# This file is a part of Julia. License is MIT: https://julialang.org/license

#### Specialized matrix types ####

## (complex) symmetric tridiagonal matrices
struct SymTridiagonal{T, V<:AbstractVector{T}} <: AbstractMatrix{T}
    dv::V                        # diagonal
    ev::V                        # superdiagonal
    function SymTridiagonal{T, V}(dv, ev) where {T, V<:AbstractVector{T}}
        require_one_based_indexing(dv, ev)
        if !(length(dv) - 1 <= length(ev) <= length(dv))
            throw(DimensionMismatch(lazy"subdiagonal has wrong length. Has length $(length(ev)), but should be either $(length(dv) - 1) or $(length(dv))."))
        end
        new{T, V}(dv, ev)
    end
end

"""
    SymTridiagonal(dv::V, ev::V) where V <: AbstractVector

Construct a symmetric tridiagonal matrix from the diagonal (`dv`) and first
sub/super-diagonal (`ev`), respectively. The result is of type `SymTridiagonal`
and provides efficient specialized eigensolvers, but may be converted into a
regular matrix with [`convert(Array, _)`](@ref) (or `Array(_)` for short).

For `SymTridiagonal` block matrices, the elements of `dv` are symmetrized.
The argument `ev` is interpreted as the superdiagonal. Blocks from the
subdiagonal are (materialized) transpose of the corresponding superdiagonal blocks.

# Examples
```jldoctest
julia> dv = [1, 2, 3, 4]
4-element Vector{Int64}:
 1
 2
 3
 4

julia> ev = [7, 8, 9]
3-element Vector{Int64}:
 7
 8
 9

julia> SymTridiagonal(dv, ev)
4×4 SymTridiagonal{Int64, Vector{Int64}}:
 1  7  ⋅  ⋅
 7  2  8  ⋅
 ⋅  8  3  9
 ⋅  ⋅  9  4

julia> A = SymTridiagonal(fill([1 2; 3 4], 3), fill([1 2; 3 4], 2));

julia> A[1,1]
2×2 Symmetric{Int64, Matrix{Int64}}:
 1  2
 2  4

julia> A[1,2]
2×2 Matrix{Int64}:
 1  2
 3  4

julia> A[2,1]
2×2 Matrix{Int64}:
 1  3
 2  4
```
"""
SymTridiagonal(dv::V, ev::V) where {T,V<:AbstractVector{T}} = SymTridiagonal{T}(dv, ev)
SymTridiagonal{T}(dv::V, ev::V) where {T,V<:AbstractVector{T}} = SymTridiagonal{T,V}(dv, ev)
function SymTridiagonal{T}(dv::AbstractVector, ev::AbstractVector) where {T}
    d = convert(AbstractVector{T}, dv)::AbstractVector{T}
    e = convert(AbstractVector{T}, ev)::AbstractVector{T}
    typeof(d) == typeof(e) ?
        SymTridiagonal{T}(d, e) :
        throw(ArgumentError("diagonal vectors needed to be convertible to same type"))
end
SymTridiagonal(d::AbstractVector{T}, e::AbstractVector{S}) where {T,S} =
    SymTridiagonal{promote_type(T, S)}(d, e)

"""
    SymTridiagonal(A::AbstractMatrix)

Construct a symmetric tridiagonal matrix from the diagonal and first superdiagonal
of the symmetric matrix `A`.

# Examples
```jldoctest
julia> A = [1 2 3; 2 4 5; 3 5 6]
3×3 Matrix{Int64}:
 1  2  3
 2  4  5
 3  5  6

julia> SymTridiagonal(A)
3×3 SymTridiagonal{Int64, Vector{Int64}}:
 1  2  ⋅
 2  4  5
 ⋅  5  6

julia> B = reshape([[1 2; 2 3], [1 2; 3 4], [1 3; 2 4], [1 2; 2 3]], 2, 2);

julia> SymTridiagonal(B)
2×2 SymTridiagonal{Matrix{Int64}, Vector{Matrix{Int64}}}:
 [1 2; 2 3]  [1 3; 2 4]
 [1 2; 3 4]  [1 2; 2 3]
```
"""
function SymTridiagonal(A::AbstractMatrix)
    checksquare(A)
    du = diag(A, 1)
    d  = diag(A)
    dl = diag(A, -1)
    if all(((x, y),) -> x == transpose(y), zip(du, dl)) && all(issymmetric, d)
        SymTridiagonal(d, du)
    else
        throw(ArgumentError("matrix is not symmetric; cannot convert to SymTridiagonal"))
    end
end

SymTridiagonal{T,V}(S::SymTridiagonal{T,V}) where {T,V<:AbstractVector{T}} = S
SymTridiagonal{T,V}(S::SymTridiagonal) where {T,V<:AbstractVector{T}} =
    SymTridiagonal(convert(V, S.dv)::V, convert(V, S.ev)::V)
SymTridiagonal{T}(S::SymTridiagonal{T}) where {T} = S
SymTridiagonal{T}(S::SymTridiagonal) where {T} =
    SymTridiagonal(convert(AbstractVector{T}, S.dv)::AbstractVector{T},
                    convert(AbstractVector{T}, S.ev)::AbstractVector{T})
SymTridiagonal(S::SymTridiagonal) = S

AbstractMatrix{T}(S::SymTridiagonal) where {T} = SymTridiagonal{T}(S)
AbstractMatrix{T}(S::SymTridiagonal{T}) where {T} = copy(S)

function Matrix{T}(M::SymTridiagonal) where T
    n = size(M, 1)
    Mf = Matrix{T}(undef, n, n)
    n == 0 && return Mf
    if haszero(T) # optimized path for types with zero(T) defined
        n > 2 && fill!(Mf, zero(T))
        @inbounds for i = 1:n-1
            Mf[i,i] = symmetric(M.dv[i], :U)
            Mf[i+1,i] = transpose(M.ev[i])
            Mf[i,i+1] = M.ev[i]
        end
        Mf[n,n] = symmetric(M.dv[n], :U)
    else
        copyto!(Mf, M)
    end
    return Mf
end
Matrix(M::SymTridiagonal{T}) where {T} = Matrix{promote_type(T, typeof(zero(T)))}(M)
Array(M::SymTridiagonal) = Matrix(M)

size(A::SymTridiagonal) = (n = length(A.dv); (n, n))
axes(M::SymTridiagonal) = (ax = axes(M.dv, 1); (ax, ax))

similar(S::SymTridiagonal, ::Type{T}) where {T} = SymTridiagonal(similar(S.dv, T), similar(S.ev, T))
similar(S::SymTridiagonal, ::Type{T}, dims::Union{Dims{1},Dims{2}}) where {T} = similar(S.dv, T, dims)

