https://github.com/JuliaLang/julia
Tip revision: 68e911be534f84f6201cbdd5d92ef0757af1238a authored by Tony Kelman on 18 May 2017, 02:31:01 UTC
Tag v0.6.0-rc2
Tag v0.6.0-rc2
Tip revision: 68e911b
tridiag.jl
# This file is a part of Julia. License is MIT: https://julialang.org/license
#### Specialized matrix types ####
## (complex) symmetric tridiagonal matrices
struct SymTridiagonal{T} <: AbstractMatrix{T}
dv::Vector{T} # diagonal
ev::Vector{T} # subdiagonal
function SymTridiagonal{T}(dv::Vector{T}, ev::Vector{T}) where T
if !(length(dv) - 1 <= length(ev) <= length(dv))
throw(DimensionMismatch("subdiagonal has wrong length. Has length $(length(ev)), but should be either $(length(dv) - 1) or $(length(dv))."))
end
new(dv,ev)
end
end
"""
SymTridiagonal(dv, ev)
Construct a symmetric tridiagonal matrix from the diagonal and first sub/super-diagonal,
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).
# Example
```jldoctest
julia> dv = [1; 2; 3; 4]
4-element Array{Int64,1}:
1
2
3
4
julia> ev = [7; 8; 9]
3-element Array{Int64,1}:
7
8
9
julia> SymTridiagonal(dv, ev)
4×4 SymTridiagonal{Int64}:
1 7 ⋅ ⋅
7 2 8 ⋅
⋅ 8 3 9
⋅ ⋅ 9 4
```
"""
SymTridiagonal(dv::Vector{T}, ev::Vector{T}) where {T} = SymTridiagonal{T}(dv, ev)
function SymTridiagonal(dv::AbstractVector{Td}, ev::AbstractVector{Te}) where {Td,Te}
T = promote_type(Td,Te)
SymTridiagonal(convert(Vector{T}, dv), convert(Vector{T}, ev))
end
function SymTridiagonal(A::AbstractMatrix)
if diag(A,1) == diag(A,-1)
SymTridiagonal(diag(A), diag(A,1))
else
throw(ArgumentError("matrix is not symmetric; cannot convert to SymTridiagonal"))
end
end
convert(::Type{SymTridiagonal{T}}, S::SymTridiagonal) where {T} =
SymTridiagonal(convert(Vector{T}, S.dv), convert(Vector{T}, S.ev))
convert(::Type{AbstractMatrix{T}}, S::SymTridiagonal) where {T} =
SymTridiagonal(convert(Vector{T}, S.dv), convert(Vector{T}, S.ev))
function convert(::Type{Matrix{T}}, M::SymTridiagonal{T}) where T
n = size(M, 1)
Mf = zeros(T, n, n)
@inbounds begin
@simd for i = 1:n-1
Mf[i,i] = M.dv[i]
Mf[i+1,i] = M.ev[i]
Mf[i,i+1] = M.ev[i]
end
Mf[n,n] = M.dv[n]
end
return Mf
end
convert(::Type{Matrix}, M::SymTridiagonal{T}) where {T} = convert(Matrix{T}, M)
convert(::Type{Array}, M::SymTridiagonal) = convert(Matrix, M)
full(M::SymTridiagonal) = convert(Array, M)
size(A::SymTridiagonal) = (length(A.dv), length(A.dv))
function size(A::SymTridiagonal, d::Integer)
if d < 1
throw(ArgumentError("dimension must be ≥ 1, got $d"))
elseif d<=2
return length(A.dv)
else
return 1
end
end
similar(S::SymTridiagonal, ::Type{T}) where {T} = SymTridiagonal{T}(similar(S.dv, T), similar(S.ev, T))
#Elementary operations
broadcast(::typeof(abs), M::SymTridiagonal) = SymTridiagonal(abs.(M.dv), abs.(M.ev))
broadcast(::typeof(round), M::SymTridiagonal) = SymTridiagonal(round.(M.dv), round.(M.ev))
broadcast(::typeof(trunc), M::SymTridiagonal) = SymTridiagonal(trunc.(M.dv), trunc.(M.ev))
broadcast(::typeof(floor), M::SymTridiagonal) = SymTridiagonal(floor.(M.dv), floor.(M.ev))
broadcast(::typeof(ceil), M::SymTridiagonal) = SymTridiagonal(ceil.(M.dv), ceil.(M.ev))
for func in (:conj, :copy, :real, :imag)
@eval ($func)(M::SymTridiagonal) = SymTridiagonal(($func)(M.dv), ($func)(M.ev))
end
broadcast(::typeof(round), ::Type{T}, M::SymTridiagonal) where {T<:Integer} = SymTridiagonal(round.(T, M.dv), round.(T, M.ev))
broadcast(::typeof(trunc), ::Type{T}, M::SymTridiagonal) where {T<:Integer} = SymTridiagonal(trunc.(T, M.dv), trunc.(T, M.ev))
broadcast(::typeof(floor), ::Type{T}, M::SymTridiagonal) where {T<:Integer} = SymTridiagonal(floor.(T, M.dv), floor.(T, M.ev))
broadcast(::typeof(ceil), ::Type{T}, M::SymTridiagonal) where {T<:Integer} = SymTridiagonal(ceil.(T, M.dv), ceil.(T, M.ev))
transpose(M::SymTridiagonal) = M #Identity operation
ctranspose(M::SymTridiagonal) = conj(M)
function diag(M::SymTridiagonal{T}, n::Integer=0) where T
absn = abs(n)
if absn == 0
return M.dv
elseif absn==1
return M.ev
elseif absn<size(M,1)
return zeros(T,size(M,1)-absn)
else
throw(ArgumentError("$n-th diagonal of a $(size(M)) matrix doesn't exist!"))
