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parametrize Tridiagonal on the wrapped vector type #23154

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parametrize Tridiagonal on the wrapped vector type #23154

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21 changes: 12 additions & 9 deletions NEWS.md
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Expand Up @@ -121,10 +121,11 @@ This section lists changes that do not have deprecation warnings.
longer present. Use `first(R)` and `last(R)` to obtain
start/stop. ([#20974])

* The `Diagonal`, `Bidiagonal` and `SymTridiagonal` type definitions have changed from
`Diagonal{T}`, `Bidiagonal{T}` and `SymTridiagonal{T}` to `Diagonal{T,V<:AbstractVector{T}}`,
`Bidiagonal{T,V<:AbstractVector{T}}` and `SymTridiagonal{T,V<:AbstractVector{T}}`
respectively ([#22718], [#22925], [#23035]).
* The `Diagonal`, `Bidiagonal`, `Tridiagonal` and `SymTridiagonal` type definitions have
changed from `Diagonal{T}`, `Bidiagonal{T}`, `Tridiagonal{T}` and `SymTridiagonal{T}`
to `Diagonal{T,V<:AbstractVector{T}}`, `Bidiagonal{T,V<:AbstractVector{T}}`,
`Tridiagonal{T,V<:AbstractVector{T}}` and `SymTridiagonal{T,V<:AbstractVector{T}}`
respectively ([#22718], [#22925], [#23035], [#23154]).

* `isapprox(x,y)` now tests `norm(x-y) <= max(atol, rtol*max(norm(x), norm(y)))`
rather than `norm(x-y) <= atol + ...`, and `rtol` defaults to zero
Expand Down Expand Up @@ -196,9 +197,10 @@ Library improvements

* `Char`s can now be concatenated with `String`s and/or other `Char`s using `*` ([#22532]).

* `Diagonal`, `Bidiagonal` and `SymTridiagonal` are now parameterized on the type of the
wrapped vectors, allowing `Diagonal`, `Bidiagonal` and `SymTridiagonal` matrices with
arbitrary `AbstractVector`s ([#22718], [#22925], [#23035]).
* `Diagonal`, `Bidiagonal`, `Tridiagonal` and `SymTridiagonal` are now parameterized on
the type of the wrapped vectors, allowing `Diagonal`, `Bidiagonal`, `Tridiagonal` and
`SymTridiagonal` matrices with arbitrary `AbstractVector`s
([#22718], [#22925], [#23035], [#23154]).

* Mutating versions of `randperm` and `randcycle` have been added:
`randperm!` and `randcycle!` ([#22723]).
Expand Down Expand Up @@ -261,8 +263,9 @@ Deprecated or removed
* `Bidiagonal` constructors now use a `Symbol` (`:U` or `:L`) for the upper/lower
argument, instead of a `Bool` or a `Char` ([#22703]).

* `Bidiagonal` and `SymTridiagonal` constructors that automatically converted the input
vectors to the same type are deprecated in favor of explicit conversion ([#22925], [#23035]).
* `Bidiagonal`, `Tridiagonal` and `SymTridiagonal` constructors that automatically
converted the input vectors to the same type are deprecated in favor of explicit
conversion ([#22925], [#23035], [#23154].

* Calling `nfields` on a type to find out how many fields its instances have is deprecated.
Use `fieldcount` instead. Use `nfields` only to get the number of fields in a specific object ([#22350]).
Expand Down
8 changes: 8 additions & 0 deletions base/deprecated.jl
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Expand Up @@ -1652,6 +1652,14 @@ function SymTridiagonal(dv::AbstractVector{T}, ev::AbstractVector{S}) where {T,S
SymTridiagonal(convert(Vector{R}, dv), convert(Vector{R}, ev))
end

# PR #23154
# also uncomment constructor tests in test/linalg/tridiag.jl
function Tridiagonal(dl::AbstractVector{Tl}, d::AbstractVector{Td}, du::AbstractVector{Tu}) where {Tl,Td,Tu}
depwarn(string("Tridiagonal(dl::AbstractVector{Tl}, d::AbstractVector{Td}, du::AbstractVector{Tu}) ",
"where {Tl, Td, Tu} is deprecated; convert all vectors to the same type instead."), :Tridiagonal)
Tridiagonal(map(v->convert(Vector{promote_type(Tl,Td,Tu)}, v), (dl, d, du))...)
end