# copyto! for matching axes
_copyto_banded!(dest::SymTridiagonal, src::SymTridiagonal) =
    (copyto!(dest.dv, src.dv); copyto!(dest.ev, _evview(src)); dest)

#Elementary operations
for func in (:conj, :copy, :real, :imag)
    @eval ($func)(M::SymTridiagonal) = SymTridiagonal(($func)(M.dv), ($func)(M.ev))
end

transpose(S::SymTridiagonal) = S
adjoint(S::SymTridiagonal{<:Number}) = SymTridiagonal(vec(adjoint(S.dv)), vec(adjoint(S.ev)))
adjoint(S::SymTridiagonal{<:Number, <:Base.ReshapedArray{<:Number,1,<:Adjoint}}) =
    SymTridiagonal(adjoint(parent(S.dv)), adjoint(parent(S.ev)))

permutedims(S::SymTridiagonal) = S
function permutedims(S::SymTridiagonal, perm)
    Base.checkdims_perm(axes(S), axes(S), perm)
    NTuple{2}(perm) == (2, 1) ? permutedims(S) : S
end
Base.copy(S::Adjoint{<:Any,<:SymTridiagonal}) = SymTridiagonal(map(x -> copy.(adjoint.(x)), (S.parent.dv, S.parent.ev))...)

ishermitian(S::SymTridiagonal) = isreal(S.dv) && isreal(_evview(S))
issymmetric(S::SymTridiagonal) = true

tr(S::SymTridiagonal) = sum(symmetric, S.dv)

_diagiter(M::SymTridiagonal{<:Number}) = M.dv
_diagiter(M::SymTridiagonal) = (symmetric(x, :U) for x in M.dv)
_eviter_transposed(M::SymTridiagonal{<:Number}) = _evview(M)
_eviter_transposed(M::SymTridiagonal) = (transpose(x) for x in _evview(M))

function diag(M::SymTridiagonal, n::Integer=0)
    # every branch call similar(..., ::Int) to make sure the
    # same vector type is returned independent of n
    v = similar(M.dv, max(0, length(M.dv)-abs(n)))
    if n == 0
        return copyto!(v, _diagiter(M))
    elseif n == 1
        return copyto!(v, _evview(M))
    elseif n == -1
        return copyto!(v, _eviter_transposed(M))
    else
        for i in eachindex(v)
            v[i] = M[BandIndex(n,i)]
        end
    end
    return v
end

+(A::SymTridiagonal, B::SymTridiagonal) = SymTridiagonal(A.dv+B.dv, _evview(A)+_evview(B))
-(A::SymTridiagonal, B::SymTridiagonal) = SymTridiagonal(A.dv-B.dv, _evview(A)-_evview(B))
-(A::SymTridiagonal) = SymTridiagonal(-A.dv, -A.ev)
*(A::SymTridiagonal, B::Number) = SymTridiagonal(A.dv*B, A.ev*B)
*(B::Number, A::SymTridiagonal) = SymTridiagonal(B*A.dv, B*A.ev)
function rmul!(A::SymTridiagonal, x::Number)
    if size(A,1) > 2
        # ensure that zeros are preserved on scaling
        y = A[3,1] * x
        iszero(y) || throw(ArgumentError(LazyString("cannot set index (3, 1) off ",
            lazy"the tridiagonal band to a nonzero value ($y)")))
    end
    A.dv .*= x
    _evview(A) .*= x
    return A
end
function lmul!(x::Number, B::SymTridiagonal)
    if size(B,1) > 2
        # ensure that zeros are preserved on scaling
        y = x * B[3,1]
        iszero(y) || throw(ArgumentError(LazyString("cannot set index (3, 1) off ",
            lazy"the tridiagonal band to a nonzero value ($y)")))
    end
    @. B.dv = x * B.dv
    ev = _evview(B)
    @. ev = x * ev
    return B
end
/(A::SymTridiagonal, B::Number) = SymTridiagonal(A.dv/B, A.ev/B)
\(B::Number, A::SymTridiagonal) = SymTridiagonal(B\A.dv, B\A.ev)
==(A::SymTridiagonal{<:Number}, B::SymTridiagonal{<:Number}) =
    (A.dv == B.dv) && (_evview(A) == _evview(B))
==(A::SymTridiagonal, B::SymTridiagonal) =
    size(A) == size(B) && all(i -> A[i,i] == B[i,i], axes(A, 1)) && (_evview(A) == _evview(B))

function dot(x::AbstractVector, S::SymTridiagonal, y::AbstractVector)
    require_one_based_indexing(x, y)
    nx, ny = length(x), length(y)
    (nx == size(S, 1) == ny) || throw(DimensionMismatch("dot"))
    if nx ≤ 1
        nx == 0 && return zero(dot(zero(eltype(x)), zero(eltype(S)), zero(eltype(y))))
        return dot(x[1], S.dv[1], y[1])
    end
    dv, ev = S.dv, S.ev
    @inbounds begin
        x₀ = x[1]
        x₊ = x[2]
        sub = transpose(ev[1])
        r = dot(adjoint(dv[1])*x₀ + adjoint(sub)*x₊, y[1])
        for j in 2:nx-1
            x₋, x₀, x₊ = x₀, x₊, x[j+1]
            sup, sub = transpose(sub), transpose(ev[j])
            r += dot(adjoint(sup)*x₋ + adjoint(dv[j])*x₀ + adjoint(sub)*x₊, y[j])
        end
        r += dot(adjoint(transpose(sub))*x₀ + adjoint(dv[nx])*x₊, y[nx])
    end
    return r
end

