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\begin{document}
%
\title{User Guide for LDL, a concise sparse Cholesky package}
\author{Timothy A. Davis\thanks{
Dept.~of Computer and Information Science and Engineering,
Univ.~of Florida, Gainesville, FL, USA.
email: davis@cise.ufl.edu.
http://www.cise.ufl.edu/$\sim$davis.
This work was supported by the National
Science Foundation, under grant CCR0203270.
Portions of the work were done while on sabbatical at Stanford University
and Lawrence Berkeley National Laboratory (with funding from Stanford
University and the SciDAC program).
}}
\date{Aug. 30, 2005}
\maketitle
%
\begin{abstract}
The {\tt LDL} software package is a set of short, concise routines for
factorizing symmetric positivedefinite sparse matrices, with some
applicability to symmetric indefinite matrices. Its primary purpose is
to illustrate much of the basic theory of sparse matrix algorithms in as
concise a code as possible, including an elegant method
of sparse symmetric factorization that computes the factorization rowbyrow
but stores it columnbycolumn. The entire symbolic and numeric factorization
consists of less than 50 lines of code. The package is written in C,
and includes a MATLAB interface.
\end{abstract}
%
%
\section{Overview}
%
{\tt LDL} is a set of short, concise routines that compute the $\m{LDL}\tr$
factorization of a sparse symmetric matrix $\m{A}$. Its primary purpose is
to illustrate much of the basic theory of sparse matrix algorithms in as
compact a code as possible, including an elegant method of
sparse symmetric factorization (related to \cite{Liu86c,Liu91}).
The lower triangular factor $\m{L}$ is computed rowbyrow, in contrast to the
conventional columnbycolumn method.
Although it does not achieve the same level of performance
as methods based on dense matrix kernels
(such as \cite{NgPeyton93,RothbergGupta91}),
its performance is competitive with columnbycolumn methods that do not
use dense kernels \cite{GeorgeLiu79, GeorgeLiu, GilbertMolerSchreiber}.
Section~\ref{Algorithm} gives a brief description of the algorithm
used in the symbolic and numeric factorization. A more detailed tutoriallevel
discussion may be found in \cite{Stewart03}. Details
of the concise implementation of this method are given in
Section~\ref{Implementation}. Sections~\ref{MATLAB}~and~\ref{C} give an
overview of how to use the package in MATLAB and in a standalone C program.
%
\section{Algorithm}
\label{Algorithm}
%
The underlying numerical algorithm is described below. The $k$th
step solves a lower triangular system of dimension $k1$ to compute the
$k$th row of $\m{L}$ and the $d_{kk}$ entry of the diagonal matrix $\m{D}$.
Colon notation is used for submatrices. For example,
$\m{L}_{k,1:k1}$ refers to the first $k1$ columns of
the $k$th row of $\m{L}$. Similarly, $\m{L}_{1:k1,1:k1}$ refers to
the leading $(k1)$by$(k1)$ submatrix of $\m{L}$.
%
\vspace{0.2in}
\begin{tabbing}
\hspace{2em} \= \hspace{2em} \= \hspace{2em} \= \\
{\bf Algorithm~1
($\m{LDL}\tr$ factorization of a $n$by$n$ symmetric matrix $\m{A}$)} \\
\> {\bf for} $k = 1$ {\bf to} $n$ \\
\>\> (step 1) Solve $\m{L}_{1:k1,1:k1}\m{y} = \m{A}_{1:k1,k}$ for $\m{y}$ \\
\>\> (step 2) $\m{L}_{k,1:k1} = (\m{D}_{1:k1,1:k1}^{1} \m{y})\tr$ \\
\>\> (step 3) $l_{kk} = 1$ \\
\>\> (step 4) $d_{kk} = a_{kk}  \m{L}_{k,1:k1}\m{y}$ \\
\> {\bf end for}
\end{tabbing}
%
The algorithm computes an $\m{LDL}\tr$ factorization without numerical pivoting.
It can thus factorize any symmetric positive definite matrix, and any
symmetric indefinite matrix whose leading minors are all wellconditioned.
When $\m{A}$ and $\m{L}$ are sparse, step 1 of Algorithm~1 requires a
triangular solve of the form $\m{Lx}=\m{b}$, where all three terms in
the equation are sparse. This is the most costly step of the Algorithm.
Steps 2 through 4 are fairly straightforward.