end
end
+(A::SymTridiagonal, B::SymTridiagonal) = SymTridiagonal(A.dv+B.dv, A.ev+B.ev)
-(A::SymTridiagonal, B::SymTridiagonal) = SymTridiagonal(A.dv-B.dv, A.ev-B.ev)
*(A::SymTridiagonal, B::Number) = SymTridiagonal(A.dv*B, A.ev*B)
*(B::Number, A::SymTridiagonal) = A*B
/(A::SymTridiagonal, B::Number) = SymTridiagonal(A.dv/B, A.ev/B)
==(A::SymTridiagonal, B::SymTridiagonal) = (A.dv==B.dv) && (A.ev==B.ev)
function A_mul_B!(C::StridedVecOrMat, S::SymTridiagonal, B::StridedVecOrMat)
m, n = size(B, 1), size(B, 2)
if !(m == size(S, 1) == size(C, 1))
throw(DimensionMismatch("A has first dimension $(size(S,1)), B has $(size(B,1)), C has $(size(C,1)) but all must match"))
end
if n != size(C, 2)
throw(DimensionMismatch("second dimension of B, $n, doesn't match second dimension of C, $(size(C,2))"))
end
α = S.dv
β = S.ev
@inbounds begin
for j = 1:n
x₀, x₊ = B[1, j], B[2, j]
β₀ = β[1]
C[1, j] = α[1]*x₀ + x₊*β₀
for i = 2:m - 1
x₋, x₀, x₊ = x₀, x₊, B[i + 1, j]
β₋, β₀ = β₀, β[i]
C[i, j] = β₋*x₋ + α[i]*x₀ + β₀*x₊
end
C[m, j] = β₀*x₀ + α[m]*x₊
end
end
return C
end
(\)(T::SymTridiagonal, B::StridedVecOrMat) = ldltfact(T)\B
eigfact!(A::SymTridiagonal{<:BlasReal}) = Eigen(LAPACK.stegr!('V', A.dv, A.ev)...)
function eigfact(A::SymTridiagonal{T}) where T
S = promote_type(Float32, typeof(zero(T)/norm(one(T))))
eigfact!(copy_oftype(A, S))
end
eigfact!(A::SymTridiagonal{<:BlasReal}, irange::UnitRange) =
Eigen(LAPACK.stegr!('V', 'I', A.dv, A.ev, 0.0, 0.0, irange.start, irange.stop)...)
function eigfact(A::SymTridiagonal{T}, irange::UnitRange) where T
S = promote_type(Float32, typeof(zero(T)/norm(one(T))))
return eigfact!(copy_oftype(A, S), irange)
end
eigfact!(A::SymTridiagonal{<:BlasReal}, vl::Real, vu::Real) =
Eigen(LAPACK.stegr!('V', 'V', A.dv, A.ev, vl, vu, 0, 0)...)
