symildl
1.2
Incomplete LDL' factorizations of indefinite symmetric and skewsymmetric matrices.

symildl is a C++ package for solving and producing fast incomplete factorizations of symmetric indefinite or skewsymmetric matrices. Given an symmetric indefinite or skewsymmetric matrix , this package produces an incomplete factorization. Prior to factorization, symildl first scales the matrix to be equilibrated in the maxnorm [2], and then preorders the matrix using either the Reverse CuthillMcKee (RCM) algorithm or the Approximate Minimum Degree algorithm (AMD) [1]. To maintain stability, the user can use BunchKaufman or rook partial pivoting during the factorization process. The factorization produced is of the form
where is a permutation matrix, a scaling matrix, and and are the unit lower triangular and block diagonal factors respectively. The user can also optionally solve the given linear system, using symildl's incomplete factorization to precondition the builtin solver or using the full factorization as a direct solver.
This package is based on and extends an incomplete factorization approach proposed by Li and Saad [3] (which itself builds on Li, Saad, and Chow [4]).
[Latest release: 11/15/2015]
To begin using the package, first download the files above or compile the code hosted at https://github.com/inutard/matrixfactor. The GitHub repository contains the most up to date source code as well as an "experimental" branch that we release new features to. The package works under most Unix distributions as well as Cygwin under Windows. The default compiler used is g++
, simply type make
at the command line to compile the entire package. In addition to usage as a standalone program, the package also has a Matlab interface.
For using symildl as a software library, see the API here. symildl is a header only library, so one only needs to include source/solver.h
for everything to work.
The compiled program ldl_driver
takes in through the command line one parameter (the matrix) as well as several additional optional parameters.
The format of execution is:
The parameters can be given in any order, and will use a default value when not specified.
A description of each of these parameters can be accessed by typing
Generally speaking, the code operates in two phases: generating the preconditioner (factorization) and solving the linear system (solver). The factorization parameters can be used to specify how the LDL' preconditioner is built. If a right hand side is specified, the builtin solver attempts to solve the given linear system. The solver takes the preconditioner and uses it in either preconditioned MINRES or a direct solve (full factorization with BunchKaufman or rook pivoting).
For convenience, the parameters are listed below.
filename  The filename of the matrix to be loaded. Several test matrices exist in the test_matrices folder. All matrices loaded are required to be in matrix market (.mtx) form. 
fill  Controls memory usage. Each column of is guaranteed to have fewer than elements. Each column of has at most 2 elements. When this argument is not given, the default value for fill is 3.0 . 
tol  Controls agressiveness of dropping. In each column k, elements less than are dropped. The default value for tol is 0.001 . 
pp_tol  A parameter to aggressiveness of BunchKaufman pivoting (BKP). When pp_tol is 0, there is no partial pivoting. Values between 0 and 1 vary the number of pivots of BKP makes. When pp_tol is equal to 1, standard BKP is used. The pp_tol parameter is ignored if the pivot parameter is set to 'rook'. See the pivot parameters for more details. The default is 'rook'. 
reordering  Determines what sort of preordering will be used on the matrix. Choices are 'amd', 'rcm', and 'none'. The default is 'amd'. 
inplace  Indicates whether the factorization should be performed inplace, leading to roughly a 33% saving in memory. This memory comes out of extra information used in the solver. If the solver is needed, then inplace should not be used. y indicates yes, n indicates no. The default flag is n . 
pivot  Indicates the pivoting algorithm used. Choices are 'rook' and 'bunch'. If rook is used, the pp_tol parameter is ignored. The default is 'rook'. 
save  Indicates whether the output matrices should be saved. y indicates yes, n indicates no. The default flag is y . All matrices are saved in matrix market (.mtx) form. The matrices are saved into a folder named output_matrices . There are five saved files: outA.mtx, outL.mtx, outD.mtx, outS.mtx , and outP.mtx . outB.mtx is the matrix . The rest of the outputs should be clear from the description above. 
display  Indicates whether the output matrices should be displayed to the command line, used for debugging purposes. y indicates yes, n indicates no. The default flag is n . For this parameter to appear, symildl must be compiled with SYM_ILDL_DEBUG defined. 
solver  This chooses the solver for symildl if given a right hand side. The options are 'sqmr', 'minres', 'full', and 'none'. If the 'full' solver is chosen, the full factorization is produced and a straightforward direct solve is done. Setting this parameter to y overrides all other solver parameters as well as fill and tol (since we will no longer produce an incomplete factorization). The default solver is 'sqmr'. 
max_iters  Number of iterations that the builtin iterative solvers (SQMR or MINRES) can use. The default is 1 (i.e. iterative solver is not applied). The output solution is written to output_matrices\outsol.mtx . 