Let ${\cal X}$ and ${\cal B}$ refer to the set of indices of nonzero entries
in $\m{x}$ and $\m{b}$, respectively, in the lower triangular system
$\m{Lx}=\m{b}$. To compute $\m{x}$ efficiently
the nonzero pattern ${\cal X}$ must be found first.
In the general case when $\m{L}$ is arbitrary \cite{GilbertPeierls88},
the nonzero
pattern ${\cal X}$ is the set of nodes reachable via paths in the graph $G_L$
from all nodes in the set ${\cal B}$, and where the graph $G_L$ has
$n$ nodes and a directed edge $(j,i)$ if and only if $l_{ij}$ is nonzero.
To compute the numerical solution to $\m{Lx}=\m{b}$ by accessing the columns of
$\m{L}$ one at a time, ${\cal X}$ can be traversed
in any topological order of the subgraph of $G_L$ consisting of nodes in
${\cal X}$. That is, $x_j$ must be computed before $x_i$ if there is a path
from $j$ to $i$ in $G_L$. The natural order ($1, 2, \ldots, n$) is one such
ordering, but that requires a costly sort of ${\cal X}$.
With a graph traversal and topological sort, the solution of $\m{Lx}=\m{b}$
can be computed using Algorithm~2 below.
The computation of ${\cal X}$ and $\m{x}$ both take
time proportional to the floatingpoint operation count.
%
\vspace{0.2in}
\begin{tabbing}
\hspace{2em} \= \hspace{2em} \= \hspace{2em} \= \\
{\bf Algorithm~2
(Solve $\m{Lx}=\m{b}$, where $\m{L}$ is lower triangular with unit diagonal)} \\
\> ${\cal X} = \mbox{Reach}_{G_L} ({\cal B})$ \\
\> $\m{x} = \m{b}$ \\
\> {\bf for} $i \in {\cal X}$ in any topological order \\
\>\> $\m{x}_{i+1:n} = \m{x}_{i+1:n}  \m{L}_{i+1:n,i} x_i$ \\
\> {\bf end for}
\end{tabbing}
%
The general result also governs the pattern of $\m{y}$ in Algorithm~1.
However, in this case $\m{L}$ arises from a sparse Cholesky factorization,
and is governed by the elimination tree \cite{Liu90a}.
A general graph traversal is not required.
In the elimination tree, the parent of node $i$ is the smallest $j > i$
such that $l_{ji}$ is nonzero. Node $i$ has no parent if column $i$ of
$\m{L}$ is completely zero below the diagonal; $i$ is a root of the
elimination tree in this case. The nonzero pattern of $\m{x}$ is the
union of all the nodes on the paths from any node $i$ (where $b_i$ is nonzero) to the
root of the elimination tree \cite[Thm 2.4]{Liu86c}. It is referred to here as a tree,
but in general it can be a forest.
Rather than a general topological sort of the subgraph of $G_L$ consisting
nodes reachable from nodes in ${\cal B}$, a simpler
tree traversal can be used. First, select any nonzero entry $b_i$
and follow the path from $i$ to the root of tree.
Nodes along this path are marked and placed in a stack,
with $i$ at the top of the
stack and the root at the bottom.
Repeat for every other nonzero entry in $b_i$, in arbitrary order, but stop
just before reaching a marked node (the result can be empty if $i$ is already
in the stack). The stack now contains ${\cal X}$, a topological ordering of
the nonzero pattern of $\m{x}$, which can be used in Algorithm~2 to solve
$\m{Lx}=\m{b}$. The time to compute ${\cal X}$
using an elimination tree traversal is much faster than the general graph
traversal, taking time proportional to the size of ${\cal X}$ rather than the
number of floatingpoint operations required to compute $\m{x}$.
In the $k$th step of the factorization, the set ${\cal X}$ becomes the
nonzero pattern of row $k$ of $\m{L}$. This step requires the elimination
tree of $\m{L}_{1:k1,1:k1}$, and must construct the elimination tree of
$\m{L}_{1:k,1:k}$ for step $k+1$. Recall that the parent of $i$ in the
tree is the smallest $j$ such that $i < j$ and $l_{ji} \ne 0$.
Thus, if any node $i$ already has a parent $j$, then $j$ will remain the
parent of $i$ in the elimination trees of all other larger leading submatrices
of $\m{L}$, and in the elimination tree of $\m{L}$ itself.