# PR #23092
@eval LibGit2 begin
function prompt(msg::AbstractString; default::AbstractString="", password::Bool=false)
Expand Down
6 changes: 4 additions & 2 deletions base/linalg/bidiag.jl
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Expand Up @@ -153,8 +153,10 @@ promote_rule(::Type{Matrix{T}}, ::Type{Bidiagonal{S}}) where {T,S} = Matrix{prom
#Converting from Bidiagonal to Tridiagonal
Tridiagonal(M::Bidiagonal{T}) where {T} = convert(Tridiagonal{T}, M)
function convert(::Type{Tridiagonal{T}}, A::Bidiagonal) where T
z = zeros(T, size(A)[1]-1)
A.uplo == 'U' ? Tridiagonal(z, convert(Vector{T},A.dv), convert(Vector{T},A.ev)) : Tridiagonal(convert(Vector{T},A.ev), convert(Vector{T},A.dv), z)
dv = convert(AbstractVector{T}, A.dv)
ev = convert(AbstractVector{T}, A.ev)
z = fill!(similar(ev), zero(T))
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To check my understanding, converting a Bidiagonal over Ranges to a Tridiagonal should fail?

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Right..., Yea I guess that’s a consequence of this series of PR's? 😄

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If we're converting from Diagonal or Bidiagonal or SymTridiagonal that have fewer diagonal vectors and they already have to be a consistent type within those structs, can't we use that same parameter to know which new diagonal vector type to fill in when converting to Tridiagonal?

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In the case of a Range: You can not create a Range of a given length with only zeros, right? That is the only failure case here I think...
Using Ranges results in other problems as well, so I guess you will just have to convert to a vector format that supports a larger set of functions.

A.uplo == 'U' ? Tridiagonal(z, dv, ev) : Tridiagonal(ev, dv, z)
end
promote_rule(::Type{Tridiagonal{T}}, ::Type{Bidiagonal{S}}) where {T,S} = Tridiagonal{promote_type(T,S)}

Expand Down
2 changes: 1 addition & 1 deletion base/linalg/bunchkaufman.jl
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Expand Up @@ -116,7 +116,7 @@ julia> A = [1 2 3; 2 1 2; 3 2 1]
julia> F = bkfact(Symmetric(A, :L))
Base.LinAlg.BunchKaufman{Float64,Array{Float64,2}}
D factor:
3×3 Tridiagonal{Float64}:
3×3 Tridiagonal{Float64,Array{Float64,1}}:
1.0 3.0 ⋅
3.0 1.0 0.0
⋅ 0.0 -1.0
Expand Down
27 changes: 14 additions & 13 deletions base/linalg/lu.jl
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Expand Up @@ -327,14 +327,14 @@ end
# Tridiagonal

# See dgttrf.f
function lufact!(A::Tridiagonal{T}, pivot::Union{Val{false}, Val{true}} = Val(true)) where T
function lufact!(A::Tridiagonal{T,V}, pivot::Union{Val{false}, Val{true}} = Val(true)) where {T,V}
n = size(A, 1)
info = 0
ipiv = Vector{BlasInt}(n)
dl = A.dl
d = A.d
du = A.du
du2 = A.du2
du2 = fill!(similar(d, n-2), 0)::V

@inbounds begin
for i = 1:n
Expand Down Expand Up @@ -389,12 +389,13 @@ function lufact!(A::Tridiagonal{T}, pivot::Union{Val{false}, Val{true}} = Val(tr
end
end
end
LU{T,Tridiagonal{T}}(A, ipiv, convert(BlasInt, info))
B = Tridiagonal{T,V}(dl, d, du, du2)
LU{T,Tridiagonal{T,V}}(B, ipiv, convert(BlasInt, info))
end

factorize(A::Tridiagonal) = lufact(A)

function getindex(F::Base.LinAlg.LU{T,Tridiagonal{T}}, d::Symbol) where T
function getindex(F::LU{T,Tridiagonal{T,V}}, d::Symbol) where {T,V}
m, n = size(F)
if d == :L
L = Array(Bidiagonal(ones(T, n), F.factors.dl, d))
Expand All @@ -419,7 +420,7 @@ function getindex(F::Base.LinAlg.LU{T,Tridiagonal{T}}, d::Symbol) where T
end

