOLVER, :ROCSOLVER, :Native, :GLA)
     algtype = Symbol(drivertype, :_HouseholderQR)
     @eval begin
         Base.@deprecate(

From f52edc83655e5ebed517d3210771c73b64193842 Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Fri, 27 Feb 2026 09:13:07 -0500
Subject: [PATCH 11/14] some cleanup

---
 src/implementations/qr.jl | 36 +++++++++++++++++++++++++-----------
 1 file changed, 25 insertions(+), 11 deletions(-)

diff --git a/src/implementations/qr.jl b/src/implementations/qr.jl
index 4cac2e003..0cae7d272 100644
--- a/src/implementations/qr.jl
+++ b/src/implementations/qr.jl
@@ -119,8 +119,8 @@ end
 
 function householder_qr!(
         driver::Union{LAPACK, CUSOLVER, ROCSOLVER}, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
-        positive = true, pivoted = false,
-        blocksize = ((driver !== LAPACK() || pivoted || A === Q) ? 1 : YALAPACK.default_qr_blocksize(A))
+        positive::Bool = true, pivoted::Bool = false,
+        blocksize::Int = ((driver !== LAPACK() || pivoted || A === Q) ? 1 : YALAPACK.default_qr_blocksize(A))
     )
     # error messages for disallowing driver - setting combinations
     (blocksize == 1 || driver === LAPACK()) ||
@@ -160,7 +160,6 @@ function householder_qr!(
         end
     end
 
-
     if positive # already fix Q even if we do not need R
         if driver === LAPACK()
             @inbounds for j in 1:minmn
@@ -201,9 +200,16 @@ function householder_qr!(
     return Q, R
 end
 function householder_qr!(
-        ::Native, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
-        positive::Bool = true # always true regardless of setting
+        driver::Native, A::AbstractMatrix, Q::AbstractMatrix, R::AbstractMatrix;
+        positive::Bool = true, pivoted::Bool = false, blocksize::Int = 1
     )
+    # error messages for disallowing driver - setting combinations
+    blocksize == 1 ||
+        throw(ArgumentError(lazy"$driver does not provide a blocked QR decomposition"))
+    pivoted &&
+        throw(ArgumentError(lazy"$driver does not provide a pivoted QR decomposition"))
+    # positive = true regardless of setting
+
     m, n = size(A)
     minmn = min(m, n)
     @inbounds for j in 1:minmn
@@ -238,8 +244,8 @@ end
 
 function householder_qr_null!(
         driver::Union{LAPACK, CUSOLVER, ROCSOLVER}, A::AbstractMatrix, N::AbstractMatrix;
-        positive = true, pivoted = false,
-        blocksize = ((driver !== LAPACK() || pivoted) ? 1 : YALAPACK.default_qr_blocksize(A))
+        positive::Bool = true, pivoted::Bool = false,
+        blocksize::Int = ((driver !== LAPACK() || pivoted) ? 1 : YALAPACK.default_qr_blocksize(A))
     )
     # error messages for disallowing driver - setting combinations
     (blocksize == 1 || driver === LAPACK()) ||
@@ -265,20 +271,28 @@ function householder_qr_null!(
     return N
 end
 function householder_qr_null!(
-        ::Native, A::AbstractMatrix, N::AbstractMatrix;
-        positive::Bool = true
+        driver::Native, A::AbstractMatrix, N::AbstractMatrix;
+        positive::Bool = true, pivoted::Bool = false, blocksize::Int = 1
     )
+    # error messages for disallowing driver - setting combinations
+    blocksize == 1 ||
+        throw(ArgumentError(lazy"$driver does not provide a blocked QR decomposition"))
+    pivoted &&
+        throw(ArgumentError(lazy"$driver does not provide a pivoted QR decomposition"))
+
     m, n = size(A)
     minmn = min(m, n)
+
     @inbounds for j in 1:minmn
         β, v, ν = _householder!(view(A, j:m, j), 1)
         H = HouseholderReflection(β, v, j:m)
         lmul!(H, A; cols = (j + 1):n)
-        # A[j,j] == 1; store β instead
+        # A[j, j] == 1; store β instead
         A[j, j] = β
     end
+
     # build N
-    fill!(N, zero(eltype(N)))
+    zero!(N)
     one!(view(N, (minmn + 1):m, 1:(m - minmn)))
     @inbounds for j in minmn:-1:1
         β = A[j, j]