If $l_{ki} \ne 0$ and $i$ does not have a parent in the elimination tree of
$\m{L}_{1:k1,1:k1}$, then the parent of $i$ is $k$
in the elimination tree of $\m{L}_{1:k,1:k}$.
Node $k$ becomes the parent of any node $i \in {\cal X}$ that does not yet
have a parent.
Since Algorithm~2 traverses $\m{L}$ in column order, $\m{L}$ is stored in a
conventional sparse column representation. Each column $j$ is stored as a list
of nonzero values and their corresponding row indices. When row $k$ is
computed, the new entries can be placed at the end of each list. As
a byproduct of computing $\m{L}$ one row at a time,
the columns of $\m{L}$ are computed in a sorted manner. This is a convenient
form of the output.
MATLAB requires the columns of its sparse matrices to be sorted, for example.
Sorted columns improve the speed of Algorithm~2, since the memory access
pattern is more regular. The conventional columnbycolumn algorithm
\cite{GeorgeLiu79,GeorgeLiu} does not produce columns of $\m{L}$ with
sorted row indices.
A simple symbolic preanalysis can be obtained by repeating the subtree traversals.
All that is required to compute the nonzero pattern of
the $k$th row of $\m{L}$ is the partially constructed elimination tree
and the nonzero pattern of the $k$th column of $\m{A}$. This is computed
in time proportional to the size of this set, using the elimination tree
traversal. Once constructed, the number of nonzeros in each column of
$\m{L}$ is incremented, for each entry in ${\cal X}$, and then ${\cal X}$
is discarded. The set ${\cal X}$ need not be constructed in topological
order, so no stack is required. The run time of the symbolic analysis
algorithm is thus proportional to the number of nonzeros in $\m{L}$.
This is more costly than the optimal algorithm \cite{GilbertNgPeyton94},
which takes time essentially proportional to the number of nonzeros in $\m{A}$.
The memory requirements are just the matrix $\m{A}$ and a few size$n$ integer
arrays. The result of the algorithm is the elimination tree, a count
of the number of nonzeros in each column of $\m{L}$, and
the cumulative sum of the column counts.
%
\section{Implementation}
\label{Implementation}
%
Because of its simplicity, the implementation of this algorithm leads to
a very short, concise code. The symbolic analysis routine {\tt ldl\_symbolic}
shown in Figure~\ref{ldlsymbolic}
consists of only 18 lines of executable C code.
This includes 5 lines of code to allow for a
sparsitypreserving ordering $\m{P}$ so that either $\m{A}$ or $\m{PAP}\tr$
can be analyzed, 3 lines of code to compute the cumulative sum of
the column counts, and one line of code to speed up a {\tt for} loop.
An additional line of code allows for a more general form of the input
sparse matrix $\m{A}$.
The {\tt n}by{\tt n} sparse matrix $\m{A}$ is provided in compressed column
form as an {\tt int} array {\tt Ap} of length {\tt n+1},
an {\tt int} array {\tt Ai} of length {\tt nz},
and a {\tt double} array {\tt Ax} also of length {\tt nz},
where {\tt nz} is the number of entries in the matrix.
The numerical values of entries in column $j$ are stored in
{\tt Ax[Ap[j]} $\ldots$ {\tt Ap[j+1]1]}
and the corresponding row indices are in
{\tt Ai[Ap[j]} $\ldots$ {\tt Ap[j+1]1]}.
With {\tt Ap[0] = 0}, the number of entries in the matrix is {\tt nz = Ap[n]}.
If no fillreducing ordering {\tt P} is provided,
only entries in the upper triangular part of $\m{A}$ are considered.
If {\tt P} is provided and row/column {\tt i} of the
matrix $\m{A}$ is the {\tt k}th row/column of $\m{PAP}\tr$, then {\tt P[k]=i}.
Only entries in the upper
triangular part of $\m{PAP}\tr$ are considered. These entries may be
in the lower triangular part of $\m{A}$, so to ensure that the correct matrix
is factorized, all entries of $\m{A}$ should be provided when using the
permutation input {\tt P}.
The outputs of {\tt ldl\_symbolic} are three size{\tt n} arrays:
{\tt Parent} holds the elimination tree,
{\tt Lnz} holds the counts of the number of entries in each column of
$\m{L}$, and
{\tt Lp} holds the cumulative sum of {\tt Lnz}.