# See dgtts2.f
function A_ldiv_B!(A::LU{T,Tridiagonal{T}}, B::AbstractVecOrMat) where T
function A_ldiv_B!(A::LU{T,Tridiagonal{T,V}}, B::AbstractVecOrMat) where {T,V}
n = size(A,1)
if n != size(B,1)
throw(DimensionMismatch("matrix has dimensions ($n,$n) but right hand side has $(size(B,1)) rows"))
Expand Down Expand Up @@ -450,7 +451,7 @@ function A_ldiv_B!(A::LU{T,Tridiagonal{T}}, B::AbstractVecOrMat) where T
return B
end

function At_ldiv_B!(A::LU{T,Tridiagonal{T}}, B::AbstractVecOrMat) where T
function At_ldiv_B!(A::LU{T,Tridiagonal{T,V}}, B::AbstractVecOrMat) where {T,V}
n = size(A,1)
if n != size(B,1)
throw(DimensionMismatch("matrix has dimensions ($n,$n) but right hand side has $(size(B,1)) rows"))
Expand Down Expand Up @@ -485,7 +486,7 @@ function At_ldiv_B!(A::LU{T,Tridiagonal{T}}, B::AbstractVecOrMat) where T
end

# Ac_ldiv_B!(A::LU{T,Tridiagonal{T}}, B::AbstractVecOrMat) where {T<:Real} = At_ldiv_B!(A,B)
function Ac_ldiv_B!(A::LU{T,Tridiagonal{T}}, B::AbstractVecOrMat) where T
function Ac_ldiv_B!(A::LU{T,Tridiagonal{T,V}}, B::AbstractVecOrMat) where {T,V}
n = size(A,1)
if n != size(B,1)
throw(DimensionMismatch("matrix has dimensions ($n,$n) but right hand side has $(size(B,1)) rows"))
Expand Down Expand Up @@ -528,7 +529,7 @@ convert(::Type{Matrix}, F::LU) = convert(Array, convert(AbstractArray, F))
convert(::Type{Array}, F::LU) = convert(Matrix, F)
full(F::LU) = convert(AbstractArray, F)

function convert(::Type{Tridiagonal}, F::Base.LinAlg.LU{T,Tridiagonal{T}}) where T
function convert(::Type{Tridiagonal}, F::Base.LinAlg.LU{T,Tridiagonal{T,V}}) where {T,V}
n = size(F, 1)

dl = copy(F.factors.dl)
Expand Down Expand Up @@ -562,12 +563,12 @@ function convert(::Type{Tridiagonal}, F::Base.LinAlg.LU{T,Tridiagonal{T}}) where
end
return Tridiagonal(dl, d, du)
end
convert(::Type{AbstractMatrix}, F::Base.LinAlg.LU{T,Tridiagonal{T}}) where {T} =
convert(::Type{AbstractMatrix}, F::LU{T,Tridiagonal{T,V}}) where {T,V} =
convert(Tridiagonal, F)
convert(::Type{AbstractArray}, F::Base.LinAlg.LU{T,Tridiagonal{T}}) where {T} =
convert(::Type{AbstractArray}, F::LU{T,Tridiagonal{T,V}}) where {T,V} =
convert(AbstractMatrix, F)
convert(::Type{Matrix}, F::Base.LinAlg.LU{T,Tridiagonal{T}}) where {T} =
convert(::Type{Matrix}, F::LU{T,Tridiagonal{T,V}}) where {T,V} =
convert(Array, convert(AbstractArray, F))
convert(::Type{Array}, F::Base.LinAlg.LU{T,Tridiagonal{T}}) where {T} =
convert(::Type{Array}, F::LU{T,Tridiagonal{T,V}}) where {T,V} =
convert(Matrix, F)
full(F::Base.LinAlg.LU{T,Tridiagonal{T}}) where {T} = convert(AbstractArray, F)
full(F::LU{T,Tridiagonal{T,V}}) where {T,V} = convert(AbstractArray, F)
100 changes: 43 additions & 57 deletions base/linalg/tridiag.jl
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Expand Up @@ -399,71 +399,58 @@ function setindex!(A::SymTridiagonal, x, i::Integer, j::Integer)
end

