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qr.jl
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qr.jl
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# This file is a part of Julia. License is MIT: https://julialang.org/license
# QR Factorization
"""
QR <: Factorization
A QR matrix factorization stored in a packed format, typically obtained from
[`qr`](@ref). If ``A`` is an `m`×`n` matrix, then
```math
A = Q R
```
where ``Q`` is an orthogonal/unitary matrix and ``R`` is upper triangular.
The matrix ``Q`` is stored as a sequence of Householder reflectors ``v_i``
and coefficients ``\\tau_i`` where:
```math
Q = \\prod_{i=1}^{\\min(m,n)} (I - \\tau_i v_i v_i^T).
```
Iterating the decomposition produces the components `Q` and `R`.
The object has two fields:
* `factors` is an `m`×`n` matrix.
- The upper triangular part contains the elements of ``R``, that is `R =
triu(F.factors)` for a `QR` object `F`.
- The subdiagonal part contains the reflectors ``v_i`` stored in a packed format where
``v_i`` is the ``i``th column of the matrix `V = I + tril(F.factors, -1)`.
* `τ` is a vector of length `min(m,n)` containing the coefficients ``\tau_i``.
"""
struct QR{T,S<:AbstractMatrix{T},C<:AbstractVector{T}} <: Factorization{T}
factors::S
τ::C
function QR{T,S,C}(factors, τ) where {T,S<:AbstractMatrix{T},C<:AbstractVector{T}}
require_one_based_indexing(factors)
new{T,S,C}(factors, τ)
end
end
QR(factors::AbstractMatrix{T}, τ::AbstractVector{T}) where {T} =
QR{T,typeof(factors),typeof(τ)}(factors, τ)
QR{T}(factors::AbstractMatrix, τ::AbstractVector) where {T} =
QR(convert(AbstractMatrix{T}, factors), convert(AbstractVector{T}, τ))
# backwards-compatible constructors (remove with Julia 2.0)
@deprecate(QR{T,S}(factors::AbstractMatrix{T}, τ::AbstractVector{T}) where {T,S},
QR{T,S,typeof(τ)}(factors, τ), false)
# iteration for destructuring into components
Base.iterate(S::QR) = (S.Q, Val(:R))
Base.iterate(S::QR, ::Val{:R}) = (S.R, Val(:done))
Base.iterate(S::QR, ::Val{:done}) = nothing
# Note. For QRCompactWY factorization without pivoting, the WY representation based method introduced in LAPACK 3.4
"""
QRCompactWY <: Factorization
A QR matrix factorization stored in a compact blocked format, typically obtained from
[`qr`](@ref). If ``A`` is an `m`×`n` matrix, then
```math
A = Q R
```
where ``Q`` is an orthogonal/unitary matrix and ``R`` is upper triangular. It is similar
to the [`QR`](@ref) format except that the orthogonal/unitary matrix ``Q`` is stored in
*Compact WY* format [^Schreiber1989]. For the block size ``n_b``, it is stored as
a `m`×`n` lower trapezoidal matrix ``V`` and a matrix ``T = (T_1 \\; T_2 \\; ... \\;
T_{b-1} \\; T_b')`` composed of ``b = \\lceil \\min(m,n) / n_b \\rceil`` upper triangular
matrices ``T_j`` of size ``n_b``×``n_b`` (``j = 1, ..., b-1``) and an upper trapezoidal
``n_b``×``\\min(m,n) - (b-1) n_b`` matrix ``T_b'`` (``j=b``) whose upper square part
denoted with ``T_b`` satisfying
```math
Q = \\prod_{i=1}^{\\min(m,n)} (I - \\tau_i v_i v_i^T)
= \\prod_{j=1}^{b} (I - V_j T_j V_j^T)
```
such that ``v_i`` is the ``i``th column of ``V``, ``\\tau_i`` is the ``i``th element
of `[diag(T_1); diag(T_2); …; diag(T_b)]`, and ``(V_1 \\; V_2 \\; ... \\; V_b)``
is the left `m`×`min(m, n)` block of ``V``. When constructed using [`qr`](@ref),
the block size is given by ``n_b = \\min(m, n, 36)``.
Iterating the decomposition produces the components `Q` and `R`.
The object has two fields:
* `factors`, as in the [`QR`](@ref) type, is an `m`×`n` matrix.
- The upper triangular part contains the elements of ``R``, that is `R =
triu(F.factors)` for a `QR` object `F`.
- The subdiagonal part contains the reflectors ``v_i`` stored in a packed format such
that `V = I + tril(F.factors, -1)`.
* `T` is a ``n_b``-by-``\\min(m,n)`` matrix as described above. The subdiagonal elements
for each triangular matrix ``T_j`` are ignored.
!!! note
This format should not to be confused with the older *WY* representation
[^Bischof1987].
