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matrix.jl
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matrix.jl
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# Types of all Flint-backed matrices
const _FieldMatTypes = Union{QQMatrix, fpMatrix, FpMatrix, FqMatrix, fqPolyRepMatrix, FqPolyRepMatrix}
const _MatTypes = Union{_FieldMatTypes, ZZMatrix, zzModMatrix, ZZModMatrix}
################################################################################
#
# Support for view(A, :, i) and view(A, i, :)
#
################################################################################
# MatrixView{MatrixType, ElemType}
struct MatrixView{S, T} <: AbstractVector{T}
A::S
end
Base.length(V::MatrixView) = length(V.A)
Base.getindex(V::MatrixView, i::Int) = V.A[i]
Base.setindex!(V::MatrixView, z::ZZRingElem, i::Int) = (V.A[i] = z)
Base.setindex!(V::MatrixView, z, i::Int) = setindex!(V, ZZ(z), i)
Base.size(V::MatrixView) = (length(V.A), )
function Base.view(x::_MatTypes, r::Int, c::UnitRange{Int})
A = view(x, r:r, c)
return MatrixView{typeof(x), elem_type(base_ring(x))}(A)
end
function Base.view(x::_MatTypes, r::UnitRange{Int}, c::Int)
A = view(x, r, c:c)
return MatrixView{typeof(x), elem_type(base_ring(x))}(A)
end
################################################################################
#
# Generic kernel (calling nullspace in flint)
#
################################################################################
function kernel(A::_FieldMatTypes; side::Symbol = :left)
Solve.check_option(side, [:right, :left], "side")
if side === :left
K = kernel(transpose(A), side = :right)
return transpose(K)
end
return nullspace(A)[2]
end
################################################################################
#
# Solve context functionality
#
################################################################################
function AbstractAlgebra.solve_context_type(::Type{T}) where {T <: Union{QQFieldElem, fpFieldElem, FpFieldElem, FqFieldElem, fqPolyRepFieldElem, FqPolyRepFieldElem}}
MatType = dense_matrix_type(T)
return Solve.SolveCtx{T, MatType, MatType, MatType}
end
################################################################################
#
# Solve context for matrices over finite fields
#
################################################################################
function Solve._init_reduce(C::Solve.SolveCtx{T}) where {T <: Union{fpFieldElem, FpFieldElem, FqFieldElem, fqPolyRepFieldElem, FqPolyRepFieldElem}}
if isdefined(C, :red) && isdefined(C, :lu_perm)
return nothing
end
LU = deepcopy(matrix(C))
p = Generic.Perm(1:nrows(LU))
r = lu!(p, LU)
Solve.set_rank!(C, r)
C.red = LU
C.lu_perm = p
if r < nrows(C)
pA = p*matrix(C)
C.permuted_matrix = view(pA, r + 1:nrows(C), :)
else
C.permuted_matrix = zero(matrix(C), 0, ncols(C))
end
return nothing
end
function Solve._init_reduce_transpose(C::Solve.SolveCtx{T}) where {T <: Union{fpFieldElem, FpFieldElem, FqFieldElem, fqPolyRepFieldElem, FqPolyRepFieldElem}}
if isdefined(C, :red_transp) && isdefined(C, :lu_perm_transp)
return nothing
end
LU = transpose(matrix(C))
p = Generic.Perm(1:nrows(LU))
r = lu!(p, LU)
Solve.set_rank!(C, r)
C.red_transp = LU
C.lu_perm_transp = p
if r < ncols(C)
Ap = matrix(C)*p
C.permuted_matrix_transp = view(Ap, :, r + 1:ncols(C))
else
C.permuted_matrix_transp = zero(matrix(C), nrows(C), 0)
end
return nothing
end
function Solve._can_solve_internal_no_check(C::Solve.SolveCtx{T}, b::MatElem{T}, task::Symbol; side::Symbol = :left) where {T <: Union{fpFieldElem, FpFieldElem, FqFieldElem, fqPolyRepFieldElem, FqPolyRepFieldElem}}
# Split up in separate functions to make the compiler happy
if side === :right
return Solve._can_solve_internal_no_check_right(C, b, task)
else
return Solve._can_solve_internal_no_check_left(C, b, task)
end
end
function Solve._can_solve_internal_no_check_right(C::Solve.SolveCtx{T}, b::MatElem{T}, task::Symbol) where {T <: Union{fpFieldElem, FpFieldElem, FqFieldElem, fqPolyRepFieldElem, FqPolyRepFieldElem}}
LU = Solve.reduced_matrix(C)
p = Solve.lu_permutation(C)
pb = p*b
r = rank(C)
