ldl.c
/* ========================================================================== */
/* === ldl.c: sparse LDL' factorization and solve package =================== */
/* ========================================================================== */
/* LDL: a simple set of routines for sparse LDL' factorization. These routines
* are not terrifically fast (they do not use dense matrix kernels), but the
* code is very short. The purpose is to illustrate the algorithms in a very
* concise manner, primarily for educational purposes. Although the code is
* very concise, this package is slightly faster than the built-in sparse
* Cholesky factorization in MATLAB 6.5 (chol), when using the same input
* permutation.
*
* The routines compute the LDL' factorization of a real sparse symmetric
* matrix A (or PAP' if a permutation P is supplied), and solve upper
* and lower triangular systems with the resulting L and D factors. If A is
* positive definite then the factorization will be accurate. A can be
* indefinite (with negative values on the diagonal D), but in this case no
* guarantee of accuracy is provided, since no numeric pivoting is performed.
*
* The n-by-n sparse matrix A is in compressed-column form. The nonzero values
* in column j are stored in Ax [Ap [j] ... Ap [j+1]-1], with corresponding row
* indices in Ai [Ap [j] ... Ap [j+1]-1]. Ap [0] = 0 is required, and thus
* nz = Ap [n] is the number of nonzeros in A. Ap is an int array of size n+1.
* The int array Ai and the double array Ax are of size nz. This data structure
* is identical to the one used by MATLAB, except for the following
* generalizations. The row indices in each column of A need not be in any
* particular order, although they must be in the range 0 to n-1. Duplicate
* entries can be present; any duplicates are summed. That is, if row index i
* appears twice in a column j, then the value of A (i,j) is the sum of the two
* entries. The data structure used here for the input matrix A is more
* flexible than MATLAB's, which requires sorted columns with no duplicate
* entries.
*
* Only the diagonal and upper triangular part of A (or PAP' if a permutation
* P is provided) is accessed. The lower triangular parts of the matrix A or
* PAP' can be present, but they are ignored.
*
* The optional input permutation is provided as an array P of length n. If
* P [k] = j, the row and column j of A is the kth row and column of PAP'.
* If P is present then the factorization is LDL' = PAP' or L*D*L' = A(P,P) in
* 0-based MATLAB notation. If P is not present (a null pointer) then no
* permutation is performed, and the factorization is LDL' = A.
*
* The lower triangular matrix L is stored in the same compressed-column
* form (an int Lp array of size n+1, an int Li array of size Lp [n], and a
* double array Lx of the same size as Li). It has a unit diagonal, which is
* not stored. The row indices in each column of L are always returned in
* ascending order, with no duplicate entries. This format is compatible with
* MATLAB, except that it would be more convenient for MATLAB to include the
* unit diagonal of L. Doing so here would add additional complexity to the
* code, and is thus omitted in the interest of keeping this code short and
* readable.
*
* The elimination tree is held in the Parent [0..n-1] array. It is normally
* not required by the user, but it is required by ldl_numeric. The diagonal
* matrix D is held as an array D [0..n-1] of size n.
*
* --------------------
* C-callable routines:
* --------------------
*
* ldl_symbolic: Given the pattern of A, computes the Lp and Parent arrays
* required by ldl_numeric. Takes time proportional to the number of
* nonzeros in L. Computes the inverse Pinv of P if P is provided.
* Also returns Lnz, the count of nonzeros in each column of L below
* the diagonal (this is not required by ldl_numeric).
* ldl_numeric: Given the pattern and numerical values of A, the Lp array,
* the Parent array, and P and Pinv if applicable, computes the
* pattern and numerical values of L and D.
* ldl_lsolve: Solves Lx=b for a dense vector b.
* ldl_dsolve: Solves Dx=b for a dense vector b.
* ldl_ltsolve: Solves L'x=b for a dense vector b.
* ldl_perm: Computes x=Pb for a dense vector b.
* ldl_permt: Computes x=P'b for a dense vector b.
* ldl_valid_perm: checks the validity of a permutation vector
* ldl_valid_matrix: checks the validity of the sparse matrix A
*
* ----------------------------
* Limitations of this package:
* ----------------------------
*
* In the interest of keeping this code simple and readable, ldl_symbolic and
* ldl_numeric assume their inputs are valid. You can check your own inputs
* prior to calling these routines with the ldl_valid_perm and ldl_valid_matrix
* routines. Except for the two ldl_valid_* routines, no routine checks to see
* if the array arguments are present (non-NULL). Like all C routines, no
* routine can determine if the arrays are long enough and don't overlap.
*
* The ldl_numeric does check the numerical factorization, however. It returns
* n if the factorization is successful. If D (k,k) is zero, then k is
* returned, and L is only partially computed.
