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Elementary operations for Tridiagonal and SymTridiagonal matrices #2611

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Mar 19, 2013
Merged

Elementary operations for Tridiagonal and SymTridiagonal matrices #2611

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f7436eb
Extends ctranspose() to Tridiagonal matrices
jiahao Mar 19, 2013
8988251
Extends transpose() and ctranspose() to Tridiagonal matrices
jiahao Mar 19, 2013
decaa52
Elementary transformations of Tridiagonal matrices
jiahao Mar 19, 2013
764f89e
Fleshed out elementary +,-,x,\ operations on Tridiagonal and SymTridi…
jiahao Mar 19, 2013
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52 changes: 45 additions & 7 deletions base/linalg/tridiag.jl
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Original file line number Diff line number Diff line change
@@ -1,5 +1,7 @@
#### Specialized matrix types ####

import Base.conj, Base.transpose, Base.ctranspose

## Hermitian tridiagonal matrices
type SymTridiagonal{T<:BlasFloat} <: AbstractMatrix{T}
dv::Vector{T} # diagonal
Expand All @@ -23,8 +25,6 @@ end

SymTridiagonal(A::AbstractMatrix) = SymTridiagonal(diag(A), diag(A,1))

copy(S::SymTridiagonal) = SymTridiagonal(S.dv,S.ev)

function full(S::SymTridiagonal)
M = diagm(S.dv)
for i in 1:length(S.ev)
Expand All @@ -45,9 +45,30 @@ end
size(m::SymTridiagonal) = (length(m.dv), length(m.dv))
size(m::SymTridiagonal, d::Integer) = d<1 ? error("dimension out of range") : (d<2 ? length(m.dv) : 1)

eig(m::SymTridiagonal) = LAPACK.stegr!('V', copy(m.dv), copy(m.ev))
#Elementary operations
copy(S::SymTridiagonal) = SymTridiagonal(copy(S.dv), copy(S.ev))
round(M::SymTridiagonal) = SymTridiagonal(round(M.dv), round(M.ev))
iround(M::SymTridiagonal) = SymTridiagonal(iround(M.dv), iround(M.ev))

conj(M::SymTridiagonal) = SymTridiagonal(conj(M.dv), conj(M.ev))
transpose(M::SymTridiagonal) = M #Identity operation
ctranspose(M::SymTridiagonal) = conj(M)

+(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)
#XXX Returns dense matrix but really should be banded
*(A::SymTridiagonal, B::SymTridiagonal) = full(A)*full(B)

## Solver
function \{T<:BlasFloat}(M::SymTridiagonal{T}, rhs::StridedVecOrMat{T})
if stride(rhs, 1) == 1
return LAPACK.gtsv!(copy(M.dv), copy(M.ev), copy(M.dv), copy(rhs))
end
solve(Tridiagonal(M), rhs) # use the Julia "fallback"
end

#Wrap LAPACK DSTEBZ to compute eigenvalues
#Wrap LAPACK DSTE{GR,BZ} to compute eigenvalues
eig(m::SymTridiagonal) = LAPACK.stegr!('V', copy(m.dv), copy(m.ev))
eigvals(m::SymTridiagonal, il::Int, iu::Int) = LAPACK.stebz!('I', 'E', 0.0, 0.0, il, iu, -1.0, copy(m.dv), copy(m.ev))[1]
eigvals(m::SymTridiagonal, vl::Float64, vu::Float64) = LAPACK.stebz!('V', 'E', vl, vu, 0, 0, -1.0, copy(m.dv), copy(m.ev))[1]
eigvals(m::SymTridiagonal) = LAPACK.stebz!('A', 'E', 0.0, 0.0, 0, 0, -1.0, copy(m.dv), copy(m.ev))[1]
Expand Down Expand Up @@ -83,8 +104,6 @@ function Tridiagonal{Tl<:Number, Td<:Number, Tu<:Number}(dl::Vector{Tl}, d::Vect
Tridiagonal(convert(Vector{R}, dl), convert(Vector{R}, d), convert(Vector{R}, du))
end

copy(A::Tridiagonal) = Tridiagonal(copy(A.dl), copy(A.d), copy(A.du))

size(M::Tridiagonal) = (length(M.d), length(M.d))
function show(io::IO, M::Tridiagonal)
println(io, summary(M), ":")
Expand Down Expand Up @@ -113,12 +132,31 @@ function similar(M::Tridiagonal, T, dims::Dims)
end
return Tridiagonal{T}(dims[1])
end
copy(M::Tridiagonal) = Tridiagonal(M.dl, M.d, M.du)

# Operations on Tridiagonal matrices
copy(A::Tridiagonal) = Tridiagonal(copy(A.dl), copy(A.d), copy(A.du))
round(M::Tridiagonal) = Tridiagonal(round(M.dl), round(M.d), round(M.du))
iround(M::Tridiagonal) = Tridiagonal(iround(M.dl), iround(M.d), iround(M.du))

conj(M::Tridiagonal) = Tridiagonal(conj(M.du), conj(M.d), conj(M.dl))
transpose(M::Tridiagonal) = Tridiagonal(M.du, M.d, M.dl)
ctranspose(M::Tridiagonal) = conj(transpose(M))

+(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)
#XXX Returns dense matrix but really should be banded
*(A::Tridiagonal, B::Tridiagonal) = full(A)*full(B)

# Elementary operations that mix Tridiagonal and SymTridiagonal matrices
Tridiagonal(A::SymTridiagonal) = Tridiagonal(A.dv, A.ev, A.dv)
+(A::Tridiagonal, B::SymTridiagonal) = Tridiagonal(A.dl+B.dv, A.d+B.ev, A.du+B.dv)
+(A::SymTridiagonal, B::Tridiagonal) = Tridiagonal(A.dv+B.dl, A.ev+B.d, A.dv+B.du)
-(A::Tridiagonal, B::SymTridiagonal) = Tridiagonal(A.dl-B.dv, A.d-B.ev, A.du-B.dv)
-(A::SymTridiagonal, B::Tridiagonal) = Tridiagonal(A.dv-B.dl, A.ev-B.d, A.dv-B.du)
#XXX Returns dense matrix but really should be banded
*(A::SymTridiagonal, B::Tridiagonal) = full(A)*full(B)
*(A::Tridiagonal, B::SymTridiagonal) = full(A)*full(B)

## Solvers

#### Tridiagonal matrix routines ####
Expand Down
5 changes: 5 additions & 0 deletions test/linalg.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -280,6 +280,11 @@ for elty in (Float32, Float64, Complex64, Complex128)
F[i+1,i] = dl[i]
end
@test full(T) == F
# elementary operations on tridiagonals
@test conj(T) == Tridiagonal(conj(dl), conj(d), conj(du))
@test transpose(T) == Tridiagonal(du, d, du)
@test ctranspose(T) == Tridiagonal(conj(du), conj(d), conj(dl))

# tridiagonal linear algebra
v = convert(Vector{elty}, v)
@test_approx_eq T*v F*v
Expand Down