(\)(T::SymTridiagonal, B::AbstractVecOrMat) = ldlt(T)\B

# division with optional shift for use in shifted-Hessenberg solvers (hessenberg.jl):
ldiv!(A::SymTridiagonal, B::AbstractVecOrMat; shift::Number=false) = ldiv!(ldlt(A, shift=shift), B)
rdiv!(B::AbstractVecOrMat, A::SymTridiagonal; shift::Number=false) = rdiv!(B, ldlt(A, shift=shift))

eigen!(A::SymTridiagonal{<:BlasReal,<:StridedVector}) = Eigen(LAPACK.stegr!('V', A.dv, A.ev)...)
eigen(A::SymTridiagonal{T}) where T = eigen!(copymutable_oftype(A, eigtype(T)))

eigen!(A::SymTridiagonal{<:BlasReal,<:StridedVector}, irange::UnitRange) =
    Eigen(LAPACK.stegr!('V', 'I', A.dv, A.ev, 0.0, 0.0, irange.start, irange.stop)...)
eigen(A::SymTridiagonal{T}, irange::UnitRange) where T =
    eigen!(copymutable_oftype(A, eigtype(T)), irange)

eigen!(A::SymTridiagonal{<:BlasReal,<:StridedVector}, vl::Real, vu::Real) =
    Eigen(LAPACK.stegr!('V', 'V', A.dv, A.ev, vl, vu, 0, 0)...)
eigen(A::SymTridiagonal{T}, vl::Real, vu::Real) where T =
    eigen!(copymutable_oftype(A, eigtype(T)), vl, vu)

eigvals!(A::SymTridiagonal{<:BlasReal,<:StridedVector}) = LAPACK.stev!('N', A.dv, A.ev)[1]
eigvals(A::SymTridiagonal{T}) where T = eigvals!(copymutable_oftype(A, eigtype(T)))

eigvals!(A::SymTridiagonal{<:BlasReal,<:StridedVector}, irange::UnitRange) =
    LAPACK.stegr!('N', 'I', A.dv, A.ev, 0.0, 0.0, irange.start, irange.stop)[1]
eigvals(A::SymTridiagonal{T}, irange::UnitRange) where T =
    eigvals!(copymutable_oftype(A, eigtype(T)), irange)

eigvals!(A::SymTridiagonal{<:BlasReal,<:StridedVector}, vl::Real, vu::Real) =
    LAPACK.stegr!('N', 'V', A.dv, A.ev, vl, vu, 0, 0)[1]
eigvals(A::SymTridiagonal{T}, vl::Real, vu::Real) where T =
    eigvals!(copymutable_oftype(A, eigtype(T)), vl, vu)

#Computes largest and smallest eigenvalue
eigmax(A::SymTridiagonal) = eigvals(A, size(A, 1):size(A, 1))[1]
eigmin(A::SymTridiagonal) = eigvals(A, 1:1)[1]

#Compute selected eigenvectors only corresponding to particular eigenvalues
"""
    eigvecs(A::SymTridiagonal[, eigvals]) -> Matrix

Return a matrix `M` whose columns are the eigenvectors of `A`. (The `k`th eigenvector can
be obtained from the slice `M[:, k]`.)

If the optional vector of eigenvalues `eigvals` is specified, `eigvecs`
returns the specific corresponding eigenvectors.

# Examples
```jldoctest
julia> A = SymTridiagonal([1.; 2.; 1.], [2.; 3.])
3×3 SymTridiagonal{Float64, Vector{Float64}}:
 1.0  2.0   ⋅
 2.0  2.0  3.0
  ⋅   3.0  1.0

julia> eigvals(A)
3-element Vector{Float64}:
 -2.1400549446402604
  1.0000000000000002
  5.140054944640259

julia> eigvecs(A)
3×3 Matrix{Float64}:
  0.418304  -0.83205      0.364299
 -0.656749  -7.39009e-16  0.754109
  0.627457   0.5547       0.546448

julia> eigvecs(A, [1.])
3×1 Matrix{Float64}:
  0.8320502943378438
  4.263514128092366e-17
 -0.5547001962252291
```
"""
eigvecs(A::SymTridiagonal{<:BlasFloat,<:StridedVector}, eigvals::Vector{<:Real}) = LAPACK.stein!(A.dv, A.ev, eigvals)

function svdvals!(A::SymTridiagonal)
    vals = eigvals!(A)
    return sort!(map!(abs, vals, vals); rev=true)
end

# tril and triu

Base.@constprop :aggressive function istriu(M::SymTridiagonal, k::Integer=0)
    if k <= -1
        return true
    elseif k == 0
        return iszero(_evview(M))
    else # k >= 1
        return iszero(_evview(M)) && iszero(M.dv)
    end
end
Base.@constprop :aggressive istril(M::SymTridiagonal, k::Integer) = istriu(M, -k)
iszero(M::SymTridiagonal) =  iszero(_evview(M)) && iszero(M.dv)
isone(M::SymTridiagonal) =  iszero(_evview(M)) && all(isone, M.dv)
isdiag(M::SymTridiagonal) =  iszero(_evview(M))


function tril!(M::SymTridiagonal{T}, k::Integer=0) where T
    n = length(M.dv)
    if !(-n - 1 <= k <= n - 1)
        throw(ArgumentError(LazyString(lazy"the requested diagonal, $k, must be at least ",
            lazy"$(-n - 1) and at most $(n - 1) in an $n-by-$n matrix")))
    elseif k < -1
        fill!(M.ev, zero(T))
        fill!(M.dv, zero(T))
        return Tridiagonal(M.ev,M.dv,copy(M.ev))
    elseif k == -1
        fill!(M.dv, zero(T))
        return Tridiagonal(M.ev,M.dv,zero(M.ev))
    elseif k == 0
        return Tridiagonal(M.ev,M.dv,zero(M.ev))
    else # if k >= 1
        return Tridiagonal(M.ev,M.dv,copy(M.ev))
    end
end

function triu!(M::SymTridiagonal{T}, k::Integer=0) where T
    n = length(M.dv)
    if !(-n + 1 <= k <= n + 1)
        throw(ArgumentError(LazyString(lazy"the requested diagonal, $k, must be at least ",
            lazy"$(-n + 1) and at most $(n + 1) in an $n-by-$n matrix")))
    elseif k > 1
        fill!(M.ev, zero(T))
        fill!(M.dv, zero(T))
        return Tridiagonal(M.ev,M.dv,copy(M.ev))
    elseif k == 1
        fill!(M.dv, zero(T))
        return Tridiagonal(zero(M.ev),M.dv,M.ev)
    elseif k == 0
        return Tridiagonal(zero(M.ev),M.dv,M.ev)
    else # if k <= -1
        return Tridiagonal(M.ev,M.dv,copy(M.ev))
    end
end