## Tridiagonal matrices ##
struct Tridiagonal{T} <: AbstractMatrix{T}
dl::Vector{T} # sub-diagonal
d::Vector{T} # diagonal
du::Vector{T} # sup-diagonal
du2::Vector{T} # supsup-diagonal for pivoting
struct Tridiagonal{T,V<:AbstractVector{T}} <: AbstractMatrix{T}
dl::V # sub-diagonal
d::V # diagonal
du::V # sup-diagonal
du2::V # supsup-diagonal for pivoting in LU
function Tridiagonal{T}(dl::V, d::V, du::V) where {T,V<:AbstractVector{T}}
n = length(d)
if (length(dl) != n-1 || length(du) != n-1)
throw(ArgumentError(string("cannot construct Tridiagonal from incompatible ",
"lengths of subdiagonal, diagonal and superdiagonal: ",
"($(length(dl)), $(length(d)), $(length(du)))")))
end
new{T,V}(dl, d, du)
end
# constructor used in lufact!
function Tridiagonal{T,V}(dl::V, d::V, du::V, du2::V) where {T,V<:AbstractVector{T}}
new{T,V}(dl, d, du, du2)
end
end

"""
Tridiagonal(dl, d, du)
Tridiagonal(dl::V, d::V, du::V) where V <: AbstractVector

Construct a tridiagonal matrix from the first subdiagonal, diagonal, and first superdiagonal,
respectively. The result is of type `Tridiagonal` and provides efficient specialized linear
respectively. The result is of type `Tridiagonal` and provides efficient specialized linear
solvers, but may be converted into a regular matrix with
[`convert(Array, _)`](@ref) (or `Array(_)` for short).
The lengths of `dl` and `du` must be one less than the length of `d`.

# Examples
```jldoctest
julia> dl = [1; 2; 3]
3-element Array{Int64,1}:
1
2
3
julia> dl = [1, 2, 3];

julia> du = [4; 5; 6]
3-element Array{Int64,1}:
4
5
6
julia> du = [4, 5, 6];

julia> d = [7; 8; 9; 0]
4-element Array{Int64,1}:
7
8
9
0
julia> d = [7, 8, 9, 0];

julia> Tridiagonal(dl, d, du)
4×4 Tridiagonal{Int64}:
4×4 Tridiagonal{Int64,Array{Int64,1}}:
7 4 ⋅ ⋅
1 8 5 ⋅
⋅ 2 9 6
⋅ ⋅ 3 0
```
"""
# Basic constructor takes in three dense vectors of same type
function Tridiagonal(dl::Vector{T}, d::Vector{T}, du::Vector{T}) where T
n = length(d)
if (length(dl) != n-1 || length(du) != n-1)
throw(ArgumentError("cannot make Tridiagonal from incompatible lengths of subdiagonal, diagonal and superdiagonal: ($(length(dl)), $(length(d)), $(length(du))"))
end
Tridiagonal(dl, d, du, zeros(T,n-2))
end

# Construct from diagonals of any abstract vector, any eltype
function Tridiagonal(dl::AbstractVector{Tl}, d::AbstractVector{Td}, du::AbstractVector{Tu}) where {Tl,Td,Tu}
Tridiagonal(map(v->convert(Vector{promote_type(Tl,Td,Tu)}, v), (dl, d, du))...)
end
Tridiagonal(dl::V, d::V, du::V) where {T,V<:AbstractVector{T}} = Tridiagonal{T}(dl, d, du)

# Provide a constructor Tridiagonal(A) similar to the triangulars, diagonal, symmetric
"""
Tridiagonal(A)

returns a `Tridiagonal` array based on (abstract) matrix `A`, using its first lower diagonal,
main diagonal, and first upper diagonal.
Construct a tridiagonal matrix from the first sub-diagonal,
diagonal and first super-diagonal of the matrix `A`.

# Examples
```jldoctest
Expand All @@ -475,16 +462,14 @@ julia> A = [1 2 3 4; 1 2 3 4; 1 2 3 4; 1 2 3 4]
1 2 3 4

julia> Tridiagonal(A)
4×4 Tridiagonal{Int64}:
4×4 Tridiagonal{Int64,Array{Int64,1}}:
1 2 ⋅ ⋅
1 2 3 ⋅
⋅ 2 3 4
⋅ ⋅ 3 4
```
"""
function Tridiagonal(A::AbstractMatrix)
return Tridiagonal(diag(A,-1), diag(A), diag(A,+1))
end
Tridiagonal(A::AbstractMatrix) = Tridiagonal(diag(A,-1), diag(A,0), diag(A,1))

size(M::Tridiagonal) = (length(M.d), length(M.d))
function size(M::Tridiagonal, d::Integer)
Expand Down Expand Up @@ -512,31 +497,31 @@ convert(::Type{Matrix}, M::Tridiagonal{T}) where {T} = convert(Matrix{T}, M)
convert(::Type{Array}, M::Tridiagonal) = convert(Matrix, M)
full(M::Tridiagonal) = convert(Array, M)
function similar(M::Tridiagonal, ::Type{T}) where T
Tridiagonal{T}(similar(M.dl, T), similar(M.d, T), similar(M.du, T), similar(M.du2, T))
Tridiagonal{T}(similar(M.dl, T), similar(M.d, T), similar(M.du, T))
end