[^Bischof1987]: C Bischof and C Van Loan, "The WY representation for products of Householder matrices", SIAM J Sci Stat Comput 8 (1987), s2-s13. [doi:10.1137/0908009](https://doi.org/10.1137/0908009)
[^Schreiber1989]: R Schreiber and C Van Loan, "A storage-efficient WY representation for products of Householder transformations", SIAM J Sci Stat Comput 10 (1989), 53-57. [doi:10.1137/0910005](https://doi.org/10.1137/0910005)
"""
struct QRCompactWY{S,M<:AbstractMatrix{S},C<:AbstractMatrix{S}} <: Factorization{S}
factors::M
T::C
function QRCompactWY{S,M,C}(factors, T) where {S,M<:AbstractMatrix{S},C<:AbstractMatrix{S}}
require_one_based_indexing(factors)
new{S,M,C}(factors, T)
end
end
QRCompactWY(factors::AbstractMatrix{S}, T::AbstractMatrix{S}) where {S} =
QRCompactWY{S,typeof(factors),typeof(T)}(factors, T)
QRCompactWY{S}(factors::AbstractMatrix, T::AbstractMatrix) where {S} =
QRCompactWY(convert(AbstractMatrix{S}, factors), convert(AbstractMatrix{S}, T))
# backwards-compatible constructors (remove with Julia 2.0)
@deprecate(QRCompactWY{S,M}(factors::AbstractMatrix{S}, T::AbstractMatrix{S}) where {S,M},
QRCompactWY{S,M,typeof(T)}(factors, T), false)
# iteration for destructuring into components
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
A QR matrix factorization with column pivoting in a packed format, typically obtained from
[`qr`](@ref). If ``A`` is an `m`×`n` matrix, then
```math
A P = Q R
```
where ``P`` is a permutation matrix, ``Q`` is an orthogonal/unitary matrix and ``R`` is
upper triangular. The matrix ``Q`` is stored as a sequence of Householder reflectors:
```math
Q = \\prod_{i=1}^{\\min(m,n)} (I - \\tau_i v_i v_i^T).
```
Iterating the decomposition produces the components `Q`, `R`, and `p`.
The object has three fields:
* `factors` is an `m`×`n` matrix.
- The upper triangular part contains the elements of ``R``, that is `R =
triu(F.factors)` for a `QR` object `F`.
- The subdiagonal part contains the reflectors ``v_i`` stored in a packed format where
``v_i`` is the ``i``th column of the matrix `V = I + tril(F.factors, -1)`.
* `τ` is a vector of length `min(m,n)` containing the coefficients ``\tau_i``.
* `jpvt` is an integer vector of length `n` corresponding to the permutation ``P``.
"""
struct QRPivoted{T,S<:AbstractMatrix{T},C<:AbstractVector{T},P<:AbstractVector{<:Integer}} <: Factorization{T}
factors::S
τ::C
jpvt::P
function QRPivoted{T,S,C,P}(factors, τ, jpvt) where {T,S<:AbstractMatrix{T},C<:AbstractVector{T},P<:AbstractVector{<:Integer}}
require_one_based_indexing(factors, τ, jpvt)
new{T,S,C,P}(factors, τ, jpvt)
end
end
QRPivoted(factors::AbstractMatrix{T}, τ::AbstractVector{T},
jpvt::AbstractVector{<:Integer}) where {T} =
QRPivoted{T,typeof(factors),typeof(τ),typeof(jpvt)}(factors, τ, jpvt)
QRPivoted{T}(factors::AbstractMatrix, τ::AbstractVector,
jpvt::AbstractVector{<:Integer}) where {T} =
QRPivoted(convert(AbstractMatrix{T}, factors), convert(AbstractVector{T}, τ), jpvt)
# backwards-compatible constructors (remove with Julia 2.0)
@deprecate(QRPivoted{T,S}(factors::AbstractMatrix{T}, τ::AbstractVector{T},
jpvt::AbstractVector{<:Integer}) where {T,S},
QRPivoted{T,S,typeof(τ),typeof(jpvt)}(factors, τ, jpvt), false)
# iteration for destructuring into components
Base.iterate(S::QRPivoted) = (S.Q, Val(:R))
Base.iterate(S::QRPivoted, ::Val{:R}) = (S.R, Val(:p))
Base.iterate(S::QRPivoted, ::Val{:p}) = (S.p, Val(:done))
Base.iterate(S::QRPivoted, ::Val{:done}) = nothing
function qrfactUnblocked!(A::AbstractMatrix{T}) where {T}
require_one_based_indexing(A)
m, n = size(A)
τ = zeros(T, min(m,n))
for k = 1:min(m - 1 + !(T<:Real), n)
x = view(A, k:m, k)
τk = reflector!(x)
τ[k] = τk
reflectorApply!(x, τk, view(A, k:m, k + 1:n))
end
QR(A, τ)
end
# Find index for columns with largest two norm
function indmaxcolumn(A::AbstractMatrix)
mm = norm(view(A, :, 1))
ii = 1
for i = 2:size(A, 2)
mi = norm(view(A, :, i))
if abs(mi) > mm
mm = mi
ii = i
end
end
return ii
end
function qrfactPivotedUnblocked!(A::AbstractMatrix)
m, n = size(A)
piv = Vector(UnitRange{BlasInt}(1,n))
τ = Vector{eltype(A)}(undef, min(m,n))
for j = 1:min(m,n)
# Find column with maximum norm in trailing submatrix
jm = indmaxcolumn(view(A, j:m, j:n)) + j - 1
if jm != j
# Flip elements in pivoting vector
tmpp = piv[jm]
piv[jm] = piv[j]
piv[j] = tmpp
# Update matrix with
for i = 1:m
tmp = A[i,jm]
A[i,jm] = A[i,j]
A[i,j] = tmp
end
end
# Compute reflector of columns j
x = view(A, j:m, j)
τj = reflector!(x)
τ[j] = τj
# Update trailing submatrix with reflector
reflectorApply!(x, τj, view(A, j:m, j+1:n))
end
return QRPivoted{eltype(A), typeof(A), typeof(τ), typeof(piv)}(A, τ, piv)
end
# LAPACK version
qr!(A::StridedMatrix{<:BlasFloat}, ::NoPivot; blocksize=36) =
QRCompactWY(LAPACK.geqrt!(A, min(min(size(A)...), blocksize))...)
qr!(A::StridedMatrix{<:BlasFloat}, ::ColumnNorm) = QRPivoted(LAPACK.geqp3!(A)...)