###################
# Generic methods #
###################

## structured matrix methods ##
function Base.replace_in_print_matrix(A::SymTridiagonal, i::Integer, j::Integer, s::AbstractString)
    i==j-1||i==j||i==j+1 ? s : Base.replace_with_centered_mark(s)
end

# Implements the determinant using principal minors
# a, b, c are assumed to be the subdiagonal, diagonal, and superdiagonal of
# a tridiagonal matrix.
#Reference:
#    R. Usmani, "Inversion of a tridiagonal Jacobi matrix",
#    Linear Algebra and its Applications 212-213 (1994), pp.413-414
#    doi:10.1016/0024-3795(94)90414-6
function det_usmani(a::V, b::V, c::V, shift::Number=0) where {T,V<:AbstractVector{T}}
    require_one_based_indexing(a, b, c)
    n = length(b)
    θa = oneunit(T)+zero(shift)
    if n == 0
        return θa
    end
    θb = b[1]+shift
    for i in 2:n
        θb, θa = (b[i]+shift)*θb - a[i-1]*c[i-1]*θa, θb
    end
    return θb
end

# det with optional diagonal shift for use with shifted Hessenberg factorizations
det(A::SymTridiagonal; shift::Number=false) = det_usmani(A.ev, A.dv, A.ev, shift)
logabsdet(A::SymTridiagonal; shift::Number=false) = logabsdet(ldlt(A; shift=shift))

@inline function Base.isassigned(A::SymTridiagonal, i::Int, j::Int)
    @boundscheck checkbounds(Bool, A, i, j) || return false
    if i == j
        return @inbounds isassigned(A.dv, i)
    elseif i == j + 1
        return @inbounds isassigned(A.ev, j)
    elseif i + 1 == j
        return @inbounds isassigned(A.ev, i)
    else
        return true
    end
end

@inline function Base.isstored(A::SymTridiagonal, i::Int, j::Int)
    @boundscheck checkbounds(A, i, j)
    if i == j
        return @inbounds Base.isstored(A.dv, i)
    elseif i == j + 1
        return @inbounds Base.isstored(A.ev, j)
    elseif i + 1 == j
        return @inbounds Base.isstored(A.ev, i)
    else
        return false
    end
end

@inline function getindex(A::SymTridiagonal{T}, i::Int, j::Int) where T
    @boundscheck checkbounds(A, i, j)
    if i == j
        return symmetric((@inbounds A.dv[i]), :U)::symmetric_type(eltype(A.dv))
    elseif i == j + 1
        return copy(transpose(@inbounds A.ev[j])) # materialized for type stability
    elseif i + 1 == j
        return @inbounds A.ev[i]
    else
        return zero(T)
    end
end

Base._reverse(A::SymTridiagonal, dims) = reverse!(Matrix(A); dims)
Base._reverse(A::SymTridiagonal, dims::Colon) = SymTridiagonal(reverse(A.dv), reverse(A.ev))
Base._reverse!(A::SymTridiagonal, dims::Colon) = (reverse!(A.dv); reverse!(A.ev); A)

@inline function setindex!(A::SymTridiagonal, x, i::Integer, j::Integer)
    @boundscheck checkbounds(A, i, j)
    if i == j
        issymmetric(x) || throw(ArgumentError("cannot set a diagonal entry of a SymTridiagonal to an asymmetric value"))
        @inbounds A.dv[i] = x
    else
        throw(ArgumentError(lazy"cannot set off-diagonal entry ($i, $j)"))
    end
    return A
end

## Tridiagonal matrices ##
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,V}(dl, d, du) where {T,V<:AbstractVector{T}}
        require_one_based_indexing(dl, d, du)
        n = length(d)
        if (length(dl) != n-1 || length(du) != n-1) && !(length(d) == 0 && length(dl) == 0 && length(du) == 0)
            throw(ArgumentError(LazyString("cannot construct Tridiagonal from incompatible ",
                "lengths of subdiagonal, diagonal and superdiagonal: ",
                lazy"($(length(dl)), $(length(d)), $(length(du)))")))
        end
        new{T,V}(dl, d, Base.unalias(dl, du))
    end
    # constructor used in lu!
    function Tridiagonal{T,V}(dl, d, du, du2) where {T,V<:AbstractVector{T}}
        require_one_based_indexing(dl, d, du, du2)
        # length checks?
        new{T,V}(dl, d, Base.unalias(dl, du), du2)
    end
end

"""
    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
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`.

!!! note
    The subdiagonal `dl` and the superdiagonal `du` must not be aliased to each other.
    If aliasing is detected, the constructor will use a copy of `du` as its argument.