function eigfact(A::SymTridiagonal{T}, vl::Real, vu::Real) where T
S = promote_type(Float32, typeof(zero(T)/norm(one(T))))
return eigfact!(copy_oftype(A, S), vl, vu)
end
eigvals!(A::SymTridiagonal{<:BlasReal}) = LAPACK.stev!('N', A.dv, A.ev)[1]
function eigvals(A::SymTridiagonal{T}) where T
S = promote_type(Float32, typeof(zero(T)/norm(one(T))))
return eigvals!(copy_oftype(A, S))
end
eigvals!(A::SymTridiagonal{<:BlasReal}, irange::UnitRange) =
LAPACK.stegr!('N', 'I', A.dv, A.ev, 0.0, 0.0, irange.start, irange.stop)[1]
function eigvals(A::SymTridiagonal{T}, irange::UnitRange) where T
S = promote_type(Float32, typeof(zero(T)/norm(one(T))))
return eigvals!(copy_oftype(A, S), irange)
end
eigvals!(A::SymTridiagonal{<:BlasReal}, vl::Real, vu::Real) =
LAPACK.stegr!('N', 'V', A.dv, A.ev, vl, vu, 0, 0)[1]
function eigvals(A::SymTridiagonal{T}, vl::Real, vu::Real) where T
S = promote_type(Float32, typeof(zero(T)/norm(one(T))))
return eigvals!(copy_oftype(A, S), vl, vu)
end
#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) = eigfact(A)[:vectors]
"""
eigvecs(A::SymTridiagonal[, eigvals]) -> Matrix
Returns 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.
# Example
```jldoctest
julia> A = SymTridiagonal([1.; 2.; 1.], [2.; 3.])
3×3 SymTridiagonal{Float64}:
1.0 2.0 ⋅
2.0 2.0 3.0
⋅ 3.0 1.0
julia> eigvals(A)
3-element Array{Float64,1}:
-2.14005
1.0
5.14005
julia> eigvecs(A)
3×3 Array{Float64,2}:
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 Array{Float64,2}:
0.83205
4.26351e-17
-0.5547
```
"""
eigvecs(A::SymTridiagonal{<:BlasFloat}, eigvals::Vector{<:Real}) = LAPACK.stein!(A.dv, A.ev, eigvals)
#tril and triu
istriu(M::SymTridiagonal) = iszero(M.ev)
istril(M::SymTridiagonal) = iszero(M.ev)
function tril!(M::SymTridiagonal, k::Integer=0)
n = length(M.dv)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif k < -1
fill!(M.ev,0)
fill!(M.dv,0)
return Tridiagonal(M.ev,M.dv,copy(M.ev))
elseif k == -1
fill!(M.dv,0)
return Tridiagonal(M.ev,M.dv,zeros(M.ev))
elseif k == 0
return Tridiagonal(M.ev,M.dv,zeros(M.ev))
elseif k >= 1
return Tridiagonal(M.ev,M.dv,copy(M.ev))
end
end
function triu!(M::SymTridiagonal, k::Integer=0)
n = length(M.dv)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif k > 1
fill!(M.ev,0)
fill!(M.dv,0)
return Tridiagonal(M.ev,M.dv,copy(M.ev))
elseif k == 1
fill!(M.dv,0)
return Tridiagonal(zeros(M.ev),M.dv,M.ev)
elseif k == 0
return Tridiagonal(zeros(M.ev),M.dv,M.ev)
elseif k <= -1
return Tridiagonal(M.ev,M.dv,copy(M.ev))
end
end
###################
# Generic methods #
###################
#Needed for inv_usmani()
mutable struct ZeroOffsetVector
data::Vector
end
getindex(a::ZeroOffsetVector, i) = a.data[i+1]
setindex!(a::ZeroOffsetVector, x, i) = a.data[i+1]=x
## 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 inverse using the recurrence relation between 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 inv_usmani(a::Vector{T}, b::Vector{T}, c::Vector{T}) where T
n = length(b)
θ = ZeroOffsetVector(zeros(T, n+1)) #principal minors of A
θ[0] = 1
n>=1 && (θ[1] = b[1])
for i=2:n
θ[i] = b[i]*θ[i-1]-a[i-1]*c[i-1]*θ[i-2]
end
φ = zeros(T, n+1)
φ[n+1] = 1
n>=1 && (φ[n] = b[n])
for i=n-1:-1:1
φ[i] = b[i]*φ[i+1]-a[i]*c[i]*φ[i+2]
end
α = Matrix{T}(n, n)
for i=1:n, j=1:n
sign = (i+j)%2==0 ? (+) : (-)
if i<j
α[i,j]=(sign)(prod(c[i:j-1]))*θ[i-1]*φ[j+1]/θ[n]
elseif i==j
α[i,i]= θ[i-1]*φ[i+1]/θ[n]
else #i>j