# Let A = matrix(C) be m x n of rank r. Then LU is build as follows (modulo
# the permutation p).
# For example, m = 5, n = 6, r = 3:
# (d b b b b b)
# (a d b b b b)
# (a a d b b b)
# (a a a 0 0 0)
# (a a a 0 0 0)
#
# L is the m x r matrix
# (1 0 0)
# (a 1 0)
# (a a 1)
# (a a a)
# (a a a)
#
# and U is the r x n matrix
# (d b b b b b)
# (0 d b b b b)
# (0 0 d b b b)
# Notice that the diagonal entries d need not be non-zero!
# Would be great if there were a `*_solve_lu_precomp` for the finite field
# types in flint.
x = similar(b, r, ncols(b))
# Solve L x = b for the first r rows. We tell flint to pretend that there
# are ones on the diagonal of LU (fourth argument)
_solve_tril_right_flint!(x, view(LU, 1:r, 1:r), view(pb, 1:r, :), true)
# Check whether x solves Lx = b also for the lower part of L
if r < nrows(C) && view(LU, r + 1:nrows(LU), 1:r)*x != view(pb, r + 1:nrows(b), :)
return false, zero(b, 0, 0), zero(b, 0, 0)
end
# Solve U y = x. We need to take extra care as U might have non-pivot columns.
y = _solve_triu_right(view(LU, 1:r, :), x)
fl = true
if r < nrows(C)
fl = Solve.permuted_matrix(C)*y == view(pb, r + 1:nrows(C), :)
end
if task !== :with_kernel
return fl, y, zero(b, 0, 0)
else
return fl, y, kernel(C, side = :right)
end
end
function Solve._can_solve_internal_no_check_left(C::Solve.SolveCtx{T}, b::MatElem{T}, task::Symbol) where {T <: Union{fpFieldElem, FpFieldElem, FqFieldElem, fqPolyRepFieldElem, FqPolyRepFieldElem}}
LU = Solve.reduced_matrix_of_transpose(C)
p = Solve.lu_permutation_of_transpose(C)
pbt = p*transpose(b)
r = rank(C)
x = similar(b, r, nrows(b))
_solve_tril_right_flint!(x, view(LU, 1:r, 1:r), view(pbt, 1:r, :), true)
# Check whether x solves Lx = b also for the lower part of L
if r < ncols(C) && view(LU, r + 1:nrows(LU), 1:r)*x != view(pbt, r + 1:nrows(pbt), :)
return false, zero(b, 0, 0), zero(b, 0, 0)
end
# Solve U y = x. We need to take extra care as U might have non-pivot columns.
yy = _solve_triu_right(view(LU, 1:r, :), x)
y = transpose(yy)
fl = true
if r < ncols(C)
bp = b*p
fl = y*Solve.permuted_matrix_of_transpose(C) == view(bp, :, r + 1:ncols(C))
end
if task !== :with_kernel
return fl, y, zero(b, 0, 0)
else
return fl, y, kernel(C, side = :left)
end
end
# Solves A x = B with A in upper triangular shape of full rank, so something
# like
# ( + * * * * )
# ( 0 0 + * * )
# ( 0 0 0 + * )
# where + is non-zero and * is anything.
# This is a helper functions because flint can only do the case where the
# diagonal entries are non-zero.
function _solve_triu_right(A::MatT, B::MatT) where {MatT <: Union{fpMatrix, FpMatrix, FqMatrix, fqPolyRepMatrix, FqPolyRepMatrix}}
@assert nrows(A) == nrows(B)
pivot_cols = Int[]
next_pivot_col = ncols(A) + 1
@inbounds for r in nrows(A):-1:1
for c in r:next_pivot_col - 1
if !is_zero_entry(A, r, c)
push!(pivot_cols, c)
next_pivot_col = c
break
end
if c == next_pivot_col - 1
error("Matrix is not in upper triangular shape")
end
end
end
reverse!(pivot_cols)
AA = reduce(hcat, view(A, 1:nrows(A), c:c) for c in pivot_cols; init = zero(A, nrows(A), 0))
xx = similar(B, nrows(A), ncols(B))
_solve_triu_right_flint!(xx, AA, B, false)
x = zero(B, ncols(A), ncols(B))
for i in 1:nrows(xx)
view(x, pivot_cols[i]:pivot_cols[i], :) .= view(xx, i:i, :)
end
return x
end
################################################################################
#
# Eigenvalues and eigenspaces
#
################################################################################
@doc raw"""
eigenvalues(M::MatElem{T}) where T <: FieldElem
Return the eigenvalues of `M`.
