SymmetricFormats.jl defines the SymmetricPacked type which
concatenates the columns
of either the upper or lower triangle into a vector thereby using only a
little more than half the memory.
Other formats to efficiently store symmetric matrices, which are not (yet) implemented here, include recursive packed (RP; Andersen et al 2001, 2002), square block packed (SBP; Gustavson 2003), block packed hybrid (BPH; Andersen et al 2005, Gustavson et al 2007), and rectangular full packed (RFP; Gustavson et al 2009). See GenericLinearAlgebra.jl for an Julian implementation of RFP.
julia> using SymmetricFormats
julia> A = rand(5,5)
5×5 Matrix{Float64}:
0.364856 0.736451 0.704095 0.842738 0.207618
0.0797498 0.63549 0.861638 0.834968 0.0660772
0.152769 0.00271082 0.131074 0.727387 0.0995039
0.201612 0.0429332 0.682401 0.96513 0.616451
0.283603 0.818754 0.76548 0.284176 0.369353
julia> AP = SymmetricPacked(A) # defaults to :U -- the upper triangle
5×5 SymmetricPacked{Float64, Matrix{Float64}}:
0.364856 0.736451 0.704095 0.842738 0.207618
0.736451 0.63549 0.861638 0.834968 0.0660772
0.704095 0.861638 0.131074 0.727387 0.0995039
0.842738 0.834968 0.727387 0.96513 0.616451
0.207618 0.0660772 0.0995039 0.616451 0.369353
julia> AP.tri
15-element Vector{Float64}:
0.364855947737336
0.7364505269442262
0.63549015394607
0.7040945131597321
0.8616375176332387
0.13107440168239382
0.8427381656704127
0.8349681382276347
0.7273865854008484
0.9651301546768161
0.20761807799000453
0.066077207744069
0.09950392155913346
0.6164510864085191
0.3693525034730706
julia> packedsize(AP)
15
julia> Base.summarysize(A)
240
julia> Base.summarysize(AP)
184
Many BLAS routines input this packed storage format:
julia> using LinearAlgebra
julia> y = rand(5)
5-element Vector{Float64}:
0.47717272627749296
0.11563714323328722
0.7096777607286059
0.2632097580818683
0.28688074092456195
julia> x = rand(5)
5-element Vector{Float64}:
0.14292999002604434
0.8641487402596497
0.914860119103189
0.06250499506408325
0.14202984533447915
julia> α = 0.5
0.5
julia> β = 0.5
0.5
julia> BLAS.spmv!(AP.uplo, α, AP.tri, x, β, copy(y)) # directly call BLAS
5-element Vector{Float64}:
0.946017838465631
0.809954221322169
0.867204761074523
0.9592679333345454
0.27783932357851576
julia> mul!(copy(y), AP, x, α, β) # or use the equivalent julian interface
5-element Vector{Float64}:
0.946017838465631
0.809954221322169
0.867204761074523
0.9592679333345454
0.27783932357851576