The size{\tt n} array {\tt Flag} is used as workspace.
None of the output or workspace arrays need to be initialized.
\begin{figure}
\caption{{\tt ldl\_symbolic:} finding the elimination tree and column counts}
\label{ldlsymbolic}
{\scriptsize
\begin{verbatim}
void ldl_symbolic
(
int n, /* A and L are nbyn, where n >= 0 */
int Ap [ ], /* input of size n+1, not modified */
int Ai [ ], /* input of size nz=Ap[n], not modified */
int Lp [ ], /* output of size n+1, not defined on input */
int Parent [ ], /* output of size n, not defined on input */
int Lnz [ ], /* output of size n, not defined on input */
int Flag [ ], /* workspace of size n, not defn. on input or output */
int P [ ], /* optional input of size n */
int Pinv [ ] /* optional output of size n (used if P is not NULL) */
)
{
int i, k, p, kk, p2 ;
if (P)
{
/* If P is present then compute Pinv, the inverse of P */
for (k = 0 ; k < n ; k++)
{
Pinv [P [k]] = k ;
}
}
for (k = 0 ; k < n ; k++)
{
/* L(k,:) pattern: all nodes reachable in etree from nz in A(0:k1,k) */
Parent [k] = 1 ; /* parent of k is not yet known */
Flag [k] = k ; /* mark node k as visited */
Lnz [k] = 0 ; /* count of nonzeros in column k of L */
kk = (P) ? (P [k]) : (k) ; /* kth original, or permuted, column */
p2 = Ap [kk+1] ;
for (p = Ap [kk] ; p < p2 ; p++)
{
/* A (i,k) is nonzero (original or permuted A) */
i = (Pinv) ? (Pinv [Ai [p]]) : (Ai [p]) ;
if (i < k)
{
/* follow path from i to root of etree, stop at flagged node */
for ( ; Flag [i] != k ; i = Parent [i])
{
/* find parent of i if not yet determined */
if (Parent [i] == 1) Parent [i] = k ;
Lnz [i]++ ; /* L (k,i) is nonzero */
Flag [i] = k ; /* mark i as visited */
}
}
}
}
/* construct Lp index array from Lnz column counts */
Lp [0] = 0 ;
for (k = 0 ; k < n ; k++)
{
Lp [k+1] = Lp [k] + Lnz [k] ;
}
}
\end{verbatim}
}
\end{figure}
The {\tt ldl\_numeric} numeric factorization routine shown
in Figure~\ref{ldlnumeric} consists of only 31 lines of
executable code. It includes this same subtree traversal algorithm
as {\tt ldl\_symbolic},
except that each path is placed on a stack that holds
nonzero pattern of the $k$th row of $\m{L}$.
This traversal is followed by a sparse forward solve
using this pattern, and all of the nonzero entries in
the resulting $k$th row of $\m{L}$ are appended to their respective columns
in the data structure of $\m{L}$.
\begin{figure}
\caption{{\tt ldl\_numeric:} numeric factorization}
\label{ldlnumeric}
{\scriptsize
\begin{verbatim}
int ldl_numeric /* returns n if successful, k if D (k,k) is zero */
(
int n, /* A and L are nbyn, where n >= 0 */
int Ap [ ], /* input of size n+1, not modified */
int Ai [ ], /* input of size nz=Ap[n], not modified */
double Ax [ ], /* input of size nz=Ap[n], not modified */
int Lp [ ], /* input of size n+1, not modified */
int Parent [ ], /* input of size n, not modified */
int Lnz [ ], /* output of size n, not defn. on input */
int Li [ ], /* output of size lnz=Lp[n], not defined on input */
double Lx [ ], /* output of size lnz=Lp[n], not defined on input */
double D [ ], /* output of size n, not defined on input */
double Y [ ], /* workspace of size n, not defn. on input or output */
int Pattern [ ], /* workspace of size n, not defn. on input or output */
int Flag [ ], /* workspace of size n, not defn. on input or output */
int P [ ], /* optional input of size n */