*
* No pivoting to control fill-in is performed, which is often critical for
* obtaining good performance. I recommend that you compute the permutation P
* using AMD or SYMAMD (approximate minimum degree ordering routines), or an
* appropriate graph-partitioning based ordering. See the ldldemo.m routine for
* an example in MATLAB, and the ldlmain.c stand-alone C program for examples of
* how to find P. Routines for manipulating compressed-column matrices are
* available in UMFPACK. AMD, SYMAMD, UMFPACK, and this LDL package are all
* available at http://www.cise.ufl.edu/research/sparse.
*
* -------------------------
* Possible simplifications:
* -------------------------
*
* These routines could be made even simpler with a few additional assumptions.
* If no input permutation were performed, the caller would have to permute the
* matrix first, but the computation of Pinv, and the use of P and Pinv could be
* removed. If only the diagonal and upper triangular part of A or PAP' are
* present, then the tests in the "if (i < k)" statement in ldl_symbolic and
* "if (i <= k)" in ldl_numeric, are always true, and could be removed (i can
* equal k in ldl_symbolic, but then the body of the if statement would
* correctly do no work since Flag [k] == k). If we could assume that no
* duplicate entries are present, then the statement Y [i] += Ax [p] could be
* replaced with Y [i] = Ax [p] in ldl_numeric.
*
* --------------------------
* Description of the method:
* --------------------------
*
* LDL computes the symbolic factorization by finding the pattern of L one row
* at a time. It does this based on the following theory. Consider a sparse
* system Lx=b, where L, x, and b, are all sparse, and where L comes from a
* Cholesky (or LDL') factorization. The elimination tree (etree) of L is
* defined as follows. The parent of node j is the smallest k > j such that
* L (k,j) is nonzero. Node j has no parent if column j of L is completely zero
* below the diagonal (j is a root of the etree in this case). The nonzero
* pattern of x is the union of the paths from each node i to the root, for
* each nonzero b (i). To compute the numerical solution to Lx=b, we can
* traverse the columns of L corresponding to nonzero values of x. This
* traversal does not need to be done in the order 0 to n-1. It can be done in
* any "topological" order, such that x (i) is computed before x (j) if i is a
* descendant of j in the elimination tree.
*
* The row-form of the LDL' factorization is shown in the MATLAB function
* ldlrow.m in this LDL package. Note that row k of L is found via a sparse
* triangular solve of L (1:k-1, 1:k-1) \ A (1:k-1, k), to use 1-based MATLAB
* notation. Thus, we can start with the nonzero pattern of the kth column of
* A (above the diagonal), follow the paths up to the root of the etree of the
* (k-1)-by-(k-1) leading submatrix of L, and obtain the pattern of the kth row
* of L. Note that we only need the leading (k-1)-by-(k-1) submatrix of L to
* do this. The elimination tree can be constructed as we go.
*
* The symbolic factorization does the same thing, except that it discards the
* pattern of L as it is computed. It simply counts the number of nonzeros in
* each column of L and then constructs the Lp index array when it's done. The
* symbolic factorization does not need to do this in topological order.
* Compare ldl_symbolic with the first part of ldl_numeric, and note that the
* while (len > 0) loop is not present in ldl_symbolic.
*
* LDL Version 1.0 (Dec. 31, 2003), Copyright (c) 2003 by Timothy A Davis,
* University of Florida. All Rights Reserved. Developed while on sabbatical
* at Stanford University and Lawrence Berkeley National Laboratory. Refer to
* the README file for the License. Available at
* http://www.cise.ufl.edu/research/sparse.
*/
#include "ldl.h"
/* ========================================================================== */
/* === ldl_symbolic ========================================================= */
/* ========================================================================== */
/* The input to this routine is a sparse matrix A, stored in column form, and
* an optional permutation P. The output is the elimination tree
* and the number of nonzeros in each column of L. Parent [i] = k if k is the
* parent of i in the tree. The Parent array is required by ldl_numeric.
* Lnz [k] gives the number of nonzeros in the kth column of L, excluding the
* diagonal.
*
* One workspace vector (Flag) of size n is required.
*
* If P is NULL, then it is ignored. The factorization will be LDL' = A.
* Pinv is not computed. In this case, neither P nor Pinv are required by
* ldl_numeric.
*
* If P is not NULL, then it is assumed to be a valid permutation. If
* row and column j of A is the kth pivot, the P [k] = j. The factorization
* will be LDL' = PAP', or A (p,p) in MATLAB notation. The inverse permutation
* Pinv is computed, where Pinv [j] = k if P [k] = j. In this case, both P
* and Pinv are required as inputs to ldl_numeric.
*
* The floating-point operation count of the subsequent call to ldl_numeric
* is not returned, but could be computed after ldl_symbolic is done. It is
* the sum of (Lnz [k]) * (Lnz [k] + 2) for k = 0 to n-1.