# Generic fallbacks
"""
qr!(A, pivot = NoPivot(); blocksize)
`qr!` is the same as [`qr`](@ref) when `A` is a subtype of [`StridedMatrix`](@ref),
but saves space by overwriting the input `A`, instead of creating a copy.
An [`InexactError`](@ref) exception is thrown if the factorization produces a number not
representable by the element type of `A`, e.g. for integer types.
!!! compat "Julia 1.4"
The `blocksize` keyword argument requires Julia 1.4 or later.
# Examples
```jldoctest
julia> a = [1. 2.; 3. 4.]
2×2 Matrix{Float64}:
1.0 2.0
3.0 4.0
julia> qr!(a)
LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}
Q factor:
2×2 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}:
-0.316228 -0.948683
-0.948683 0.316228
R factor:
2×2 Matrix{Float64}:
-3.16228 -4.42719
0.0 -0.632456
julia> a = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> qr!(a)
ERROR: InexactError: Int64(3.1622776601683795)
Stacktrace:
[...]
```
"""
qr!(A::AbstractMatrix, ::NoPivot) = qrfactUnblocked!(A)
qr!(A::AbstractMatrix, ::ColumnNorm) = qrfactPivotedUnblocked!(A)
qr!(A::AbstractMatrix) = qr!(A, NoPivot())
# TODO: Remove in Julia v2.0
@deprecate qr!(A::AbstractMatrix, ::Val{true}) qr!(A, ColumnNorm())
@deprecate qr!(A::AbstractMatrix, ::Val{false}) qr!(A, NoPivot())
_qreltype(::Type{T}) where T = typeof(zero(T)/sqrt(abs2(one(T))))
"""
qr(A, pivot = NoPivot(); blocksize) -> F
Compute the QR factorization of the matrix `A`: an orthogonal (or unitary if `A` is
complex-valued) matrix `Q`, and an upper triangular matrix `R` such that
```math
A = Q R
```
The returned object `F` stores the factorization in a packed format:
- if `pivot == ColumnNorm()` then `F` is a [`QRPivoted`](@ref) object,
- otherwise if the element type of `A` is a BLAS type ([`Float32`](@ref), [`Float64`](@ref),
`ComplexF32` or `ComplexF64`), then `F` is a [`QRCompactWY`](@ref) object,
- otherwise `F` is a [`QR`](@ref) object.
The individual components of the decomposition `F` can be retrieved via property accessors:
- `F.Q`: the orthogonal/unitary matrix `Q`
- `F.R`: the upper triangular matrix `R`
- `F.p`: the permutation vector of the pivot ([`QRPivoted`](@ref) only)
- `F.P`: the permutation matrix of the pivot ([`QRPivoted`](@ref) only)
Iterating the decomposition produces the components `Q`, `R`, and if extant `p`.
The following functions are available for the `QR` objects: [`inv`](@ref), [`size`](@ref),
and [`\\`](@ref). When `A` is rectangular, `\\` will return a least squares
solution and if the solution is not unique, the one with smallest norm is returned. When
`A` is not full rank, factorization with (column) pivoting is required to obtain a minimum
norm solution.
Multiplication with respect to either full/square or non-full/square `Q` is allowed, i.e. both `F.Q*F.R`
and `F.Q*A` are supported. A `Q` matrix can be converted into a regular matrix with
[`Matrix`](@ref). This operation returns the "thin" Q factor, i.e., if `A` is `m`×`n` with `m>=n`, then
`Matrix(F.Q)` yields an `m`×`n` matrix with orthonormal columns. To retrieve the "full" Q factor, an
`m`×`m` orthogonal matrix, use `F.Q*I`. If `m<=n`, then `Matrix(F.Q)` yields an `m`×`m`
orthogonal matrix.
The block size for QR decomposition can be specified by keyword argument
`blocksize :: Integer` when `pivot == NoPivot()` and `A isa StridedMatrix{<:BlasFloat}`.
It is ignored when `blocksize > minimum(size(A))`. See [`QRCompactWY`](@ref).
!!! compat "Julia 1.4"
The `blocksize` keyword argument requires Julia 1.4 or later.
# Examples
```jldoctest
julia> A = [3.0 -6.0; 4.0 -8.0; 0.0 1.0]
3×2 Matrix{Float64}:
3.0 -6.0
4.0 -8.0
0.0 1.0
julia> F = qr(A)
LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}, Matrix{Float64}}
Q factor:
3×3 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}:
-0.6 0.0 0.8
-0.8 0.0 -0.6
0.0 -1.0 0.0
R factor:
2×2 Matrix{Float64}:
-5.0 10.0
0.0 -1.0
julia> F.Q * F.R == A
true
```
!!! note
`qr` returns multiple types because LAPACK uses several representations
that minimize the memory storage requirements of products of Householder
elementary reflectors, so that the `Q` and `R` matrices can be stored
compactly rather as two separate dense matrices.
