Matrix Decompositions

UDT (QR)

Based on the (pivoted) QR decomposition, we introduce a $UDT$ factorization,

\[\begin{aligned} B = QR = UDT, \end{aligned}\]

where we have split $R$ into a diagonal piece $D$ and upper triangular piece $T$. Hence, $U = Q$ is unitary, $D = \textrm{diag}(R)$ is a real diagonal matrix, and $T$ is upper triangular.

To decompose a given matrix M the udt function is exported.

julia> M = rand(10,10);

julia> udt(M)
UDT{Float64,Float64,Array{Float64,2}}([-0.246588 0.12668 … 0.582208 0.206435; -0.373953 -0.300804 … 0.152994 0.0523203; … ; -0.214686 -0.403362 … -0.124248 -0.390502; -0.40412 -0.147009 … 0.1839 0.197964], [2.15087, 1.47129, 1.14085, 0.911765, 0.850504, 0.620149, 0.545588, 0.412213, 0.305983, 0.148787], [-0.597235 -1.0 … -0.678767 -0.59054; -0.385741 0.0 … -1.0 -0.361263; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0])

In our tests (see paper/), this decomposition turns out to be superior to SVD for DQMC.

SVD factorization

A singular value decomposition (SVD) is given by

\[\begin{aligned} B = USV^\dagger, \end{aligned}\]

where $U$ is unitary, $S$ is a real diagonal matrix, and $V^\dagger$ is unitary.

The package provides convenient access to several LAPACK algorithms for calculating singular value decompositions (SVDs):

gesdd, gesdd!: Divide and conquer
gesvd, gesvd!: Regular
gesvj, gesvj!: Jacobi (based on JacobiSVD.jl)

Furthermore, you can access a type-generic, pure Julia implementation,

genericsvd, genericsvd! (based on GenericSVD.jl)

However, to keep the dependencies of the package minimal, only the first two are available by default and loading of the Jacobi or type-generic SVD is opt-in. We provide convenience functions StableDQMC.addJacobiSVD() and StableDQMC.addGenericSVD() to facilitate this process. See below for a quick demonstration.

Automagically opt-in to JacobiSVD/GenericSVD

julia> using StableDQMC

julia> gesvj
ERROR: UndefVarError: gesvj not defined

julia> StableDQMC.addJacobiSVD()
  Updating registry at C:\Users\carsten\.julia\registries\General
  Updating git-repo https://github.com/JuliaRegistries/General.git
  Updating git-repo https://github.com/RalphAS/JacobiSVD.jl
 Resolving package versions...
  Updating C:\Users\carsten\Desktop\stabledqmctest\Project.toml
  [2ca068c6] + JacobiSVD v0.0.0 #master (https://github.com/RalphAS/JacobiSVD.jl)
  Updating C:\Users\carsten\Desktop\stabledqmctest\Manifest.toml
  [2ca068c6] + JacobiSVD v0.0.0 #master (https://github.com/RalphAS/JacobiSVD.jl)
┌ Warning: Package StableDQMC does not have JacobiSVD in its dependencies:
│ - If you have StableDQMC checked out for development and have
│   added JacobiSVD as a dependency but haven't updated your primary
│   environment's manifest file, try Pkg.resolve().
│ - Otherwise you may need to report an issue with StableDQMC
└ Loading JacobiSVD into StableDQMC from project dependency, future warnings for StableDQMC are suppressed.
[ Info: Recompiling stale cache file C:\Users\carsten.julia\compiled\v1.1\JacobiSVD\Frhox.ji for JacobiSVD [2ca068c6-2156-5cf0-8317-c67edf277a2c]

julia> gesvj # gesvj and gesvj! are available now
gesvj (generic function with 1 method)

Manual opt-in

Provided that you have JacobiSVD.jl (or GenericSVD.jl) installed, you can get the LAPACK access functions gesvj, gesvj! (or genericsvd, genericsvd!) simply by import JacobiSVD (import GenericSVD).

julia> using StableDQMC

julia> gesvj
ERROR: UndefVarError: gesvj not defined

julia> import JacobiSVD # using might lead to name conflicts

julia> gesvj # gesvj and gesvj! are available now
gesvj (generic function with 1 method)