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Provide inv and det of Tridiagonal and SymTridiagonal matrices #5358

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Jan 12, 2014
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Provide inv and det of Tridiagonal and SymTridiagonal matrices #5358

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417389c
Relax eigenvector test slightly
jiahao Jan 11, 2014
a0fd83b
Provide inv and det of Tridiagonal and SymTridiagonal matrices
jiahao Jan 11, 2014
964731a
Update NEWS.md
jiahao Jan 11, 2014
e5c0f69
Consolidate det and inv for Tridiagonal and SymTridiagonal
jiahao Jan 12, 2014
5de5fb0
Restore tridiagonal LU factorization for Rational
jiahao Jan 12, 2014
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2 changes: 2 additions & 0 deletions NEWS.md
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Original file line number Diff line number Diff line change
Expand Up @@ -88,6 +88,7 @@ Library improvements
matrices of generic types ([#5263])
- new algorithms for linear solvers and eigensystems of `Bidiagonal`
matrices of generic types ([#5277])
- specialized `inv` and `det` for `Tridiagonal` and `SymTridiagonal` ([#5358])
- specialized methods `transpose`, `ctranspose`, `istril`, `istriu` for
`Triangular` ([#5255]) and `Bidiagonal` ([#5277])
- new LAPACK wrappers
Expand Down Expand Up @@ -140,6 +141,7 @@ Deprecated or removed
[#2345]: https://github.com/JuliaLang/julia/issues/2345
[#5330]: https://github.com/JuliaLang/julia/issues/5330
[#4882]: https://github.com/JuliaLang/julia/issues/4882
[#5358]: https://github.com/JuliaLang/julia/pull/5358

Julia v0.2.0 Release Notes
==========================
Expand Down
73 changes: 71 additions & 2 deletions base/linalg/tridiag.jl
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Expand Up @@ -89,6 +89,68 @@ eigmin(m::SymTridiagonal) = eigvals(m, 1, 1)[1]
eigvecs(m::SymTridiagonal) = eig(m)[2]
eigvecs{Eigenvalue<:Real}(m::SymTridiagonal, eigvals::Vector{Eigenvalue}) = LAPACK.stein!(m.dv, m.ev, eigvals)

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

#Needed for inv_usmani()
type ZeroOffsetVector
data::Vector
end
getindex (a::ZeroOffsetVector, i) = a.data[i+1]
setindex!(a::ZeroOffsetVector, x, i) = a.data[i+1]=x

#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.
#Ref:
# 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{T}(a::Vector{T}, b::Vector{T}, c::Vector{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 = Array(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{T}(a::Vector{T}, b::Vector{T}, c::Vector{T})
n = length(b)
θa = one(T)
n==0 && return θa
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)

## Tridiagonal matrices ##
type Tridiagonal{T} <: AbstractMatrix{T}
dl::Vector{T} # sub-diagonal
Expand Down Expand Up @@ -175,6 +237,10 @@ function diag{T}(M::Tridiagonal{T}, n::Integer=0)
end
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)
Expand All @@ -185,6 +251,9 @@ end
==(A::Tridiagonal, B::SymTridiagonal) = (A.dl==A.du==B.ev) && (A.d==B.dv)
==(A::SymTridiagonal, B::SymTridiagonal) = B==A

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

# Elementary operations that mix Tridiagonal and SymTridiagonal matrices
convert(::Type{Tridiagonal}, A::SymTridiagonal) = Tridiagonal(A.ev, A.dv, A.ev)
+(A::Tridiagonal, B::SymTridiagonal) = Tridiagonal(A.dl+B.ev, A.d+B.dv, A.du+B.ev)
Expand Down Expand Up @@ -351,7 +420,7 @@ type LUTridiagonal{T} <: Factorization{T}
# end
end
lufact!{T<:BlasFloat}(A::Tridiagonal{T}) = LUTridiagonal{T}(LAPACK.gttrf!(A.dl,A.d,A.du)...)
lufact!(A::Tridiagonal) = lufact!(float(A))
lufact!{T<:Integer}(A::Tridiagonal{T}) = lufact!(float(A))
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This excludes rational matrices, but I think we'll have LU implementations in julia soon, so maybe it doesn't matter.

lufact(A::Tridiagonal) = lufact!(copy(A))
factorize!(A::Tridiagonal) = lufact!(A)
#show(io, lu::LUTridiagonal) = print(io, "LU decomposition of ", summary(lu.lu))
Expand All @@ -361,7 +430,7 @@ function det{T}(lu::LUTridiagonal{T})
prod(lu.d) * (bool(sum(lu.ipiv .!= 1:n) % 2) ? -one(T) : one(T))
end

det(A::Tridiagonal) = det(lufact(A))
det{T<:BlasFloat}(A::Tridiagonal{T}) = det(lufact(A))

A_ldiv_B!{T<:BlasFloat}(lu::LUTridiagonal{T}, B::StridedVecOrMat{T}) =
LAPACK.gttrs!('N', lu.dl, lu.d, lu.du, lu.du2, lu.ipiv, B)
Expand Down
36 changes: 35 additions & 1 deletion test/linalg.jl
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Expand Up @@ -553,7 +553,7 @@ end
function test_approx_eq_vecs{S<:Real,T<:Real}(a::StridedVecOrMat{S}, b::StridedVecOrMat{T}, error=nothing)
n = size(a, 1)
@test n==size(b,1) && size(a,2)==size(b,2)
if error==nothing error=n^2*(eps(S)+eps(T)) end
error==nothing && (error=n^3*(eps(S)+eps(T)))
for i=1:n
ev1, ev2 = a[:,i], b[:,i]
deviation = min(abs(norm(ev1-ev2)),abs(norm(ev1+ev2)))
Expand Down Expand Up @@ -603,7 +603,41 @@ for relty in (Float16, Float32, Float64, BigFloat), elty in (relty, Complex{relt
end
end

#Tridiagonal matrices
for relty in (Float16, Float32, Float64), elty in (relty, Complex{relty})
a = convert(Vector{elty}, randn(n-1))
b = convert(Vector{elty}, randn(n))
c = convert(Vector{elty}, randn(n-1))
if elty <: Complex
a += im*convert(Vector{elty}, randn(n-1))
b += im*convert(Vector{elty}, randn(n))
c += im*convert(Vector{elty}, randn(n-1))
end

A=Tridiagonal(a, b, c)
fA=(elty<:Complex?complex128:float64)(full(A))
for func in (det, inv)
@test_approx_eq_eps func(A) func(fA) n^2*sqrt(eps(relty))
end
end

#SymTridiagonal (symmetric tridiagonal) matrices
for relty in (Float16, Float32, Float64), elty in (relty, )#XXX Complex{relty}) doesn't work
a = convert(Vector{elty}, randn(n))
b = convert(Vector{elty}, randn(n-1))
if elty <: Complex
relty==Float16 && continue
a += im*convert(Vector{elty}, randn(n))
b += im*convert(Vector{elty}, randn(n-1))
end

A=SymTridiagonal(a, b)
fA=(elty<:Complex?complex128:float64)(full(A))
for func in (det, inv)
@test_approx_eq_eps func(A) func(fA) n^2*sqrt(eps(relty))
end
end

Ainit = randn(n)
Binit = randn(n-1)
for elty in (Float32, Float64)
Expand Down