*/
void ldl_symbolic
(
int n, /* A and L are n-by-n, 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:k-1,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] ;
}
}
/* ========================================================================== */
/* === ldl_numeric ========================================================== */
/* ========================================================================== */
/* Given a sparse matrix A (the arguments n, Ap, Ai, and Ax) and its symbolic
* analysis (Lp and Parent, and optionally P and Pinv), compute the numeric LDL'
* factorization of A or PAP'. The outputs of this routine are arguments Li,
* Lx, and D. It also requires three size-n workspaces (Y, Pattern, and Flag).
*/
int ldl_numeric /* returns n if successful, k if D (k,k) is zero */
(
int n, /* A and L are n-by-n, 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) */
/* follow path from i to root of etree, stop at flagged node */
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) /* push path on top of stack */
{
Pattern [--top] = Pattern [--len] ;
}
}
}
/* Pattern [top ... n-1] now contains nonzero pattern of L (:,k) */
/* 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] ;
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,k )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 */
}
/* ========================================================================== */
/* === ldl_lsolve: solve Lx=b ============================================== */
/* ========================================================================== */
void ldl_lsolve
(
int n, /* L is n-by-n, where n >= 0 */
double X [ ], /* size n. right-hand-side 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] ;
}
}
}
/* ========================================================================== */
/* === ldl_dsolve: solve Dx=b ============================================== */
/* ========================================================================== */
void ldl_dsolve
(
int n, /* D is n-by-n, where n >= 0 */
double X [ ], /* size n. right-hand-side 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] ;
}
}
/* ========================================================================== */
/* === ldl_ltsolve: solve L'x=b ============================================ */
/* ========================================================================== */
void ldl_ltsolve
(
int n, /* L is n-by-n, where n >= 0 */
double X [ ], /* size n. right-hand-side 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 = n-1 ; j >= 0 ; j--)
{
p2 = Lp [j+1] ;
for (p = Lp [j] ; p < p2 ; p++)
{
X [j] -= Lx [p] * X [Li [p]] ;
}
}
}
/* ========================================================================== */
/* === ldl_perm: permute a vector, x=Pb ===================================== */
/* ========================================================================== */
void ldl_perm
(
int n, /* size of X, B, and P */
double X [ ], /* output of size n. */
double B [ ], /* input of size n. */
int P [ ] /* input permutation array of size n. */
)
{
int j ;
for (j = 0 ; j < n ; j++)
{
X [j] = B [P [j]] ;
}
}
/* ========================================================================== */
/* === ldl_permt: permute a vector, x=P'b =================================== */
/* ========================================================================== */
void ldl_permt
(
int n, /* size of X, B, and P */
double X [ ], /* output of size n. */
double B [ ], /* input of size n. */
int P [ ] /* input permutation array of size n. */
)
{
int j ;
for (j = 0 ; j < n ; j++)
{
X [P [j]] = B [j] ;
}
}
/* ========================================================================== */
/* === ldl_valid_perm: check if a permutation vector is valid =============== */
/* ========================================================================== */
int ldl_valid_perm /* returns 1 if valid, 0 otherwise */
(
int n,
int P [ ], /* input of size n, a permutation of 0:n-1 */
int Flag [ ] /* workspace of size n */
)
{
int j, k ;
if (n < 0 || !Flag)
{
return (0) ; /* n must be >= 0, and Flag must be present */
}
if (!P)
{
return (1) ; /* If NULL, P is assumed to be the identity perm. */
}
for (j = 0 ; j < n ; j++)
{
Flag [j] = 0 ; /* clear the Flag array */
}
for (k = 0 ; k < n ; k++)
{
j = P [k] ;
if (j < 0 || j >= n || Flag [j] != 0)
{
return (0) ; /* P is not valid */
}
Flag [j] = 1 ;
}
return (1) ; /* P is valid */
}
/* ========================================================================== */
/* === ldl_valid_matrix: check if a sparse matrix is valid ================== */
/* ========================================================================== */
/* This routine checks to see if a sparse matrix A is valid for input to
* ldl_symbolic and ldl_numeric. It returns 1 if the matrix is valid, 0
* otherwise. A is in sparse column form. The numerical values in column j
* are stored in Ax [Ap [j] ... Ap [j+1]-1], with row indices in
* Ai [Ap [j] ... Ap [j+1]-1]. The Ax array is not checked.
*/
int ldl_valid_matrix
(
int n,
int Ap [ ],
int Ai [ ]
)
{
int j, p ;
if (n < 0 || !Ap || !Ai || Ap [0] != 0)
{
return (0) ; /* n must be >= 0, and Ap and Ai must be present */
}
for (j = 0 ; j < n ; j++)
{
if (Ap [j] > Ap [j+1])
{
return (0) ; /* Ap must be monotonically nondecreasing */
}
}
for (p = 0 ; p < Ap [n] ; p++)
{
if (Ai [p] < 0 || Ai [p] >= n)
{
return (0) ; /* row indices must be in the range 0 to n-1 */
}
}
return (1) ; /* matrix is valid */
}