# Examples
```jldoctest
julia> dl = [1, 2, 3];

julia> du = [4, 5, 6];

julia> d = [7, 8, 9, 0];

julia> Tridiagonal(dl, d, du)
4×4 Tridiagonal{Int64, Vector{Int64}}:
 7  4  ⋅  ⋅
 1  8  5  ⋅
 ⋅  2  9  6
 ⋅  ⋅  3  0
```
"""
Tridiagonal(dl::V, d::V, du::V) where {T,V<:AbstractVector{T}} = Tridiagonal{T,V}(dl, d, du)
Tridiagonal(dl::V, d::V, du::V, du2::V) where {T,V<:AbstractVector{T}} = Tridiagonal{T,V}(dl, d, du, du2)
Tridiagonal(dl::AbstractVector{T}, d::AbstractVector{S}, du::AbstractVector{U}) where {T,S,U} =
    Tridiagonal{promote_type(T, S, U)}(dl, d, du)
Tridiagonal(dl::AbstractVector{T}, d::AbstractVector{S}, du::AbstractVector{U}, du2::AbstractVector{V}) where {T,S,U,V} =
    Tridiagonal{promote_type(T, S, U, V)}(dl, d, du, du2)
function Tridiagonal{T}(dl::AbstractVector, d::AbstractVector, du::AbstractVector) where {T}
    l, d, u = map(x->convert(AbstractVector{T}, x), (dl, d, du))
    typeof(l) == typeof(d) == typeof(u) ?
        Tridiagonal(l, d, u) :
        throw(ArgumentError("diagonal vectors needed to be convertible to same type"))
end
function Tridiagonal{T}(dl::AbstractVector, d::AbstractVector, du::AbstractVector, du2::AbstractVector) where {T}
    l, d, u, u2 = map(x->convert(AbstractVector{T}, x), (dl, d, du, du2))
    typeof(l) == typeof(d) == typeof(u) == typeof(u2) ?
        Tridiagonal(l, d, u, u2) :
        throw(ArgumentError("diagonal vectors needed to be convertible to same type"))
end

"""
    Tridiagonal(A)

Construct a tridiagonal matrix from the first sub-diagonal,
diagonal and first super-diagonal of the matrix `A`.