int Pinv [ ] /* optional input of size n */
)
{
double yi, l_ki ;
int i, k, p, kk, p2, len, top ;
for (k = 0 ; k < n ; k++)
{
/* compute nonzero Pattern of kth row of L, in topological order */
Y [k] = 0.0 ; /* Y(0:k) is now all zero */
top = n ; /* stack for pattern is empty */
Flag [k] = k ; /* mark node k as visited */
Lnz [k] = 0 ; /* count of nonzeros in column k of L */
kk = (P) ? (P [k]) : (k) ; /* kth original, or permuted, column */
p2 = Ap [kk+1] ;
for (p = Ap [kk] ; p < p2 ; p++)
{
i = (Pinv) ? (Pinv [Ai [p]]) : (Ai [p]) ; /* get A(i,k) */
if (i <= k)
{
Y [i] += Ax [p] ; /* scatter A(i,k) into Y (sum duplicates) */
for (len = 0 ; Flag [i] != k ; i = Parent [i])
{
Pattern [len++] = i ; /* L(k,i) is nonzero */
Flag [i] = k ; /* mark i as visited */
}
while (len > 0) Pattern [top] = Pattern [len] ;
}
}
/* compute numerical values kth row of L (a sparse triangular solve) */
D [k] = Y [k] ; /* get D(k,k) and clear Y(k) */
Y [k] = 0.0 ;
for ( ; top < n ; top++)
{
i = Pattern [top] ; /* Pattern [top:n1] is pattern of L(:,k) */
yi = Y [i] ; /* get and clear Y(i) */
Y [i] = 0.0 ;
p2 = Lp [i] + Lnz [i] ;
for (p = Lp [i] ; p < p2 ; p++)
{
Y [Li [p]] = Lx [p] * yi ;
}
l_ki = yi / D [i] ; /* the nonzero entry L(k,i) */
D [k] = l_ki * yi ;
Li [p] = k ; /* store L(k,i) in column form of L */
Lx [p] = l_ki ;
Lnz [i]++ ; /* increment count of nonzeros in col i */
}
if (D [k] == 0.0) return (k) ; /* failure, D(k,k) is zero */
}
return (n) ; /* success, diagonal of D is all nonzero */
}
\end{verbatim}
}
\end{figure}
After the matrix is factorized, the {\tt ldl\_lsolve}, {\tt ldl\_dsolve},
and {\tt ldl\_ltsolve} routines shown in Figure~\ref{ldlsolve}
are provided to solve
$\m{Lx}=\m{b}$, $\m{Dx}=\m{b}$, and $\m{L}\tr\m{x}=\m{b}$, respectively.
Together, they solve $\m{Ax}=\m{b}$, and consist of only 10 lines of executable
code. If a fillreducing permutation is used,
{\tt ldl\_perm} and {\tt ldl\_permt} must be used to permute $\m{b}$ and
$\m{x}$ accordingly.
\begin{figure}
\caption{Solve routines}
\label{ldlsolve}
{\scriptsize
\begin{verbatim}
void ldl_lsolve
(
int n, /* L is nbyn, where n >= 0 */
double X [ ], /* size n. righthandside on input, soln. on output */
int Lp [ ], /* input of size n+1, not modified */
int Li [ ], /* input of size lnz=Lp[n], not modified */
double Lx [ ] /* input of size lnz=Lp[n], not modified */
)
{
int j, p, p2 ;
for (j = 0 ; j < n ; j++)
{
p2 = Lp [j+1] ;
for (p = Lp [j] ; p < p2 ; p++)
{
X [Li [p]] = Lx [p] * X [j] ;
}
}
}
void ldl_dsolve
(
int n, /* D is nbyn, where n >= 0 */
double X [ ], /* size n. righthandside on input, soln. on output */
double D [ ] /* input of size n, not modified */
)
{
int j ;
for (j = 0 ; j < n ; j++)
{
X [j] /= D [j] ;
}
}
void ldl_ltsolve
(
int n, /* L is nbyn, where n >= 0 */
double X [ ], /* size n. righthandside on input, soln. on output */
int Lp [ ], /* input of size n+1, not modified */
int Li [ ], /* input of size lnz=Lp[n], not modified */
double Lx [ ] /* input of size lnz=Lp[n], not modified */
)
{
int j, p, p2 ;
for (j = n1 ; j >= 0 ; j)
{
p2 = Lp [j+1] ;
for (p = Lp [j] ; p < p2 ; p++)
{
X [j] = Lx [p] * X [Li [p]] ;
}
}
}
\end{verbatim}
}
\end{figure}
In addition to appearing as a Collected Algorithm of the ACM \cite{Davis05},
{\tt LDL} is available at http://www.cise.ufl.edu/research/sparse.