# Operations on Tridiagonal matrices
copy!(dest::Tridiagonal, src::Tridiagonal) = Tridiagonal(copy!(dest.dl, src.dl), copy!(dest.d, src.d), copy!(dest.du, src.du), copy!(dest.du2, src.du2))
copy!(dest::Tridiagonal, src::Tridiagonal) = (copy!(dest.dl, src.dl); copy!(dest.d, src.d); copy!(dest.du, src.du); dest)

#Elementary operations
broadcast(::typeof(abs), M::Tridiagonal) = Tridiagonal(abs.(M.dl), abs.(M.d), abs.(M.du), abs.(M.du2))
broadcast(::typeof(round), M::Tridiagonal) = Tridiagonal(round.(M.dl), round.(M.d), round.(M.du), round.(M.du2))
broadcast(::typeof(trunc), M::Tridiagonal) = Tridiagonal(trunc.(M.dl), trunc.(M.d), trunc.(M.du), trunc.(M.du2))
broadcast(::typeof(floor), M::Tridiagonal) = Tridiagonal(floor.(M.dl), floor.(M.d), floor.(M.du), floor.(M.du2))
broadcast(::typeof(ceil), M::Tridiagonal) = Tridiagonal(ceil.(M.dl), ceil.(M.d), ceil.(M.du), ceil.(M.du2))
broadcast(::typeof(abs), M::Tridiagonal) = Tridiagonal(abs.(M.dl), abs.(M.d), abs.(M.du))
broadcast(::typeof(round), M::Tridiagonal) = Tridiagonal(round.(M.dl), round.(M.d), round.(M.du))
broadcast(::typeof(trunc), M::Tridiagonal) = Tridiagonal(trunc.(M.dl), trunc.(M.d), trunc.(M.du))
broadcast(::typeof(floor), M::Tridiagonal) = Tridiagonal(floor.(M.dl), floor.(M.d), floor.(M.du))
broadcast(::typeof(ceil), M::Tridiagonal) = Tridiagonal(ceil.(M.dl), ceil.(M.d), ceil.(M.du))
for func in (:conj, :copy, :real, :imag)
@eval function ($func)(M::Tridiagonal)
Tridiagonal(($func)(M.dl), ($func)(M.d), ($func)(M.du), ($func)(M.du2))
Tridiagonal(($func)(M.dl), ($func)(M.d), ($func)(M.du))
end
end
broadcast(::typeof(round), ::Type{T}, M::Tridiagonal) where {T<:Integer} =
Tridiagonal(round.(T, M.dl), round.(T, M.d), round.(T, M.du), round.(T, M.du2))
Tridiagonal(round.(T, M.dl), round.(T, M.d), round.(T, M.du))
broadcast(::typeof(trunc), ::Type{T}, M::Tridiagonal) where {T<:Integer} =
Tridiagonal(trunc.(T, M.dl), trunc.(T, M.d), trunc.(T, M.du), trunc.(T, M.du2))
Tridiagonal(trunc.(T, M.dl), trunc.(T, M.d), trunc.(T, M.du))
broadcast(::typeof(floor), ::Type{T}, M::Tridiagonal) where {T<:Integer} =
Tridiagonal(floor.(T, M.dl), floor.(T, M.d), floor.(T, M.du), floor.(T, M.du2))
Tridiagonal(floor.(T, M.dl), floor.(T, M.d), floor.(T, M.du))
broadcast(::typeof(ceil), ::Type{T}, M::Tridiagonal) where {T<:Integer} =
Tridiagonal(ceil.(T, M.dl), ceil.(T, M.d), ceil.(T, M.du), ceil.(T, M.du2))
Tridiagonal(ceil.(T, M.dl), ceil.(T, M.d), ceil.(T, M.du))