From 0eb574866e9225e12badc53aa08319d4e4949da6 Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Fri, 27 Feb 2026 09:37:50 -0500
Subject: [PATCH 12/14] remove reference to undefined docstring

---
 src/algorithms.jl | 1 -
 1 file changed, 1 deletion(-)

diff --git a/src/algorithms.jl b/src/algorithms.jl
index b6e1bff25..8ffc22a5c 100644
--- a/src/algorithms.jl
+++ b/src/algorithms.jl
@@ -157,7 +157,6 @@ abstract type Driver end
     DefaultDriver <: Driver
 
 Select a default driver at runtime, based on the input matrix.
-See also [`default_householder_driver`](@ref).
 """
 struct DefaultDriver <: Driver end
 

From f0886e82c6841ac2e551a8f2d40f9a54f822f0b5 Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Fri, 27 Feb 2026 09:40:21 -0500
Subject: [PATCH 13/14] update algorithm tests

---
 test/algorithms.jl | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/test/algorithms.jl b/test/algorithms.jl
index 0a2ccfec7..a2f5ad093 100644
--- a/test/algorithms.jl
+++ b/test/algorithms.jl
@@ -2,7 +2,7 @@ using MatrixAlgebraKit
 using Test
 using TestExtras
 using MatrixAlgebraKit: LAPACK_SVDAlgorithm, PolarViaSVD, TruncatedAlgorithm,
-    default_algorithm, select_algorithm
+    default_algorithm, select_algorithm, Householder
 
 @testset "default_algorithm" begin
     A = randn(3, 3)
@@ -17,21 +17,21 @@ using MatrixAlgebraKit: LAPACK_SVDAlgorithm, PolarViaSVD, TruncatedAlgorithm,
             LAPACK_MultipleRelativelyRobustRepresentations()
     end
     for f in (lq_full!, lq_full, lq_compact!, lq_compact, lq_null!, lq_null)
-        @test @constinferred(default_algorithm(f, A)) == LAPACK_HouseholderLQ()
+        @test @constinferred(default_algorithm(f, A)) == Householder()
     end
     for f in (left_polar!, left_polar, right_polar!, right_polar)
         @test @constinferred(default_algorithm(f, A)) ==
             PolarViaSVD(LAPACK_DivideAndConquer())
     end
     for f in (qr_full!, qr_full, qr_compact!, qr_compact, qr_null!, qr_null)
-        @test @constinferred(default_algorithm(f, A)) == LAPACK_HouseholderQR()
+        @test @constinferred(default_algorithm(f, A)) == Householder()
     end
     for f in (schur_full!, schur_full, schur_vals!, schur_vals)
         @test @constinferred(default_algorithm(f, A)) === LAPACK_Expert()
     end
 
-    @test @constinferred(default_algorithm(qr_compact!, A; blocksize = 2)) ===
-        LAPACK_HouseholderQR(; blocksize = 2)
+    @test @constinferred(default_algorithm(qr_compact!, A; blocksize = 2)) ==
+        Householder(; blocksize = 2)
 end
 
 @testset "select_algorithm" begin

From 841ebacc016c7eb4aceb64d2a8ced7a5c354d89f Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Fri, 27 Feb 2026 09:44:32 -0500
Subject: [PATCH 14/14] fix JET errors

---
 src/implementations/qr.jl | 1 +
 1 file changed, 1 insertion(+)

diff --git a/src/implementations/qr.jl b/src/implementations/qr.jl
index 0cae7d272..ac60cd962 100644
--- a/src/implementations/qr.jl
+++ b/src/implementations/qr.jl
@@ -113,6 +113,7 @@ householder_qr_null!(::DefaultDriver, A, N; kwargs...) =
 # dispatch helpers
 for f in (:geqrt!, :gemqrt!, :geqp3!, :geqrf!, :ungqr!, :unmqr!)
     @eval begin
+        $f(driver::Driver, args...) = throw(MethodError($f, (driver, args...))) # make JET not complain
         $f(::LAPACK, args...) = YALAPACK.$f(args...)
     end
 end