%
\section{Using LDL in MATLAB}
\label{MATLAB}
%
The simplest way to use {\tt LDL} is within MATLAB. Once the {\tt LDL}
mexFunction is compiled and installed, the MATLAB statement
{\tt [L, D, Parent, fl] = ldl (A)} returns the sparse factorization
{\tt A = (L+I)*D*(L+I)'}, where {\tt L} is lower triangular, {\tt D} is a
diagonal matrix, and {\tt I} is the {\tt n}by{\tt n}
identity matrix ({\tt LDL} does not return the unit diagonal of {\tt L}).
The elimination tree is returned in {\tt Parent}.
If no zero on the diagonal of {\tt D} is encountered, {\tt fl} is the
floatingpoint operation count. Otherwise, {\tt D(fl,fl)} is the first
zero entry encountered. Let {\tt d=fl}. The function returns the
factorization of {\tt A (1:d,1:d)}, where rows {\tt d+1} to {\tt n} of {\tt L}
and {\tt D} are all zero. If a sparsity preserving permutation {\tt P} is
passed, {\tt [L, D, Parent, fl] = ldl (A,P)} operates on {\tt A(P,P)} without
forming it explicitly.
The statement {\tt x = ldl (A, [ ], b)} is roughly equivalent to
{\tt x = A}$\backslash${\tt b}, when {\tt A} is sparse, real, and symmetric.
The $\m{LDL}\tr$ factorization of {\tt A} is performed. If {\tt P} is
provided, {\tt x = ldl (A, P, b)} still performs
{\tt x = A}$\backslash${\tt b}, except that {\tt A(P,P)} is factorized
instead.
%
\section{Using LDL in a C program}
\label{C}
%
The Ccallable {\tt LDL} library consists of nine usercallable routines
and one include file.
\begin{itemize}
\item {\tt ldl\_symbolic}: given the nonzero pattern of a sparse symmetric
matrix $\m{A}$ and an optional permutation $\m{P}$, analyzes either
$\m{A}$ or $\m{PAP}\tr$, and returns the elimination tree, the
number of nonzeros in each column of $\m{L}$, and the {\tt Lp} array
for the sparse matrix data structure for $\m{L}$.
Duplicate entries are allowed in the columns of $\m{A}$, and the
row indices in each column need not be sorted.
Providing a sparsitypreserving ordering is critical for obtaining
good performance. A minimum degree ordering
(such as AMD \cite{AmestoyDavisDuff96,AmestoyDavisDuff03})
or a graphpartitioning based ordering are appropriate.
\item {\tt ldl\_numeric}: given {\tt Lp} and the elimination tree computed
by {\tt ldl\_symbolic}, and an optional permutation $\m{P}$,
returns the numerical factorization of $\m{A}$ or $\m{PAP}\tr$.
Duplicate entries are allowed in the columns of $\m{A}$
(any duplicate entries are summed), and the
row indices in each column need not be sorted.
The data structure for $\m{L}$ is the same as $\m{A}$, except that
no duplicates appear, and each column has sorted row indices.
\item {\tt ldl\_lsolve}: given the factor $\m{L}$ computed by
{\tt ldl\_numeric}, solves the linear system $\m{Lx}=\m{b}$, where
$\m{x}$ and $\m{b}$ are full $n$by1 vectors.
\item {\tt ldl\_dsolve}: given the factor $\m{D}$ computed by
{\tt ldl\_numeric}, solves the linear system $\m{Dx}=\m{b}$.
\item {\tt ldl\_ltsolve}: given the factor $\m{L}$ computed by
{\tt ldl\_numeric}, solves the linear system $\m{L}\tr\m{x}=\m{b}$.
\item {\tt ldl\_perm}: given a vector $\m{b}$ and a permutation $\m{P}$,
returns $\m{x}=\m{Pb}$.
\item {\tt ldl\_permt}: given a vector $\m{b}$ and a permutation $\m{P}$,
returns $\m{x}=\m{P}\tr\m{b}$.
\item {\tt ldl\_valid\_perm}: Except for checking if the diagonal of
$\m{D}$ is zero, none of the above routines check their inputs for errors.
This routine checks the validity of a permutation $\m{P}$.
\item {\tt ldl\_valid\_matrix}: checks if a matrix $\m{A}$ is valid as input
to {\tt ldl\_symbolic} and {\tt ldl\_numeric}.