"""
function eigenvalues(M::MatElem{T}) where T <: FieldElem
@assert is_square(M)
K = base_ring(M)
f = charpoly(M)
return roots(f)
end
@doc raw"""
eigenvalues_with_multiplicities(M::MatElem{T}) where T <: FieldElem
Return the eigenvalues of `M` together with their algebraic multiplicities as a
vector of tuples.
"""
function eigenvalues_with_multiplicities(M::MatElem{T}) where T <: FieldElem
@assert is_square(M)
K = base_ring(M)
Kx, x = polynomial_ring(K, "x", cached = false)
f = charpoly(Kx, M)
r = roots(f)
return [ (a, valuation(f, x - a)) for a in r ]
end
@doc raw"""
eigenvalues(L::Field, M::MatElem{T}) where T <: RingElem
Return the eigenvalues of `M` over the field `L`.
"""
function eigenvalues(L::Field, M::MatElem{T}) where T <: RingElem
@assert is_square(M)
M1 = change_base_ring(L, M)
return eigenvalues(M1)
end
@doc raw"""
eigenvalues_with_multiplicities(L::Field, M::MatElem{T}) where T <: RingElem
Return the eigenvalues of `M` over the field `L` together with their algebraic
multiplicities as a vector of tuples.
"""
function eigenvalues_with_multiplicities(L::Field, M::MatElem{T}) where T <: RingElem
@assert is_square(M)
M1 = change_base_ring(L, M)
return eigenvalues_with_multiplicities(M1)
end
@doc raw"""
eigenspace(M::MatElem{T}, lambda::T; side::Symbol = :left)
where T <: FieldElem -> Vector{MatElem{T}}
Return a basis of the eigenspace of $M$ with respect to the eigenvalue $\lambda$.
If side is `:right`, the right eigenspace is computed, i.e. vectors $v$ such that
$Mv = \lambda v$. If side is `:left`, the left eigenspace is computed, i.e. vectors
$v$ such that $vM = \lambda v$.
"""
function eigenspace(M::MatElem{T}, lambda::T; side::Symbol = :left) where T <: FieldElem
@assert is_square(M)
N = deepcopy(M)
for i = 1:ncols(N)
N[i, i] -= lambda
end
return kernel(N, side = side)
end
@doc raw"""
eigenspaces(M::MatElem{T}; side::Symbol = :left)
where T <: FieldElem -> Dict{T, MatElem{T}}
Return a dictionary containing the eigenvalues of $M$ as keys and bases for the
corresponding eigenspaces as values.
If side is `:right`, the right eigenspaces are computed, if it is `:left` then the
left eigenspaces are computed.
See also `eigenspace`.
"""
function eigenspaces(M::MatElem{T}; side::Symbol = :left) where T<:FieldElem
S = eigenvalues(M)
L = Dict{elem_type(base_ring(M)), typeof(M)}()
for k in S
push!(L, k => vcat(eigenspace(M, k, side = side)))
end
return L
end
###############################################################################
#
# Permutation
#
###############################################################################
# Unfortunately, there is no fmpq_mat_set_perm etc. in flint
function *(P::Perm, x::_FieldMatTypes)
z = similar(x)
t = base_ring(x)()
@inbounds for i = 1:nrows(x)
for j = 1:ncols(x)
z[P[i], j] = getindex!(t, x, i, j)
end
end
return z
end
function *(x::_FieldMatTypes, P::Perm)
z = similar(x)
t = base_ring(x)()
@inbounds for i = 1:nrows(x)
for j = 1:ncols(x)
z[i, P[j]] = getindex!(t, x, i, j)
end
end
return z
end
###############################################################################
#
# Norm
#
###############################################################################
function norm(v::Union{AcbMatrix, ArbMatrix, ComplexMat, RealMat})
return sqrt(sum(a^2 for a in v; init=zero(base_ring(v))))
end