solver_tol  Relative tolerance for the builtin iterative solvers. When the iterate x satisfies Axb/b < solver_tol , the iterative solver is terminated. The default is 1e6 . 
rhs_file  The filename of the right hand size we want to solve. All right hand sides loaded are required to be in matrix market (.mtx) form. If no right hand sides are given, only the preconditioner is generated. 
pivot
, equil
, and reordering
parameters are best left to the default options.aug3dcqp
matrix stored in test_matrices/aug3dcqp.mtx
. Using the parameters described above, the execution of the program may go something like this: aug3dcqp.mtx
matrix (fill=3.0, tol=0.001
) from the test_matrices
folder and saves the outputs. The time it took to preorder and equilibrate the matrix (0.047s) as well as the actual factorization (0.109s) are also given.fill=3.0 tol=0.001 reordering=amd save=y
. In general, we may give ldl_driver
the arguments in any order, and omit any number of them (except the filename
argument).rhs_file
argument) and a maximum number of solver iterations. When no right hand side is specified (but a solver iteration is), symildl assumes a right hand side of all 1's for debugging purposes: If everything is compiled correctly, simply open MATLAB in the package directory. The startup.m
script adds all necessary paths to MATLAB upon initiation. The program can now be called by its function handle, ildl
.
ildl
takes in seven arguments, six of them being optional. A full description of the parameters can be displayed by typing
For convenience, the parameters are described below:
A  The matrix to be factored. 
fill  Controls memory usage. Each column is guaranteed to have fewer than elements. When this argument is not given, the default value for fill is 3.0 . 
tol  Controls aggressiveness of dropping. In each column k, elements less than are dropped. The default value for tol is 0.001 . 
reordering  Determines what sort of preordering will be used on the matrix. Choices are 'amd', 'rcm', and 'none'. The default is 'amd'. 
equil  Determines if matrix is to be equilibriated (in the max norm) before anything. This parameter must be one of 'bunch' or 'none'. Default: 'bunch' 
pivot_type  Chooses the pivoting scheme used during the factorization. This parameter must be one of 'rook' or 'bunch'. Tbe default is 'rook'. 
pp_tol  Threshold parameter for BunchKaufman pivoting (BKP). When pp_tol >= 1, full BKP is used. When pp_tol is 0, there is no partial pivoting. As pp_tol increases from 0 to 1, we smoothly switch from no pivoting to full BKP. Low values of pp_tol can be useful as an aggressive pivoting process may damage and permute any special structure present in the input matrix. The default value is 1.0. When rook pivoting is used, this parameter has no effect. 
As with the standalone executable, the function has five outputs: L, D, p, S,
and B:
A
generated is a special type of matrix called a saddlepoint matrix. These matrices are indefinite and arise often in optimzation problems. Note that A must be a MATLAB sparse matrix.ildl
with the parameters described above: ildl
will supply some timing information to the console when used. The reordering time is the time taken to equilibrate and preorder the matrix. The factorization time is the time it took to factor and pivot the matrix with partial pivoting.ildl
only one parameter: fill=3.0
, tol=0.001
, pivot_type='rook'
, pp_tol=1.0
, and reordering=amd
are used.To use the factorization as a preconditioner, we must apply the permutation and scaling matrices returned by ildl
. For convenience here is a complete example of how ildl
can be used with GMRES:
When the matrix is skewsymmetric, almost all documentation above still applies. The only difference is that the executable is skew_ldl_driver
instead of ldl_driver
. The skew functionality of symildl can be found in the experimental branch of https://github.com/inutard/matrixfactor.
Let's factor the m3dskew50
matrix stored in test_matrices/skew/m3dskew50.mtx
. As before, this is as simple as:
As in the symmetric case, we used the default values for the parameter we did not specify. A description of each of these parameters can be accessed by typing
Within MATLAB, using symildl is even easier. As in the symmetric case, the command ildl
can be used. Everything remains the same as the symmetric case, as ildl
automatically detects whether the input is symmetric or skewsymmetric.
Let's first generate a skewsymmetric matrix for testing:
Since B is a matrix of random values between 0 and 1, A is almost certainly nonsingular. Now we can call ildl
exactly as before:
Finally, helpful information on the parameters for ildl
can be found by typing:
symildl is open source, and we're always looking for new contributions! The entire codebase is freely accessible at https://github.com/inutard/matrixfactor. We also use the MIT Licence, which essentially allow free use of this software in any way you want (see here for more details). Simply send us a pull request on GitHub to contribute.
If you have found our code useful, please consider citing us! A technical report (pdf), a BibTeX entry, and a link to the code are available here:
SYMILDL: Incomplete LDL^{T} Factorization of Symmetric Indefinite and SkewSymmetric Matrices. Chen Greif, Shiwen He, Paul Liu. UBC Computer Science Technical Report, 2016. [pdf] [bib]