\end{itemize}
Note that the primary input to the {\tt ldl\_symbolic} and
{\tt ldl\_numeric} is the sparse matrix $\m{A}$. It is provided in
columnoriented form, and only the upper triangular part is accessed.
This is slightly different than the primary output: the matrix $\m{L}$,
which is lower triangular in columnoriented form.
If you wish to factorize a symmetric matrix $\m{A}$ for which only the lower
triangular part is supplied, you would need to transpose $\m{A}$ before
passing it {\tt ldl\_symbolic} and {\tt ldl\_numeric}.
\begin{figure}
\caption{Example of use}
\label{ldlsimple}
{\scriptsize
\begin{verbatim}
#include <stdio.h>
#include "ldl.h"
#define N 10 /* A is 10by10 */
#define ANZ 19 /* # of nonzeros on diagonal and upper triangular part of A */
#define LNZ 13 /* # of nonzeros below the diagonal of L */
int main (int argc, int **argv)
{
/* only the upper triangular part of A is required */
int Ap [N+1] = {0, 1, 2, 3, 4, 6, 7, 9, 11, 15, ANZ},
Ai [ANZ] = {0, 1, 2, 3, 1,4, 5, 4,6, 4,7, 0,4,7,8, 1,4,6,9 } ;
double Ax [ANZ] = {1.7, 1., 1.5, 1.1, .02,2.6, 1.2, .16,1.3, .09,1.6,
.13,.52,.11,1.4, .01,.53,.56,3.1},
b [N] = {.287, .22, .45, .44, 2.486, .72, 1.55, 1.424, 1.621, 3.759};
double Lx [LNZ], D [N], Y [N] ;
int Li [LNZ], Lp [N+1], Parent [N], Lnz [N], Flag [N], Pattern [N], d, i ;
/* factorize A into LDL' (P and Pinv not used) */
ldl_symbolic (N, Ap, Ai, Lp, Parent, Lnz, Flag, NULL, NULL) ;
printf ("Nonzeros in L, excluding diagonal: %d\n", Lp [N]) ;
d = ldl_numeric (N, Ap, Ai, Ax, Lp, Parent, Lnz, Li, Lx, D, Y, Pattern,
Flag, NULL, NULL) ;
if (d == N)
{
/* solve Ax=b, overwriting b with the solution x */
ldl_lsolve (N, b, Lp, Li, Lx) ;
ldl_dsolve (N, b, D) ;
ldl_ltsolve (N, b, Lp, Li, Lx) ;
for (i = 0 ; i < N ; i++) printf ("x [%d] = %g\n", i, b [i]) ;
}
else
{
printf ("ldl_numeric failed, D (%d,%d) is zero\n", d, d) ;
}
return (0) ;
}
\end{verbatim}
}
\end{figure}
The program in Figure~\ref{ldlsimple}
illustrates the basic usage of the {\tt LDL} routines.
It analyzes and factorizes the sparse symmetric positivedefinite matrix
{\small
\[
\m{A} = \left[
\begin{array}{cccccccccc}
1.7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & .13 & 0 \\
0 & 1. & 0 & 0 & .02 & 0 & 0 & 0 & 0 & .01 \\
0 & 0 & 1.5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1.1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & .02 & 0 & 0 & 2.6 & 0 & .16 & .09 & .52 & .53 \\
0 & 0 & 0 & 0 & 0 & 1.2 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & .16 & 0 & 1.3 & 0 & 0 & .56 \\
0 & 0 & 0 & 0 & .09 & 0 & 0 & 1.6 & .11 & 0 \\
.13 & 0 & 0 & 0 & .52 & 0 & 0 & .11 & 1.4 & 0 \\
0 & .01 & 0 & 0 & .53 & 0 & .56 & 0 & 0 & 3.1 \\
\end{array}
\right]
\]
}
and then solves a system $\m{Ax}=\m{b}$ whose true solution is
$x_i = i/10$. Note that {\tt Li} and {\tt Lx} are statically allocated.
Normally they would be allocated after their size, {\tt Lp[n]},
is determined by {\tt ldl\_symbolic}.
More example programs are included with the {\tt LDL} package.
\section{Acknowledgments}
I would like to thank Pete Stewart for his comments on an earlier draft
of this software and its accompanying paper.
\newpage
\bibliographystyle{plain}
\bibliography{ldl}
\end{document}