"""
function qr(A::AbstractMatrix{T}, arg...; kwargs...) where T
require_one_based_indexing(A)
AA = copy_similar(A, _qreltype(T))
return qr!(AA, arg...; kwargs...)
end
# TODO: remove in Julia v2.0
@deprecate qr(A::AbstractMatrix, ::Val{false}; kwargs...) qr(A, NoPivot(); kwargs...)
@deprecate qr(A::AbstractMatrix, ::Val{true}; kwargs...) qr(A, ColumnNorm(); kwargs...)
qr(x::Number) = qr(fill(x,1,1))
function qr(v::AbstractVector)
require_one_based_indexing(v)
qr(reshape(v, (length(v), 1)))
end
# Conversions
QR{T}(A::QR) where {T} = QR(convert(AbstractMatrix{T}, A.factors), convert(Vector{T}, A.τ))
Factorization{T}(A::QR{T}) where {T} = A
Factorization{T}(A::QR) where {T} = QR{T}(A)
QRCompactWY{T}(A::QRCompactWY) where {T} = QRCompactWY(convert(AbstractMatrix{T}, A.factors), convert(AbstractMatrix{T}, A.T))
Factorization{T}(A::QRCompactWY{T}) where {T} = A
Factorization{T}(A::QRCompactWY) where {T} = QRCompactWY{T}(A)
AbstractMatrix(F::Union{QR,QRCompactWY}) = F.Q * F.R
AbstractArray(F::Union{QR,QRCompactWY}) = AbstractMatrix(F)
Matrix(F::Union{QR,QRCompactWY}) = Array(AbstractArray(F))
Array(F::Union{QR,QRCompactWY}) = Matrix(F)
QRPivoted{T}(A::QRPivoted) where {T} = QRPivoted(convert(AbstractMatrix{T}, A.factors), convert(Vector{T}, A.τ), A.jpvt)
Factorization{T}(A::QRPivoted{T}) where {T} = A
Factorization{T}(A::QRPivoted) where {T} = QRPivoted{T}(A)
AbstractMatrix(F::QRPivoted) = (F.Q * F.R)[:,invperm(F.p)]
AbstractArray(F::QRPivoted) = AbstractMatrix(F)
Matrix(F::QRPivoted) = Array(AbstractArray(F))
Array(F::QRPivoted) = Matrix(F)
function show(io::IO, mime::MIME{Symbol("text/plain")}, F::Union{QR, QRCompactWY, QRPivoted})
summary(io, F); println(io)
println(io, "Q factor:")
show(io, mime, F.Q)
println(io, "\nR factor:")
show(io, mime, F.R)
if F isa QRPivoted
println(io, "\npermutation:")
show(io, mime, F.p)
end
end
function getproperty(F::QR, d::Symbol)
m, n = size(F)
if d === :R
return triu!(getfield(F, :factors)[1:min(m,n), 1:n])
elseif d === :Q
return QRPackedQ(getfield(F, :factors), F.τ)
else
getfield(F, d)
end
end
function getproperty(F::QRCompactWY, d::Symbol)
m, n = size(F)
if d === :R
return triu!(getfield(F, :factors)[1:min(m,n), 1:n])
elseif d === :Q
return QRCompactWYQ(getfield(F, :factors), F.T)
else
getfield(F, d)
end
end
Base.propertynames(F::Union{QR,QRCompactWY}, private::Bool=false) =
(:R, :Q, (private ? fieldnames(typeof(F)) : ())...)
function getproperty(F::QRPivoted{T}, d::Symbol) where T
m, n = size(F)
if d === :R
return triu!(getfield(F, :factors)[1:min(m,n), 1:n])
elseif d === :Q
return QRPackedQ(getfield(F, :factors), F.τ)
elseif d === :p
return getfield(F, :jpvt)
elseif d === :P
p = F.p
n = length(p)
P = zeros(T, n, n)
for i in 1:n
P[p[i],i] = one(T)
end
return P
else
getfield(F, d)
end
end
Base.propertynames(F::QRPivoted, private::Bool=false) =
(:R, :Q, :p, :P, (private ? fieldnames(typeof(F)) : ())...)
adjoint(F::Union{QR,QRPivoted,QRCompactWY}) = Adjoint(F)
abstract type AbstractQ{T} <: AbstractMatrix{T} end
inv(Q::AbstractQ) = Q'
"""
QRPackedQ <: AbstractMatrix
The orthogonal/unitary ``Q`` matrix of a QR factorization stored in [`QR`](@ref) or
[`QRPivoted`](@ref) format.
"""
struct QRPackedQ{T,S<:AbstractMatrix{T},C<:AbstractVector{T}} <: AbstractQ{T}
factors::S
τ::C
function QRPackedQ{T,S,C}(factors, τ) where {T,S<:AbstractMatrix{T},C<:AbstractVector{T}}
require_one_based_indexing(factors)
new{T,S,C}(factors, τ)
end
end
QRPackedQ(factors::AbstractMatrix{T}, τ::AbstractVector{T}) where {T} =
QRPackedQ{T,typeof(factors),typeof(τ)}(factors, τ)
QRPackedQ{T}(factors::AbstractMatrix, τ::AbstractVector) where {T} =
QRPackedQ(convert(AbstractMatrix{T}, factors), convert(AbstractVector{T}, τ))
# backwards-compatible constructors (remove with Julia 2.0)
@deprecate(QRPackedQ{T,S}(factors::AbstractMatrix{T}, τ::AbstractVector{T}) where {T,S},
QRPackedQ{T,S,typeof(τ)}(factors, τ), false)
"""
QRCompactWYQ <: AbstractMatrix
The orthogonal/unitary ``Q`` matrix of a QR factorization stored in [`QRCompactWY`](@ref)
format.