α[i,j]=(sign)(prod(a[j:i-1]))*θ[j-1]*φ[i+1]/θ[n]
end
end
α
end
#Implements the determinant using principal minors
#Inputs and reference are as above for inv_usmani()
function det_usmani(a::Vector{T}, b::Vector{T}, c::Vector{T}) where T
n = length(b)
θa = one(T)
if n == 0
return θa
end
θb = b[1]
for i=2:n
θb, θa = b[i]*θb-a[i-1]*c[i-1]*θa, θb
end
return θb
end
inv(A::SymTridiagonal) = inv_usmani(A.ev, A.dv, A.ev)
det(A::SymTridiagonal) = det_usmani(A.ev, A.dv, A.ev)
function getindex(A::SymTridiagonal{T}, i::Integer, j::Integer) where T
if !(1 <= i <= size(A,2) && 1 <= j <= size(A,2))
throw(BoundsError(A, (i,j)))
end
if i == j
return A.dv[i]
elseif i == j + 1
return A.ev[j]
elseif i + 1 == j
return A.ev[i]
else
return zero(T)
end
end
function setindex!(A::SymTridiagonal, x, i::Integer, j::Integer)
@boundscheck checkbounds(A, i, j)
if i == j
@inbounds A.dv[i] = x
else
throw(ArgumentError("cannot set off-diagonal entry ($i, $j)"))
end
return x
end
## Tridiagonal matrices ##
struct Tridiagonal{T} <: AbstractMatrix{T}
dl::Vector{T} # sub-diagonal
d::Vector{T} # diagonal
du::Vector{T} # sup-diagonal
du2::Vector{T} # supsup-diagonal for pivoting
end
"""
Tridiagonal(dl, d, du)
Construct a tridiagonal matrix from the first subdiagonal, diagonal, and first superdiagonal,
respectively. The result is of type `Tridiagonal` and provides efficient specialized linear
solvers, but may be converted into a regular matrix with
[`convert(Array, _)`](@ref) (or `Array(_)` for short).
The lengths of `dl` and `du` must be one less than the length of `d`.
# Example
```jldoctest
julia> dl = [1; 2; 3]
3-element Array{Int64,1}:
1
2
3
julia> du = [4; 5; 6]
3-element Array{Int64,1}:
4
5
6
julia> d = [7; 8; 9; 0]
4-element Array{Int64,1}:
7
8
9
0
julia> Tridiagonal(dl, d, du)
4×4 Tridiagonal{Int64}:
7 4 ⋅ ⋅
1 8 5 ⋅
⋅ 2 9 6
⋅ ⋅ 3 0
```
"""
# Basic constructor takes in three dense vectors of same type
function Tridiagonal(dl::Vector{T}, d::Vector{T}, du::Vector{T}) where T
n = length(d)
if (length(dl) != n-1 || length(du) != n-1)
throw(ArgumentError("cannot make Tridiagonal from incompatible lengths of subdiagonal, diagonal and superdiagonal: ($(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
# 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.
# Example
```jldoctest
julia> A = [1 2 3 4; 1 2 3 4; 1 2 3 4; 1 2 3 4]
4×4 Array{Int64,2}:
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
julia> Tridiagonal(A)
4×4 Tridiagonal{Int64}:
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
size(M::Tridiagonal) = (length(M.d), length(M.d))
function size(M::Tridiagonal, d::Integer)
if d < 1
throw(ArgumentError("dimension d must be ≥ 1, got $d"))
elseif d <= 2
return length(M.d)
else
return 1
end
end
function convert(::Type{Matrix{T}}, M::Tridiagonal{T}) where T
A = zeros(T, size(M))
for i = 1:length(M.d)
A[i,i] = M.d[i]
end
for i = 1:length(M.d)-1
A[i+1,i] = M.dl[i]
A[i,i+1] = M.du[i]
end
A
end
convert(::Type{Matrix}, M::Tridiagonal{T}) where {T} = convert(Matrix{T}, M)
convert(::Type{Array}, M::Tridiagonal) = convert(Matrix, M)
full(M::Tridiagonal) = convert(Array, M)
function similar(M::Tridiagonal, ::Type{T}) where T
Tridiagonal{T}(similar(M.dl, T), similar(M.d, T), similar(M.du, T), similar(M.du2, T))
end
# Operations on Tridiagonal matrices
copy!(dest::Tridiagonal, src::Tridiagonal) = Tridiagonal(copy!(dest.dl, src.dl), copy!(dest.d, src.d), copy!(dest.du, src.du), copy!(dest.du2, src.du2))
