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Use a stable inverse for Tridiagonal and SymTridiagonal #29667

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Merged
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Oct 17, 2018
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Use a stable inverse for Tridiagonal and SymTridiagonal #29667

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7b7b149
Use a stable inverse for Tridiagonal and SymTridiagonal
andreasnoack Oct 16, 2018
cb9e179
Fix ldlt in the complex case
andreasnoack Oct 16, 2018
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4 changes: 2 additions & 2 deletions stdlib/LinearAlgebra/src/ldlt.jl
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Expand Up @@ -48,13 +48,13 @@ julia> S
⋅ 0.545455 3.90909
```
"""
function ldlt!(S::SymTridiagonal{T,V}) where {T<:Real,V}
function ldlt!(S::SymTridiagonal{T,V}) where {T,V}
n = size(S,1)
d = S.dv
e = S.ev
@inbounds @simd for i = 1:n-1
e[i] /= d[i]
d[i+1] -= abs2(e[i])*d[i]
d[i+1] -= e[i]^2*d[i]
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Was abs2 incorrect here?

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Yes, but the previous version was also restricted to Real so it didn't matter. Had the type been HermitianTridiagonal then abs2 would have been correct.

end
return LDLt{T,SymTridiagonal{T,V}}(S)
end
Expand Down
39 changes: 3 additions & 36 deletions stdlib/LinearAlgebra/src/tridiag.jl
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Expand Up @@ -320,44 +320,13 @@ function Base.replace_in_print_matrix(A::SymTridiagonal, i::Integer, j::Integer,
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
# 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 inv_usmani(a::V, b::V, c::V) where {T,V<:AbstractVector{T}}
@assert !has_offset_axes(a, b, c)
n = length(b)
θ = zeros(T, n+1) #principal minors of A
θ[1] = 1
n>=1 && (θ[2] = b[1])
for i=2:n
θ[i+1] = b[i]*θ[i]-a[i-1]*c[i-1]*θ[i-1]
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}(undef, 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]*φ[j+1]/θ[n+1]
elseif i==j
α[i,i]= θ[i]*φ[i+1]/θ[n+1]
else #i>j
α[i,j]=(sign)(prod(a[j:i-1]))*θ[j]*φ[i+1]/θ[n+1]
end
end
α
end

#Implements the determinant using principal minors
#Inputs and reference are as above for inv_usmani()
function det_usmani(a::V, b::V, c::V) where {T,V<:AbstractVector{T}}
@assert !has_offset_axes(a, b, c)
n = length(b)
Expand All @@ -366,13 +335,12 @@ function det_usmani(a::V, b::V, c::V) where {T,V<:AbstractVector{T}}
return θa
end
θb = b[1]
for i=2:n
θb, θa = b[i]*θb-a[i-1]*c[i-1]*θa, θb
for i in 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
Expand Down Expand Up @@ -651,7 +619,6 @@ end
==(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)

AbstractMatrix{T}(M::Tridiagonal) where {T} = Tridiagonal{T}(M)
Expand Down
20 changes: 20 additions & 0 deletions stdlib/LinearAlgebra/test/tridiag.jl
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Expand Up @@ -391,4 +391,24 @@ end
"LinearAlgebra.LU{Float64,LinearAlgebra.Tridiagonal{Float64,SparseArrays.SparseVector")
end

@testset "Issue 29630" begin
function central_difference_discretization(N; dfunc = x -> 12x^2 - 2N^2,
dufunc = x -> N^2 + 4N*x,
dlfunc = x -> N^2 - 4N*x,
bfunc = x -> 114ℯ^-x * (1 + 3x),
b0 = 0, bf = 57/ℯ,
x0 = 0, xf = 1)
h = 1/N
d, du, dl, b = map(dfunc, (x0+h):h:(xf-h)), map(dufunc, (x0+h):h:(xf-2h)),
map(dlfunc, (x0+2h):h:(xf-h)), map(bfunc, (x0+h):h:(xf-h))
b[1] -= dlfunc(x0)*b0 # subtract the boundary term
b[end] -= dufunc(xf)*bf # subtract the boundary term
Tridiagonal(dl, d, du), b
end

A90, b90 = central_difference_discretization(90)

@test A90\b90 ≈ inv(A90)*b90
end

end # module TestTridiagonal