"""
struct QRCompactWYQ{S, M<:AbstractMatrix{S}, C<:AbstractMatrix{S}} <: AbstractQ{S}
factors::M
T::C
function QRCompactWYQ{S,M,C}(factors, T) where {S,M<:AbstractMatrix{S},C<:AbstractMatrix{S}}
require_one_based_indexing(factors)
new{S,M,C}(factors, T)
end
end
QRCompactWYQ(factors::AbstractMatrix{S}, T::AbstractMatrix{S}) where {S} =
QRCompactWYQ{S,typeof(factors),typeof(T)}(factors, T)
QRCompactWYQ{S}(factors::AbstractMatrix, T::AbstractMatrix) where {S} =
QRCompactWYQ(convert(AbstractMatrix{S}, factors), convert(AbstractMatrix{S}, T))
# backwards-compatible constructors (remove with Julia 2.0)
@deprecate(QRCompactWYQ{S,M}(factors::AbstractMatrix{S}, T::AbstractMatrix{S}) where {S,M},
QRCompactWYQ{S,M,typeof(T)}(factors, T), false)
QRPackedQ{T}(Q::QRPackedQ) where {T} = QRPackedQ(convert(AbstractMatrix{T}, Q.factors), convert(Vector{T}, Q.τ))
AbstractMatrix{T}(Q::QRPackedQ{T}) where {T} = Q
AbstractMatrix{T}(Q::QRPackedQ) where {T} = QRPackedQ{T}(Q)
QRCompactWYQ{S}(Q::QRCompactWYQ) where {S} = QRCompactWYQ(convert(AbstractMatrix{S}, Q.factors), convert(AbstractMatrix{S}, Q.T))
AbstractMatrix{S}(Q::QRCompactWYQ{S}) where {S} = Q
AbstractMatrix{S}(Q::QRCompactWYQ) where {S} = QRCompactWYQ{S}(Q)
Matrix{T}(Q::AbstractQ{S}) where {T,S} = convert(Matrix{T}, lmul!(Q, Matrix{S}(I, size(Q, 1), min(size(Q.factors)...))))
Matrix(Q::AbstractQ{T}) where {T} = Matrix{T}(Q)
Array{T}(Q::AbstractQ) where {T} = Matrix{T}(Q)
Array(Q::AbstractQ) = Matrix(Q)
size(F::Union{QR,QRCompactWY,QRPivoted}, dim::Integer) = size(getfield(F, :factors), dim)
size(F::Union{QR,QRCompactWY,QRPivoted}) = size(getfield(F, :factors))
size(Q::Union{QRCompactWYQ,QRPackedQ}, dim::Integer) =
size(getfield(Q, :factors), dim == 2 ? 1 : dim)
size(Q::Union{QRCompactWYQ,QRPackedQ}) = size(Q, 1), size(Q, 2)
copymutable(Q::AbstractQ{T}) where {T} = lmul!(Q, Matrix{T}(I, size(Q)))
copy(Q::AbstractQ) = copymutable(Q)
getindex(Q::AbstractQ, inds...) = copymutable(Q)[inds...]
getindex(Q::AbstractQ, ::Colon, ::Colon) = copy(Q)
function getindex(Q::AbstractQ, ::Colon, j::Int)
y = zeros(eltype(Q), size(Q, 2))
y[j] = 1
lmul!(Q, y)
end
getindex(Q::AbstractQ, i::Int, j::Int) = Q[:, j][i]
# specialization avoiding the fallback using slow `getindex`
function copyto!(dest::AbstractMatrix, src::AbstractQ)
copyto!(dest, I)
lmul!(src, dest)
end
# needed to resolve method ambiguities
function copyto!(dest::PermutedDimsArray{T,2,perm}, src::AbstractQ) where {T,perm}
if perm == (1, 2)
copyto!(parent(dest), src)
else
@assert perm == (2, 1) # there are no other permutations of two indices
if T <: Real
copyto!(parent(dest), I)
lmul!(src', parent(dest))
else
# LAPACK does not offer inplace lmul!(transpose(Q), B) for complex Q
tmp = similar(parent(dest))
copyto!(tmp, I)
rmul!(tmp, src)
permutedims!(parent(dest), tmp, (2, 1))
end
end
return dest
end
## Multiplication by Q
### QB
lmul!(A::QRCompactWYQ{T,S}, B::StridedVecOrMat{T}) where {T<:BlasFloat, S<:StridedMatrix} =
LAPACK.gemqrt!('L', 'N', A.factors, A.T, B)
lmul!(A::QRPackedQ{T,S}, B::StridedVecOrMat{T}) where {T<:BlasFloat, S<:StridedMatrix} =
LAPACK.ormqr!('L', 'N', A.factors, A.τ, B)
function lmul!(A::QRPackedQ, B::AbstractVecOrMat)
require_one_based_indexing(B)
mA, nA = size(A.factors)
mB, nB = size(B,1), size(B,2)
if mA != mB
