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LLDL

Overview

LLDL is a modification of Lin and Moré's limited-memory Cholesky factorization code-named ICFS for symmetric positive definite matrices. LLDL implements a similar limited-memory scheme for symmetric indefinite matrices that possess a LDL^T factorization, i.e., with D diagonal. Symmetric quasi-definite matrices fall into this category.

LLDL is applicable to symmetric indefinite matrices that are not quasi definite, or more generally do not admit a LDL^T factorization. In this case, it computes an incomplete LDL^T factorization of a nearby matrix.

The main idea is that if L and D are the exact factors of A, the preconditioned matrix (L |D| L^T)^-1 A has only two eigenvalues: +1 and -1. The hope is that incomplete factors will strike a good balance between computational cost and eigenvalue clustering.

It remains possible to use MINRES or SYMMLQ on the symmetrically-preconditioned system

|D|^-1/2 L^-1 A L^-T |D|^-1/2 x = |D|^-1/2 L^-1 b.

Installing

Just do it ;)

./install

You can use non-default compilers and compiler flags by passing options to the install script. For instance

CC=clang FC=ifort CFLAGS='-g' FFLAGS='-FI' ./install

Here is the complete list:

$ ./install --help
install [--help] [--skip-matlab-check]
The following environment variables may be set to influence
the behavior of the install procedure:
  CC           C compiler executable
  FC           Fortran compiler executable
  CFLAGS       C compiler flags
  FFLAGS       Fortran compiler flags
  MATLABDIR    Path to Matlab's bin and extern subdirs

Matlab Interface

You can specify the location of Matlab's bin and extern subdirectories using the MATLABDIR variable:

MATLABDIR=/Applications/Matlab/MATLAB_R2012a.app ./install

The original Matlab interface has been fixed and modernized. The Mathworks only support version 4.3 gfortran on OSX and Linux. On Linux, gcc-4.3 should be found in your package manager. On OSX, I recommend using Homebrew. Once Homebrew is installed, gcc-4.3, including the Fortran compiler, may be installed using

brew tap homebrew/versions
brew install gcc43 --enable-fortran --enable-profiled-build

The compiler executables installed by the above commands are those used by default in LLDL's install script. If you know that your compilers will produce valid MEX files (it is the case for gcc-4.2 and gfortran-4.2 on OSX 10.6.8), then:

./install --skip-matlab-check

Here is an example Matlab session:

n = 6; m = 4; E = rand(m,n);
A = [(n+m+1)*eye(n) E' ; E -(n+m+1)*eye(m)];
Adiag = full(diag(A)); lA = sparse(tril(A,-1)); p=1;
[L, D, shift] = lldl(lA, Adiag, p);
L = L + speye(size(L));  % Diagonal was left out of L.

In practice it is more programmatically convenient to use the incomplete factors L and D as implicitly defining the preconditioner by bundling them into an abstract SPOT operator:

LLDL = opLLDL(A, p);
x = LLDL * y;  % Solves L |D| L' x = y;

Python Interface

A Python interface is included as part of NLPy.

Julia Interface

A Julia interface is included in the julia folder. You must have Julia installed. Due to a bug in version 0.2.1, you should install a more recent version. If you use Homebrew

brew install gcc
brew tap staticfloat/julia
brew tap homebrew/versions
brew install julia --HEAD  # Add --64bit if desired.

Here is an example Julia session:

n = 10; m = 6; E = sprand(m, n, .2); p = 1;
A = [(n+m+1)*speye(n) E' ; E  -(n+m+1)*speye(m)];
(L, d, shift) = lldl(K, p);     # Returns L as a matrix, not including unit diagonal
(LLDL, shift) = lldl_op(K, p);  # Returns a linear operator
x = LLDL * y;                   # Solves L |D| L' x = y;

Run lldl_test.jl for another example.

Trouble / Questions / Bugs

Search existing issues. If that does not answer your question, please open a new issue.

Relics

The test problems in the tprobs folder and the preconditioned CG implementation are relics of ICFS. The main driver is not applicable, except if the coefficient matrix is definite.

To Do List

This is a list of improvements that could be added to LLDL to improve its efficiency, particularly with Matlab.

• Drop-off threshold.
• Implicit permutation: instead of having to explicity apply a reordering, the user supplies a permutation vector. The incomplete factorization proceeds in the order given by this permutation vector.

References

• D. Orban. Limited-Memory LDL^T Factorization of Symmetric Quasi-Definite Matrices with Application to Constrained Optimization. Cahier du GERAD G-2013-87. GERAD, Montreal, Canada.
• C.-J. Lin and J. J. Moré. Incomplete Cholesky factorizations with limited memory. SIAM Journal on Scientific Computing, 21(1):24--45, 1999.

Notes

The original ICFS README is in README.orig.