#Elementary operations
broadcast(::typeof(abs), M::Tridiagonal) = Tridiagonal(abs.(M.dl), abs.(M.d), abs.(M.du), abs.(M.du2))
broadcast(::typeof(round), M::Tridiagonal) = Tridiagonal(round.(M.dl), round.(M.d), round.(M.du), round.(M.du2))
broadcast(::typeof(trunc), M::Tridiagonal) = Tridiagonal(trunc.(M.dl), trunc.(M.d), trunc.(M.du), trunc.(M.du2))
broadcast(::typeof(floor), M::Tridiagonal) = Tridiagonal(floor.(M.dl), floor.(M.d), floor.(M.du), floor.(M.du2))
broadcast(::typeof(ceil), M::Tridiagonal) = Tridiagonal(ceil.(M.dl), ceil.(M.d), ceil.(M.du), ceil.(M.du2))
for func in (:conj, :copy, :real, :imag)
@eval function ($func)(M::Tridiagonal)
Tridiagonal(($func)(M.dl), ($func)(M.d), ($func)(M.du), ($func)(M.du2))
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))
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))
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))
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))
transpose(M::Tridiagonal) = Tridiagonal(M.du, M.d, M.dl)
ctranspose(M::Tridiagonal) = conj(transpose(M))
function diag(M::Tridiagonal{T}, n::Integer=0) where T
if n == 0
return M.d
elseif n == -1
return M.dl
elseif n == 1
return M.du
elseif abs(n) < size(M,1)
return zeros(T,size(M,1)-abs(n))
else
throw(ArgumentError("$n-th diagonal of a $(size(M)) matrix doesn't exist!"))
end
end
function getindex(A::Tridiagonal{T}, i::Integer, j::Integer) where T
if !(1 <= i <= size(A,2) && 1 <= j <= size(A,2))
throw(BoundsError(A, (i,j)))
end
if i == j
return A.d[i]
elseif i == j + 1
return A.dl[j]
elseif i + 1 == j
return A.du[i]
else
return zero(T)
end
end
function setindex!(A::Tridiagonal, x, i::Integer, j::Integer)
@boundscheck checkbounds(A, i, j)
if i == j
@inbounds A.d[i] = x
elseif i - j == 1
@inbounds A.dl[j] = x
elseif j - i == 1
@inbounds A.du[i] = x
elseif !iszero(x)
throw(ArgumentError(string("cannot set entry ($i, $j) off ",
"the tridiagonal band to a nonzero value ($x)")))
end
return x
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
#tril and triu
istriu(M::Tridiagonal) = iszero(M.dl)
istril(M::Tridiagonal) = iszero(M.du)
function tril!(M::Tridiagonal, k::Integer=0)
n = length(M.d)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif k < -1
fill!(M.dl,0)
fill!(M.d,0)
fill!(M.du,0)
elseif k == -1
fill!(M.d,0)
fill!(M.du,0)
elseif k == 0
fill!(M.du,0)
end
return M
end
function triu!(M::Tridiagonal, k::Integer=0)
n = length(M.d)
if abs(k) > n
throw(ArgumentError("requested diagonal, $k, out of bounds in matrix of size ($n,$n)"))
elseif k > 1
fill!(M.dl,0)
fill!(M.d,0)
fill!(M.du,0)
elseif k == 1
fill!(M.dl,0)
fill!(M.d,0)
elseif k == 0
fill!(M.dl,0)
end
return M
end
###################
# 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, B::Number) = Tridiagonal(A.dl*B, A.d*B, A.du*B)
*(B::Number, A::Tridiagonal) = A*B
/(A::Tridiagonal, B::Number) = Tridiagonal(A.dl/B, A.d/B, A.du/B)
==(A::Tridiagonal, B::Tridiagonal) = (A.dl==B.dl) && (A.d==B.d) && (A.du==B.du)
==(A::Tridiagonal, B::SymTridiagonal) = (A.dl==A.du==B.ev) && (A.d==B.dv)
==(A::SymTridiagonal, B::Tridiagonal) = (B.dl==B.du==A.ev) && (B.d==A.dv)
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{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
if M.dl == M.du
return SymTridiagonal(convert(Vector{T},M.d), convert(Vector{T},M.dl))
else
throw(ArgumentError("Tridiagonal is not symmetric, cannot convert to SymTridiagonal"))
end
end