throw(DimensionMismatch("matrix A has dimensions ($mA,$nA) but B has dimensions ($mB, $nB)"))
end
Afactors = A.factors
@inbounds begin
for k = min(mA,nA):-1:1
for j = 1:nB
vBj = B[k,j]
for i = k+1:mB
vBj += conj(Afactors[i,k])*B[i,j]
end
vBj = A.τ[k]*vBj
B[k,j] -= vBj
for i = k+1:mB
B[i,j] -= Afactors[i,k]*vBj
end
end
end
end
B
end
function (*)(A::AbstractQ, b::StridedVector)
TAb = promote_type(eltype(A), eltype(b))
Anew = convert(AbstractMatrix{TAb}, A)
if size(A.factors, 1) == length(b)
bnew = copymutable_oftype(b, TAb)
elseif size(A.factors, 2) == length(b)
bnew = [b; zeros(TAb, size(A.factors, 1) - length(b))]
else
throw(DimensionMismatch("vector must have length either $(size(A.factors, 1)) or $(size(A.factors, 2))"))
end
lmul!(Anew, bnew)
end
function (*)(A::AbstractQ, B::StridedMatrix)
TAB = promote_type(eltype(A), eltype(B))
Anew = convert(AbstractMatrix{TAB}, A)
if size(A.factors, 1) == size(B, 1)
Bnew = copymutable_oftype(B, TAB)
elseif size(A.factors, 2) == size(B, 1)
Bnew = [B; zeros(TAB, size(A.factors, 1) - size(B,1), size(B, 2))]
else
throw(DimensionMismatch("first dimension of matrix must have size either $(size(A.factors, 1)) or $(size(A.factors, 2))"))
end
lmul!(Anew, Bnew)
end
function (*)(A::AbstractQ, b::Number)
TAb = promote_type(eltype(A), typeof(b))
dest = similar(A, TAb)
copyto!(dest, b*I)
lmul!(A, dest)
end
### QcB
lmul!(adjA::Adjoint{<:Any,<:QRCompactWYQ{T,S}}, B::StridedVecOrMat{T}) where {T<:BlasReal,S<:StridedMatrix} =
(A = adjA.parent; LAPACK.gemqrt!('L', 'T', A.factors, A.T, B))
lmul!(adjA::Adjoint{<:Any,<:QRCompactWYQ{T,S}}, B::StridedVecOrMat{T}) where {T<:BlasComplex,S<:StridedMatrix} =
(A = adjA.parent; LAPACK.gemqrt!('L', 'C', A.factors, A.T, B))
lmul!(adjA::Adjoint{<:Any,<:QRPackedQ{T,S}}, B::StridedVecOrMat{T}) where {T<:BlasReal,S<:StridedMatrix} =
(A = adjA.parent; LAPACK.ormqr!('L', 'T', A.factors, A.τ, B))
lmul!(adjA::Adjoint{<:Any,<:QRPackedQ{T,S}}, B::StridedVecOrMat{T}) where {T<:BlasComplex,S<:StridedMatrix} =
(A = adjA.parent; LAPACK.ormqr!('L', 'C', A.factors, A.τ, B))
function lmul!(adjA::Adjoint{<:Any,<:QRPackedQ}, B::AbstractVecOrMat)
require_one_based_indexing(B)
A = adjA.parent
mA, nA = size(A.factors)
mB, nB = size(B,1), size(B,2)
if mA != mB
throw(DimensionMismatch("matrix A has dimensions ($mA,$nA) but B has dimensions ($mB, $nB)"))
end
Afactors = A.factors
@inbounds begin
for k = 1:min(mA,nA)
for j = 1:nB
vBj = B[k,j]
for i = k+1:mB
vBj += conj(Afactors[i,k])*B[i,j]
end
vBj = conj(A.τ[k])*vBj
B[k,j] -= vBj
for i = k+1:mB
B[i,j] -= Afactors[i,k]*vBj
end
end
end
end
B
end
function *(adjQ::Adjoint{<:Any,<:AbstractQ}, B::StridedVecOrMat)
Q = adjQ.parent
TQB = promote_type(eltype(Q), eltype(B))
return lmul!(adjoint(convert(AbstractMatrix{TQB}, Q)), copymutable_oftype(B, TQB))
end
### QBc/QcBc
function *(Q::AbstractQ, adjB::Adjoint{<:Any,<:StridedVecOrMat})
B = adjB.parent
TQB = promote_type(eltype(Q), eltype(B))
Bc = similar(B, TQB, (size(B, 2), size(B, 1)))
adjoint!(Bc, B)
return lmul!(convert(AbstractMatrix{TQB}, Q), Bc)
end
function *(adjQ::Adjoint{<:Any,<:AbstractQ}, adjB::Adjoint{<:Any,<:StridedVecOrMat})
Q, B = adjQ.parent, adjB.parent
TQB = promote_type(eltype(Q), eltype(B))
Bc = similar(B, TQB, (size(B, 2), size(B, 1)))
adjoint!(Bc, B)
return lmul!(adjoint(convert(AbstractMatrix{TQB}, Q)), Bc)