transpose(M::Tridiagonal) = Tridiagonal(M.du, M.d, M.dl)
ctranspose(M::Tridiagonal) = conj(transpose(M))
Expand Down Expand Up @@ -646,7 +631,8 @@ end
inv(A::Tridiagonal) = inv_usmani(A.dl, A.d, A.du)
det(A::Tridiagonal) = det_usmani(A.dl, A.d, A.du)

convert(::Type{Tridiagonal{T}},M::Tridiagonal) where {T} = Tridiagonal(convert(Vector{T}, M.dl), convert(Vector{T}, M.d), convert(Vector{T}, M.du), convert(Vector{T}, M.du2))
convert(::Type{Tridiagonal{T}},M::Tridiagonal) where {T} =
Tridiagonal(convert(AbstractVector{T}, M.dl), convert(AbstractVector{T}, M.d), convert(AbstractVector{T}, M.du))
convert(::Type{AbstractMatrix{T}},M::Tridiagonal) where {T} = convert(Tridiagonal{T}, M)
convert(::Type{Tridiagonal{T}}, M::SymTridiagonal{T}) where {T} = Tridiagonal(M)
function convert(::Type{SymTridiagonal{T}}, M::Tridiagonal) where T
Expand Down
7 changes: 7 additions & 0 deletions test/linalg/tridiag.jl
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Expand Up @@ -41,10 +41,17 @@ B = randn(n,2)
@test ST == Matrix(ST)
@test ST.dv === x
@test ST.ev === y
TT = (Tridiagonal(y, x, y))::Tridiagonal{elty, typeof(x)}
@test TT == Matrix(TT)
@test TT.dl === y
@test TT.d === x
@test TT.du === y
end
# enable when deprecations for 0.7 are dropped
# @test_throws MethodError SymTridiagonal(dv, GenericArray(ev))
# @test_throws MethodError SymTridiagonal(GenericArray(dv), ev)
# @test_throws MethodError Tridiagonal(GenericArray(ev), dv, GenericArray(ev))
# @test_throws MethodError Tridiagonal(ev, GenericArray(dv), ev)
end

@testset "size and Array" begin
Expand Down
2 changes: 1 addition & 1 deletion test/show.jl
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Expand Up @@ -598,7 +598,7 @@ A = reshape(1:16,4,4)
@test replstr(Bidiagonal(A,:U)) == "4×4 Bidiagonal{$(Int),Array{$(Int),1}}:\n 1 5 ⋅ ⋅\n ⋅ 6 10 ⋅\n ⋅ ⋅ 11 15\n ⋅ ⋅ ⋅ 16"
@test replstr(Bidiagonal(A,:L)) == "4×4 Bidiagonal{$(Int),Array{$(Int),1}}:\n 1 ⋅ ⋅ ⋅\n 2 6 ⋅ ⋅\n ⋅ 7 11 ⋅\n ⋅ ⋅ 12 16"
@test replstr(SymTridiagonal(A+A')) == "4×4 SymTridiagonal{$(Int),Array{$(Int),1}}:\n 2 7 ⋅ ⋅\n 7 12 17 ⋅\n ⋅ 17 22 27\n ⋅ ⋅ 27 32"
@test replstr(Tridiagonal(diag(A,-1),diag(A),diag(A,+1))) == "4×4 Tridiagonal{$Int}:\n 1 5 ⋅ ⋅\n 2 6 10 ⋅\n ⋅ 7 11 15\n ⋅ ⋅ 12 16"
@test replstr(Tridiagonal(diag(A,-1),diag(A),diag(A,+1))) == "4×4 Tridiagonal{$(Int),Array{$(Int),1}}:\n 1 5 ⋅ ⋅\n 2 6 10 ⋅\n ⋅ 7 11 15\n ⋅ ⋅ 12 16"
@test replstr(UpperTriangular(copy(A))) == "4×4 UpperTriangular{$Int,Array{$Int,2}}:\n 1 5 9 13\n ⋅ 6 10 14\n ⋅ ⋅ 11 15\n ⋅ ⋅ ⋅ 16"
@test replstr(LowerTriangular(copy(A))) == "4×4 LowerTriangular{$Int,Array{$Int,2}}:\n 1 ⋅ ⋅ ⋅\n 2 6 ⋅ ⋅\n 3 7 11 ⋅\n 4 8 12 16"

Expand Down