# Examples
```jldoctest
julia> A = [1 2 3 4; 1 2 3 4; 1 2 3 4; 1 2 3 4]
4×4 Matrix{Int64}:
 1  2  3  4
 1  2  3  4
 1  2  3  4
 1  2  3  4

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

Tridiagonal(A::Tridiagonal) = A
Tridiagonal{T}(A::Tridiagonal{T}) where {T} = A
function Tridiagonal{T}(A::Tridiagonal) where {T}
    dl, d, du = map(x -> convert(AbstractVector{T}, x)::AbstractVector{T}, (A.dl, A.d, A.du))
    if isdefined(A, :du2)
        Tridiagonal{T}(dl, d, du, convert(AbstractVector{T}, A.du2)::AbstractVector{T})
    else
        Tridiagonal{T}(dl, d, du)
    end
end
Tridiagonal{T,V}(A::Tridiagonal{T,V}) where {T,V<:AbstractVector{T}} = A
function Tridiagonal{T,V}(A::Tridiagonal) where {T,V<:AbstractVector{T}}
    dl, d, du = map(x -> convert(V, x)::V, (A.dl, A.d, A.du))
    if isdefined(A, :du2)
        Tridiagonal{T,V}(dl, d, du, convert(V, A.du2)::V)
    else
        Tridiagonal{T,V}(dl, d, du)
    end
end

size(M::Tridiagonal) = (n = length(M.d); (n, n))
axes(M::Tridiagonal) = (ax = axes(M.d,1); (ax, ax))

function Matrix{T}(M::Tridiagonal) where {T}
    A = Matrix{T}(undef, size(M))
    if haszero(T) # optimized path for types with zero(T) defined
        size(A,1) > 2 && fill!(A, zero(T))
        copyto!(diagview(A), M.d)
        copyto!(diagview(A,1), M.du)
        copyto!(diagview(A,-1), M.dl)
    else
        copyto!(A, M)
    end
    A
end
Matrix(M::Tridiagonal{T}) where {T} = Matrix{promote_type(T, typeof(zero(T)))}(M)
Array(M::Tridiagonal) = Matrix(M)

similar(M::Tridiagonal, ::Type{T}) where {T} = Tridiagonal(similar(M.dl, T), similar(M.d, T), similar(M.du, T))
similar(M::Tridiagonal, ::Type{T}, dims::Union{Dims{1},Dims{2}}) where {T} = similar(M.d, T, dims)

# Operations on Tridiagonal matrices
# copyto! for matching axes
function _copyto_banded!(dest::Tridiagonal, src::Tridiagonal)
    copyto!(dest.dl, src.dl)
    copyto!(dest.d, src.d)
    copyto!(dest.du, src.du)
    dest
end

#Elementary operations
for func in (:conj, :copy, :real, :imag)
    @eval function ($func)(M::Tridiagonal)
        Tridiagonal(($func)(M.dl), ($func)(M.d), ($func)(M.du))
    end
end

adjoint(S::Tridiagonal{<:Number}) = Tridiagonal(vec(adjoint(S.du)), vec(adjoint(S.d)), vec(adjoint(S.dl)))
adjoint(S::Tridiagonal{<:Number, <:Base.ReshapedArray{<:Number,1,<:Adjoint}}) =
    Tridiagonal(adjoint(parent(S.du)), adjoint(parent(S.d)), adjoint(parent(S.dl)))
transpose(S::Tridiagonal{<:Number}) = Tridiagonal(S.du, S.d, S.dl)
permutedims(T::Tridiagonal) = Tridiagonal(T.du, T.d, T.dl)
function permutedims(T::Tridiagonal, perm)
    Base.checkdims_perm(axes(T), axes(T), perm)
    NTuple{2}(perm) == (2, 1) ? permutedims(T) : T
end
Base.copy(aS::Adjoint{<:Any,<:Tridiagonal}) = (S = aS.parent; Tridiagonal(map(x -> copy.(adjoint.(x)), (S.du, S.d, S.dl))...))
Base.copy(tS::Transpose{<:Any,<:Tridiagonal}) = (S = tS.parent; Tridiagonal(map(x -> copy.(transpose.(x)), (S.du, S.d, S.dl))...))

ishermitian(S::Tridiagonal) = all(ishermitian, S.d) && all(Iterators.map((x, y) -> x == y', S.du, S.dl))
issymmetric(S::Tridiagonal) = all(issymmetric, S.d) && all(Iterators.map((x, y) -> x == transpose(y), S.du, S.dl))

\(A::Adjoint{<:Any,<:Tridiagonal}, B::Adjoint{<:Any,<:AbstractVecOrMat}) = copy(A) \ B

function diag(M::Tridiagonal, n::Integer=0)
    # every branch call similar(..., ::Int) to make sure the
    # same vector type is returned independent of n
    v = similar(M.d, max(0, length(M.d)-abs(n)))
    if n == 0
        copyto!(v, M.d)
    elseif n == -1
        copyto!(v, M.dl)
    elseif n == 1
        copyto!(v, M.du)
    elseif abs(n) <= size(M,1)
        for i in eachindex(v)
            v[i] = M[BandIndex(n,i)]
        end
    end
    return v
end

@inline function Base.isassigned(A::Tridiagonal, i::Int, j::Int)
    @boundscheck checkbounds(Bool, A, i, j) || return false
    if i == j
        return @inbounds isassigned(A.d, i)
    elseif i == j + 1
        return @inbounds isassigned(A.dl, j)
    elseif i + 1 == j
        return @inbounds isassigned(A.du, i)
    else
        return true
    end
end

@inline function Base.isstored(A::Tridiagonal, i::Int, j::Int)
    @boundscheck checkbounds(A, i, j)
    if i == j
        return @inbounds Base.isstored(A.d, i)
    elseif i == j + 1
        return @inbounds Base.isstored(A.dl, j)
    elseif i + 1 == j
        return @inbounds Base.isstored(A.du, i)
    else
        return false
    end
end

@inline function getindex(A::Tridiagonal{T}, i::Int, j::Int) where T
    @boundscheck checkbounds(A, i, j)
    if i == j
        return @inbounds A.d[i]
    elseif i == j + 1
        return @inbounds A.dl[j]
    elseif i + 1 == j
        return @inbounds A.du[i]
    else
        return zero(T)
    end
end

@inline function getindex(A::Tridiagonal{T}, b::BandIndex) where T
    @boundscheck checkbounds(A, b)
    if b.band == 0
        return @inbounds A.d[b.index]
    elseif b.band == -1
        return @inbounds A.dl[b.index]
    elseif b.band == 1
        return @inbounds A.du[b.index]
    else
        return zero(T)
    end
end

@inline function setindex!(A::Tridiagonal, x, i::Integer, j::Integer)
    @boundscheck checkbounds(A, i, j)
    if i == j
        @inbounds A.d[i] = x
    elseif i - j == 1
        @inbounds A.dl[j] = x
    elseif j - i == 1
        @inbounds A.du[i] = x
    elseif !iszero(x)
        throw(ArgumentError(LazyString(lazy"cannot set entry ($i, $j) off ",
            lazy"the tridiagonal band to a nonzero value ($x)")))
    end
    return A
end

## structured matrix methods ##
function Base.replace_in_print_matrix(A::Tridiagonal,i::Integer,j::Integer,s::AbstractString)
    i==j-1||i==j||i==j+1 ? s : Base.replace_with_centered_mark(s)
end

# reverse

Base._reverse(A::Tridiagonal, dims) = reverse!(Matrix(A); dims)
Base._reverse(A::Tridiagonal, dims::Colon) = Tridiagonal(reverse(A.du), reverse(A.d), reverse(A.dl))
function Base._reverse!(A::Tridiagonal, dims::Colon)
    n = length(A.du) # == length(A.dl), & always 1-based