end
### AQ
rmul!(A::StridedVecOrMat{T}, B::QRCompactWYQ{T,S}) where {T<:BlasFloat,S<:StridedMatrix} =
LAPACK.gemqrt!('R', 'N', B.factors, B.T, A)
rmul!(A::StridedVecOrMat{T}, B::QRPackedQ{T,S}) where {T<:BlasFloat,S<:StridedMatrix} =
LAPACK.ormqr!('R', 'N', B.factors, B.τ, A)
function rmul!(A::StridedMatrix,Q::QRPackedQ)
mQ, nQ = size(Q.factors)
mA, nA = size(A,1), size(A,2)
if nA != mQ
throw(DimensionMismatch("matrix A has dimensions ($mA,$nA) but matrix Q has dimensions ($mQ, $nQ)"))
end
Qfactors = Q.factors
@inbounds begin
for k = 1:min(mQ,nQ)
for i = 1:mA
vAi = A[i,k]
for j = k+1:mQ
vAi += A[i,j]*Qfactors[j,k]
end
vAi = vAi*Q.τ[k]
A[i,k] -= vAi
for j = k+1:nA
A[i,j] -= vAi*conj(Qfactors[j,k])
end
end
end
end
A
end
function (*)(A::StridedMatrix, Q::AbstractQ)
TAQ = promote_type(eltype(A), eltype(Q))
return rmul!(copymutable_oftype(A, TAQ), convert(AbstractMatrix{TAQ}, Q))
end
function (*)(a::Number, B::AbstractQ)
TaB = promote_type(typeof(a), eltype(B))
dest = similar(B, TaB)
copyto!(dest, a*I)
rmul!(dest, B)
end
### AQc
rmul!(A::StridedVecOrMat{T}, adjB::Adjoint{<:Any,<:QRCompactWYQ{T}}) where {T<:BlasReal} =
(B = adjB.parent; LAPACK.gemqrt!('R', 'T', B.factors, B.T, A))
rmul!(A::StridedVecOrMat{T}, adjB::Adjoint{<:Any,<:QRCompactWYQ{T}}) where {T<:BlasComplex} =
(B = adjB.parent; LAPACK.gemqrt!('R', 'C', B.factors, B.T, A))
rmul!(A::StridedVecOrMat{T}, adjB::Adjoint{<:Any,<:QRPackedQ{T}}) where {T<:BlasReal} =
(B = adjB.parent; LAPACK.ormqr!('R', 'T', B.factors, B.τ, A))
rmul!(A::StridedVecOrMat{T}, adjB::Adjoint{<:Any,<:QRPackedQ{T}}) where {T<:BlasComplex} =
(B = adjB.parent; LAPACK.ormqr!('R', 'C', B.factors, B.τ, A))
function rmul!(A::StridedMatrix, adjQ::Adjoint{<:Any,<:QRPackedQ})
Q = adjQ.parent
mQ, nQ = size(Q.factors)
mA, nA = size(A,1), size(A,2)
if nA != mQ
throw(DimensionMismatch("matrix A has dimensions ($mA,$nA) but matrix Q has dimensions ($mQ, $nQ)"))
end
Qfactors = Q.factors
@inbounds begin
for k = min(mQ,nQ):-1:1
for i = 1:mA
vAi = A[i,k]
for j = k+1:mQ
vAi += A[i,j]*Qfactors[j,k]
end
vAi = vAi*conj(Q.τ[k])
A[i,k] -= vAi
for j = k+1:nA
A[i,j] -= vAi*conj(Qfactors[j,k])
end
end
end
end
A
end
function *(A::StridedMatrix, adjB::Adjoint{<:Any,<:AbstractQ})
B = adjB.parent
TAB = promote_type(eltype(A),eltype(B))
BB = convert(AbstractMatrix{TAB}, B)
if size(A,2) == size(B.factors, 1)
AA = copy_similar(A, TAB)
return rmul!(AA, adjoint(BB))
elseif size(A,2) == size(B.factors,2)
return rmul!([A zeros(TAB, size(A, 1), size(B.factors, 1) - size(B.factors, 2))], adjoint(BB))
else
throw(DimensionMismatch("matrix A has dimensions $(size(A)) but matrix B has dimensions $(size(B))"))
end
end
*(u::AdjointAbsVec, A::Adjoint{<:Any,<:AbstractQ}) = adjoint(A.parent * u.parent)
### AcQ/AcQc
function *(adjA::Adjoint{<:Any,<:StridedVecOrMat}, Q::AbstractQ)
A = adjA.parent
TAQ = promote_type(eltype(A), eltype(Q))
Ac = similar(A, TAQ, (size(A, 2), size(A, 1)))
adjoint!(Ac, A)
return rmul!(Ac, convert(AbstractMatrix{TAQ}, Q))
end
function *(adjA::Adjoint{<:Any,<:StridedVecOrMat}, adjQ::Adjoint{<:Any,<:AbstractQ})
A, Q = adjA.parent, adjQ.parent
TAQ = promote_type(eltype(A), eltype(Q))
Ac = similar(A, TAQ, (size(A, 2), size(A, 1)))
adjoint!(Ac, A)
return rmul!(Ac, adjoint(convert(AbstractMatrix{TAQ}, Q)))
