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# View of /pkg/src/R_ldl.c

Wed Feb 16 20:49:16 2005 UTC (15 years ago) by bates
File size: 19871 byte(s)
`cosmetic changes - reformating`
```/* This is a modified version of the ldl.c file released by Timothy A. Davis  */
/* in the LDL package and carrying the copyright shown below. The             */
/* modifications are to replace scratch arrays passed as arguments by         */
/* dynamically allocated arrays.                                              */
/* Douglas Bates (Nov., 2004)                                                 */

/* ========================================================================== */
/* === 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
*
* 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 R_ldl_numeric.  The diagonal
* matrix D is held as an array D [0..n-1] of size n.
*
* --------------------
* C-callable routines:
* --------------------
*
*	R_ldl_symbolic:  Given the pattern of A, computes the Lp and Parent
*	    arrays required by R_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 R_ldl_numeric).
*	R_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.
*	R_ldl_lsolve:  Solves Lx=b for a dense vector b.
*	R_ldl_dsolve:  Solves Dx=b for a dense vector b.
*	R_ldl_ltsolve: Solves L'x=b for a dense vector b.
*	R_ldl_perm:    Computes x=Pb for a dense vector b.
*	R-ldl_permt:   Computes x=P'b for a dense vector b.
*	R_ldl_valid_perm:  checks the validity of a permutation vector
*	R_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,
* R_ldl_symbolic and R_ldl_numeric assume their inputs are valid.
* You can check your own inputs prior to calling these routines with
* the R_ldl_valid_perm and R_ldl_valid_matrix routines.  Except for
* the two R_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 R_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 R_ldl_symbolic and "if
* (i <= k)" in R_ldl_numeric, are always true, and could be removed
* (i can equal k in R_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
* R_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 R_ldl_symbolic with
* the first part of R_ldl_numeric, and note that the while (len > 0)
* loop is not present in R_ldl_symbolic.
*
* LDL Version 1.0 (Dec. 31, 2003), Copyright (c) 2003 by Timothy A
* while on sabbatical at Stanford University and Lawrence Berkeley
* Available at http://www.cise.ufl.edu/research/sparse.
*/

#include "R_ldl.h"

/* ========================================================================== */
/* === R_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 R_ldl_numeric.  Lnz [k] gives the number of nonzeros in
* the kth column of L, excluding the diagonal.
*
* 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 R_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
* R_ldl_numeric.
*
* The floating-point operation count of the subsequent call to
* R_ldl_numeric is not returned, but could be computed after
* R_ldl_symbolic is done.  It is the sum of (Lnz [k]) * (Lnz [k] + 2)
* for k = 0 to n-1.
*
* @param n A and L are n-by-n, where n >= 0
* @param Ap column pointers of size n+1
* @param Ai row indices of size nz=Ap[n]
* @param Lp column pointers of size n+1
* @param Parent elimination tree of size n
* @param P optional permutation vector of size n [use (int *) NULL for none]
* @param Pinv optional inverse permutation [not used if P is NULL]
*/
void
R_ldl_symbolic(int n, const int Ap[], const int Ai[], int Lp[],
int Parent[], const int P[], int Pinv[])
{
int i, k, p, kk, p2;
int *Flag = Calloc(n, int);
int *Lnz = Calloc(n, int);
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] ;
}
Free(Flag); Free(Lnz);
}

/**
* 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.
*
* @param n A and L are n-by-n, where n >= 0
* @param Ap column pointer array of size n+1
* @param Ai row index array of size nz=Ap[n] (upper triangle only)
* @param Ax array of non-zero matrix elements of size nz=Ap[n]
* @param Lp column pointer array of size n+1
* @param Parent elimination tree of size n
* @param Li row index array of size lnz=Lp[n]
* @param Lx non-zero off-diagonal elements of L (size lnz=Lp[n])
* @param D vector of diagonal elements (size n)
* @param P optional permutation vector of size n [use (int *) NULL for none]
* @param Pinv optional inverse permutation [use (int *) NULL for none]
*
* @return n if successful, k if D (k,k) is zero
*/
int
R_ldl_numeric(int n,
const int Ap[], const int Ai[], const double Ax[],
const int Lp[], const int Parent[],
int Li[], double Lx[], double D[],
const int P[], const int Pinv[])
{
double yi, l_ki ;
int i, k, p, kk, p2, len, top ;
int *Lnz = Calloc(n, int),
*Pattern = Calloc(n, int),
*Flag = Calloc(n, int);
double *Y = Calloc(n, double);

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)
{
Free(Y); Free(Pattern); Free(Flag); Free(Lnz);
return (k) ;	/* failure, D (k,k) is zero */
}
}
Free(Y); Free(Pattern); Free(Flag); Free(Lnz);
return (n) ;	   /* success, diagonal of D is all nonzero */
}

/**
* solve Lx=b
*
* @param n L is n-by-n, where n >= 0
* @param X size n.  right-hand-side on input, soln. on output
* @param Lp column pointer array of size n+1
* @param Li row index array of size lnz=Lp[n]
* @param Lx non-zero off-diagonal elements (size lnz=Lp[n])
*/
void
R_ldl_lsolve (int n, double X[],
const int Lp[], const int Li[], const double Lx[])
{
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] ;
}
}
}

/**
* solve Dx=b
*
* @param n L is n-by-n, where n >= 0
* @param X size n.  right-hand-side on input, soln. on output
* @param D diagonal elements of size n
*/
void
R_ldl_dsolve(int n, double X[], const double D[])
{
int j ;
for (j = 0 ; j < n ; j++)
{
X [j] /= D [j] ;
}
}

/**
* solve L'x=b
*
* @param n L is n-by-n, where n >= 0
* @param X size n.  right-hand-side on input, soln. on output
* @param Lp column pointer array of size n+1
* @param Li row index array of size lnz=Lp[n]
* @param Lx non-zero off-diagonal elements (size lnz=Lp[n])
*/
void
R_ldl_ltsolve (int n, double X[],
const int Lp[], const int Li[], const double Lx[])
{
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]] ;
}
}
}

/**
* permute a vector, x=Pb
*
* @param n size of X, B, and P
* @param X output of size n
* @param B input of size n
* @param P input permutation array of size n
*/
void
R_ldl_perm(int n, double X[], const double B[], const int P[])
{
int j ;
for (j = 0 ; j < n ; j++)
{
X [j] = B [P [j]] ;
}
}

/**
* permute a vector, x=P'b
*
* @param n size of X, B, and P
* @param X output of size n
* @param B input of size n
* @param P input permutation array of size n
*/
void
R_ldl_permt(int n, double X[], const double B[], const int P[])
{
int j ;
for (j = 0 ; j < n ; j++)
{
X [P [j]] = B [j] ;
}
}

/**
* Check if a permutation vector is valid
*
* @param n size of permutation
* @param P input of size n, a permutation of 0:n-1
*
* @return 1 if valid, otherwise 0
*/
int
R_ldl_valid_perm (int n, const int P[])
{

int j, k ;
int *Flag = (int *) R_alloc(n, sizeof(int));

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 */
}

/**
* This routine checks to see if a sparse matrix A is valid for input
* to R_ldl_symbolic and R_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.
*
* @param n A is n by n (n >= 0)
* @param Ap column pointer array of size n+1
* @param Ai row index array of size nz=Ap[n]
*
* @return 1 if valid sparse matrix, otherwise 0
*/
int
R_ldl_valid_matrix (int n, const int Ap[], const 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 */
}
```