    # reverse and swap A.dl and A.du:
    @inbounds for i in 1:n
        A.dl[i], A.du[n+1-i] = A.du[n+1-i], A.dl[i]
    end
    reverse!(A.d)
    return A
end

#tril and triu

iszero(M::Tridiagonal) = iszero(M.dl) && iszero(M.d) && iszero(M.du)
isone(M::Tridiagonal) = iszero(M.dl) && all(isone, M.d) && iszero(M.du)
Base.@constprop :aggressive function istriu(M::Tridiagonal, k::Integer=0)
    if k <= -1
        return true
    elseif k == 0
        return iszero(M.dl)
    elseif k == 1
        return iszero(M.dl) && iszero(M.d)
    else # k >= 2
        return iszero(M.dl) && iszero(M.d) && iszero(M.du)
    end
end
Base.@constprop :aggressive function istril(M::Tridiagonal, k::Integer=0)
    if k >= 1
        return true
    elseif k == 0
        return iszero(M.du)
    elseif k == -1
        return iszero(M.du) && iszero(M.d)
    else # k <= -2
        return iszero(M.du) && iszero(M.d) && iszero(M.dl)
    end
end
isdiag(M::Tridiagonal) = iszero(M.dl) && iszero(M.du)

function tril!(M::Tridiagonal{T}, k::Integer=0) where T
    n = length(M.d)
    if !(-n - 1 <= k <= n - 1)
        throw(ArgumentError(LazyString(lazy"the requested diagonal, $k, must be at least ",
            lazy"$(-n - 1) and at most $(n - 1) in an $n-by-$n matrix")))
    elseif k < -1
        fill!(M.dl, zero(T))
        fill!(M.d, zero(T))
        fill!(M.du, zero(T))
    elseif k == -1
        fill!(M.d, zero(T))
        fill!(M.du, zero(T))
    elseif k == 0
        fill!(M.du, zero(T))
    end
    return M
end

function triu!(M::Tridiagonal{T}, k::Integer=0) where T
    n = length(M.d)
    if !(-n + 1 <= k <= n + 1)
        throw(ArgumentError(LazyString(lazy"the requested diagonal, $k, must be at least ",
            lazy"$(-n + 1) and at most $(n + 1) in an $n-by-$n matrix")))
    elseif k > 1
        fill!(M.dl, zero(T))
        fill!(M.d, zero(T))
        fill!(M.du, zero(T))
    elseif k == 1
        fill!(M.dl, zero(T))
        fill!(M.d, zero(T))
    elseif k == 0
        fill!(M.dl, zero(T))
    end
    return M
end

tr(M::Tridiagonal) = sum(M.d)

###################
# Generic methods #
###################

+(A::Tridiagonal, B::Tridiagonal) = Tridiagonal(A.dl+B.dl, A.d+B.d, A.du+B.du)
-(A::Tridiagonal, B::Tridiagonal) = Tridiagonal(A.dl-B.dl, A.d-B.d, A.du-B.du)
-(A::Tridiagonal) = Tridiagonal(-A.dl, -A.d, -A.du)
*(A::Tridiagonal, B::Number) = Tridiagonal(A.dl*B, A.d*B, A.du*B)
*(B::Number, A::Tridiagonal) = Tridiagonal(B*A.dl, B*A.d, B*A.du)
function rmul!(T::Tridiagonal, x::Number)
    if size(T,1) > 2
        # ensure that zeros are preserved on scaling
        y = T[3,1] * x
        iszero(y) || throw(ArgumentError(LazyString("cannot set index (3, 1) off ",
            lazy"the tridiagonal band to a nonzero value ($y)")))
    end
    T.dl .*= x
    T.d .*= x
    T.du .*= x
    return T
end
function lmul!(x::Number, T::Tridiagonal)
    if size(T,1) > 2
        # ensure that zeros are preserved on scaling
        y = x * T[3,1]
        iszero(y) || throw(ArgumentError(LazyString("cannot set index (3, 1) off ",
            lazy"the tridiagonal band to a nonzero value ($y)")))
    end
    @. T.dl = x * T.dl
    @. T.d = x * T.d
    @. T.du = x * T.du
    return T
end
/(A::Tridiagonal, B::Number) = Tridiagonal(A.dl/B, A.d/B, A.du/B)
\(B::Number, A::Tridiagonal) = Tridiagonal(B\A.dl, B\A.d, B\A.du)

==(A::Tridiagonal, B::Tridiagonal) = (A.dl==B.dl) && (A.d==B.d) && (A.du==B.du)
function ==(A::Tridiagonal, B::SymTridiagonal)
    iseq = all(Iterators.map((x, y) -> x == transpose(y), A.du, A.dl))
    iseq = iseq && A.du == _evview(B)
    iseq && all(Iterators.map((x, y) -> x == symmetric(y, :U), A.d, B.dv))
end
==(A::SymTridiagonal, B::Tridiagonal) = B == A

det(A::Tridiagonal) = det_usmani(A.dl, A.d, A.du)

AbstractMatrix{T}(M::Tridiagonal) where {T} = Tridiagonal{T}(M)
AbstractMatrix{T}(M::Tridiagonal{T}) where {T} = copy(M)
Tridiagonal{T}(M::SymTridiagonal{T}) where {T} = Tridiagonal(M)
function SymTridiagonal{T}(M::Tridiagonal) where T
    if issymmetric(M)
        return SymTridiagonal{T}(convert(AbstractVector{T},M.d), convert(AbstractVector{T},M.dl))
    else
        throw(ArgumentError("Tridiagonal is not symmetric, cannot convert to SymTridiagonal"))
    end
end

Base._sum(A::Tridiagonal, ::Colon) = sum(A.d) + sum(A.dl) + sum(A.du)
function Base._sum(A::SymTridiagonal, ::Colon)
    se = sum(_evview(A))
    symmetric(sum(A.dv), :U) + se + transpose(se)
end

function Base._sum(A::Tridiagonal, dims::Integer)
    res = Base.reducedim_initarray(A, dims, zero(eltype(A)))
    n = length(A.d)
    if n == 0
        return res
    elseif n == 1
        res[1] = A.d[1]
        return res
    end
    @inbounds begin
        if dims == 1
            res[1] = A.dl[1] + A.d[1]
            for i = 2:n-1
                res[i] = A.dl[i] + A.d[i] + A.du[i-1]
            end
            res[n] = A.d[n] + A.du[n-1]
        elseif dims == 2
            res[1] = A.d[1] + A.du[1]
            for i = 2:n-1
                res[i] = A.dl[i-1] + A.d[i] + A.du[i]
            end
            res[n] = A.dl[n-1] + A.d[n]
        elseif dims >= 3
            for i = 1:n-1
                res[i,i+1] = A.du[i]
                res[i,i]   = A.d[i]
                res[i+1,i] = A.dl[i]
            end
            res[n,n] = A.d[n]
        end
    end
    res
end

function Base._sum(A::SymTridiagonal, dims::Integer)
    res = Base.reducedim_initarray(A, dims, zero(eltype(A)))
    n = length(A.dv)
    if n == 0
        return res
    elseif n == 1
        res[1] = A.dv[1]
        return res
    end
    @inbounds begin
        if dims == 1
            res[1] = transpose(A.ev[1]) + symmetric(A.dv[1], :U)
            for i = 2:n-1
                res[i] = transpose(A.ev[i]) + symmetric(A.dv[i], :U) + A.ev[i-1]
            end
            res[n] = symmetric(A.dv[n], :U) + A.ev[n-1]
        elseif dims == 2
            res[1] = symmetric(A.dv[1], :U) + A.ev[1]
            for i = 2:n-1
                res[i] = transpose(A.ev[i-1]) + symmetric(A.dv[i], :U) + A.ev[i]