end
### mul!
function mul!(C::StridedVecOrMat{T}, Q::AbstractQ{T}, B::StridedVecOrMat{T}) where {T}
require_one_based_indexing(C, B)
mB = size(B, 1)
mC = size(C, 1)
if mB < mC
inds = CartesianIndices(B)
copyto!(C, inds, B, inds)
C[CartesianIndices((mB+1:mC, axes(C, 2)))] .= zero(T)
return lmul!(Q, C)
else
return lmul!(Q, copyto!(C, B))
end
end
mul!(C::StridedVecOrMat{T}, A::StridedVecOrMat{T}, Q::AbstractQ{T}) where {T} = rmul!(copyto!(C, A), Q)
mul!(C::StridedVecOrMat{T}, adjQ::Adjoint{<:Any,<:AbstractQ{T}}, B::StridedVecOrMat{T}) where {T} = lmul!(adjQ, copyto!(C, B))
mul!(C::StridedVecOrMat{T}, A::StridedVecOrMat{T}, adjQ::Adjoint{<:Any,<:AbstractQ{T}}) where {T} = rmul!(copyto!(C, A), adjQ)
function ldiv!(A::QRCompactWY{T}, b::AbstractVector{T}) where {T<:BlasFloat}
m,n = size(A)
ldiv!(UpperTriangular(view(A.factors, 1:min(m,n), 1:n)), view(lmul!(adjoint(A.Q), b), 1:size(A, 2)))
return b
end
function ldiv!(A::QRCompactWY{T}, B::AbstractMatrix{T}) where {T<:BlasFloat}
m,n = size(A)
ldiv!(UpperTriangular(view(A.factors, 1:min(m,n), 1:n)), view(lmul!(adjoint(A.Q), B), 1:size(A, 2), 1:size(B, 2)))
return B
end
# Julia implementation similar to xgelsy
function ldiv!(A::QRPivoted{T}, B::AbstractMatrix{T}, rcond::Real) where T<:BlasFloat
mA, nA = size(A.factors)
nr = min(mA,nA)
nrhs = size(B, 2)
if nr == 0
return B, 0
end
ar = abs(A.factors[1])
if ar == 0
B[1:nA, :] .= 0
return B, 0
end
rnk = 1
xmin = T[1]
xmax = T[1]
tmin = tmax = ar
while rnk < nr
tmin, smin, cmin = LAPACK.laic1!(2, xmin, tmin, view(A.factors, 1:rnk, rnk + 1), A.factors[rnk + 1, rnk + 1])
tmax, smax, cmax = LAPACK.laic1!(1, xmax, tmax, view(A.factors, 1:rnk, rnk + 1), A.factors[rnk + 1, rnk + 1])
tmax*rcond > tmin && break
push!(xmin, cmin)
push!(xmax, cmax)
for i = 1:rnk
xmin[i] *= smin
xmax[i] *= smax
end
rnk += 1
end
C, τ = LAPACK.tzrzf!(A.factors[1:rnk, :])
lmul!(A.Q', view(B, 1:mA, :))
ldiv!(UpperTriangular(view(C, :, 1:rnk)), view(B, 1:rnk, :))
B[rnk+1:end,:] .= zero(T)
LAPACK.ormrz!('L', eltype(B)<:Complex ? 'C' : 'T', C, τ, view(B, 1:nA, :))
B[A.p,:] = B[1:nA,:]
return B, rnk
end
ldiv!(A::QRPivoted{T}, B::AbstractVector{T}) where {T<:BlasFloat} =
vec(ldiv!(A, reshape(B, length(B), 1)))
ldiv!(A::QRPivoted{T}, B::AbstractVecOrMat{T}) where {T<:BlasFloat} =
ldiv!(A, B, min(size(A)...)*eps(real(float(one(eltype(B))))))[1]
function _wide_qr_ldiv!(A::QR{T}, B::AbstractMatrix{T}) where T
m, n = size(A)
minmn = min(m,n)
mB, nB = size(B)
lmul!(adjoint(A.Q), view(B, 1:m, :))
R = A.R # makes a copy, used as a buffer below
@inbounds begin
if n > m # minimum norm solution
τ = zeros(T,m)
for k = m:-1:1 # Trapezoid to triangular by elementary operation
x = view(R, k, [k; m + 1:n])
τk = reflector!(x)
τ[k] = conj(τk)
for i = 1:k - 1
vRi = R[i,k]
for j = m + 1:n
vRi += R[i,j]*x[j - m + 1]'
end
vRi *= τk
R[i,k] -= vRi
for j = m + 1:n
R[i,j] -= vRi*x[j - m + 1]
end
end
end
end
ldiv!(UpperTriangular(view(R, :, 1:minmn)), view(B, 1:minmn, :))
if n > m # Apply elementary transformation to solution
B[m + 1:mB,1:nB] .= zero(T)
for j = 1:nB
for k = 1:m
vBj = B[k,j]
for i = m + 1:n
vBj += B[i,j]*R[k,i]'
end
vBj *= τ[k]
B[k,j] -= vBj
for i = m + 1:n
B[i,j] -= R[k,i]*vBj
end
end
end
end
end
return B
end
function ldiv!(A::QR{T}, B::AbstractMatrix{T}) where T
m, n = size(A)
m < n && return _wide_qr_ldiv!(A, B)
lmul!(adjoint(A.Q), view(B, 1:m, :))
R = A.factors
ldiv!(UpperTriangular(view(R,1:n,:)), view(B, 1:n, :))
return B
end
function ldiv!(A::QR, B::AbstractVector)
ldiv!(A, reshape(B, length(B), 1))
return B
end
function ldiv!(A::QRPivoted, b::AbstractVector)
ldiv!(QR(A.factors,A.τ), b)
b[1:size(A.factors, 2)] = view(b, 1:size(A.factors, 2))[invperm(A.jpvt)]
b
end
function ldiv!(A::QRPivoted, B::AbstractMatrix)
ldiv!(QR(A.factors, A.τ), B)
B[1:size(A.factors, 2),:] = view(B, 1:size(A.factors, 2), :)[invperm(A.jpvt),:]
B
end
function _apply_permutation!(F::QRPivoted, B::AbstractVecOrMat)
# Apply permutation but only to the top part of the solution vector since
# it's padded with zeros for underdetermined problems
B[1:length(F.p), :] = B[F.p, :]