            end
            res[n] = transpose(A.ev[n-1]) + symmetric(A.dv[n], :U)
        elseif dims >= 3
            for i = 1:n-1
                res[i,i+1] = A.ev[i]
                res[i,i]   = symmetric(A.dv[i], :U)
                res[i+1,i] = transpose(A.ev[i])
            end
            res[n,n] = symmetric(A.dv[n], :U)
        end
    end
    res
end

function dot(x::AbstractVector, A::Tridiagonal, y::AbstractVector)
    require_one_based_indexing(x, y)
    nx, ny = length(x), length(y)
    (nx == size(A, 1) == ny) || throw(DimensionMismatch())
    if nx ≤ 1
        nx == 0 && return zero(dot(zero(eltype(x)), zero(eltype(A)), zero(eltype(y))))
        return dot(x[1], A.d[1], y[1])
    end
    @inbounds begin
        x₀ = x[1]
        x₊ = x[2]
        dl, d, du = A.dl, A.d, A.du
        r = dot(adjoint(d[1])*x₀ + adjoint(dl[1])*x₊, y[1])
        for j in 2:nx-1
            x₋, x₀, x₊ = x₀, x₊, x[j+1]
            r += dot(adjoint(du[j-1])*x₋ + adjoint(d[j])*x₀ + adjoint(dl[j])*x₊, y[j])
        end
        r += dot(adjoint(du[nx-1])*x₀ + adjoint(d[nx])*x₊, y[nx])
    end
    return r
end

function cholesky(S::SymTridiagonal, ::NoPivot = NoPivot(); check::Bool = true)
    if !ishermitian(S)
        check && checkpositivedefinite(-1)
        return Cholesky(S, 'U', convert(BlasInt, -1))
    end
    T = choltype(S)
    cholesky!(Hermitian(Bidiagonal{T}(diag(S, 0), diag(S, 1), :U)), NoPivot(); check = check)
end

# See dgtsv.f
"""
    ldiv!(A::Tridiagonal, B::AbstractVecOrMat) -> B

Compute `A \\ B` in-place by Gaussian elimination with partial pivoting and store the result
in `B`, returning the result. In the process, the diagonals of `A` are overwritten as well.

!!! compat "Julia 1.11"
    `ldiv!` for `Tridiagonal` left-hand sides requires at least Julia 1.11.
"""
function ldiv!(A::Tridiagonal, B::AbstractVecOrMat)
    LinearAlgebra.require_one_based_indexing(B)
    n = size(A, 1)
    if n != size(B,1)
        throw(DimensionMismatch(lazy"matrix has dimensions ($n,$n) but right hand side has $(size(B,1)) rows"))
    end
    nrhs = size(B, 2)

    # Initialize variables
    dl = A.dl
    d = A.d
    du = A.du

    @inbounds begin
        for i in 1:n-1
            # pivot or not?
            if abs(d[i]) >= abs(dl[i])
                # No interchange
                if d[i] != 0
                    fact = dl[i]/d[i]
                    d[i+1] -= fact*du[i]
                    for j in 1:nrhs
                        B[i+1,j] -= fact*B[i,j]
                    end
                else
                    checknonsingular(i)
                end
                i < n-1 && (dl[i] = 0)
            else
                # Interchange
                fact = d[i]/dl[i]
                d[i] = dl[i]
                tmp = d[i+1]
                d[i+1] = du[i] - fact*tmp
                du[i] = tmp
                if i < n-1
                    dl[i] = du[i+1]
                    du[i+1] = -fact*dl[i]
                end
                for j in 1:nrhs
                    temp = B[i,j]
                    B[i,j] = B[i+1,j]
                    B[i+1,j] = temp - fact*B[i+1,j]
                end
            end
        end
        iszero(d[n]) && checknonsingular(n)
        # backward substitution
        for j in 1:nrhs
            B[n,j] /= d[n]
            if n > 1
                B[n-1,j] = (B[n-1,j] - du[n-1]*B[n,j])/d[n-1]
            end
            for i in n-2:-1:1
                B[i,j] = (B[i,j] - du[i]*B[i+1,j] - dl[i]*B[i+2,j]) / d[i]
            end
        end
    end
    return B
end

# combinations of Tridiagonal and Symtridiagonal
# copyto! for matching axes
function _copyto_banded!(A::Tridiagonal, B::SymTridiagonal)
    Bev = _evview(B)
    A.du .= Bev
    # Broadcast identity for numbers to access the faster copyto! path
    # This uses the fact that transpose(x::Number) = x and symmetric(x::Number) = x
    A.dl .= (eltype(B) <: Number ? identity : transpose).(Bev)
    A.d .= (eltype(B) <: Number ? identity : symmetric).(B.dv)
    return A
end
function _copyto_banded!(A::SymTridiagonal, B::Tridiagonal)
    issymmetric(B) || throw(ArgumentError("cannot copy an asymmetric Tridiagonal matrix to a SymTridiagonal"))
    A.dv .= B.d
    _evview(A) .= B.du
    return A
end

# display
function show(io::IO, T::Tridiagonal)
    print(io, "Tridiagonal(")
    show(io, T.dl)
    print(io, ", ")
    show(io, T.d)
    print(io, ", ")
    show(io, T.du)
    print(io, ")")
end
function show(io::IO, S::SymTridiagonal)
    print(io, "SymTridiagonal(")
    show(io, _diagview(S))
    print(io, ", ")
    show(io, S.ev)
    print(io, ")")
end

###################
#     opnorms     #
###################

# Tridiagonal

function _opnorm1Inf(A::Tridiagonal, p)
    size(A, 1) == 1 && return norm(first(A.d))
    case = p == Inf
    lowerrange, upperrange = case ? (1:length(A.dl)-1, 2:length(A.dl)) : (2:length(A.dl), 1:length(A.dl)-1)
    normfirst, normend = case ? (norm(first(A.d))+norm(first(A.du)), norm(last(A.dl))+norm(last(A.d))) : (norm(first(A.d))+norm(first(A.dl)), norm(last(A.du))+norm(last(A.d)))
    size(A, 1) == 2 && return max(normfirst, normend)
    return max(
                mapreduce(t -> sum(norm, t),
                    max,
                    zip(view(A.d, (2:length(A.d)-1)), view(A.dl, lowerrange), view(A.du, upperrange))
                ),
                normfirst, normend)
end

# SymTridiagonal

function _opnorm1Inf(A::SymTridiagonal, p::Real)
    size(A, 1) == 1 && return norm(first(A.dv))
    lowerrange, upperrange = 1:length(A.ev)-1, 2:length(A.ev)
    normfirst, normend = norm(first(A.dv))+norm(first(A.ev)), norm(last(A.ev))+norm(last(A.dv))
    size(A, 1) == 2 && return max(normfirst, normend)
    return max(
                mapreduce(t -> sum(norm, t),
                    max,
                    zip(view(A.dv, (2:length(A.dv)-1)), view(A.ev, lowerrange), view(A.ev, upperrange))
                ),
                normfirst, normend)
end
