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3dpcp/3rdparty/sparse/csparse.cc

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initial commit
2012-09-16 12:33:11 +00:00
#include <cstdlib>
#include <climits>
#include <cmath>
#include <cstdio>
#include "csparse.h"
cs *cs_add ( const cs *A, const cs *B, double alpha, double beta )
/*
Purpose:
CS_ADD computes C = alpha*A + beta*B for sparse A and B.
Reference:
Timothy Davis,
Direct Methods for Sparse Linear Systems,
SIAM, Philadelphia, 2006.
*/
{
int p, j, nz = 0, anz, *Cp, *Ci, *Bp, m, n, bnz, *w, values ;
double *x, *Bx, *Cx ;
cs *C ;
if (!A || !B) return (NULL) ; /* check inputs */
m = A->m ; anz = A->p [A->n] ;
n = B->n ; Bp = B->p ; Bx = B->x ; bnz = Bp [n] ;
w = (int*)cs_calloc (m, sizeof (int)) ;
values = (A->x != NULL) && (Bx != NULL) ;
x = values ? (double*)cs_malloc (m, sizeof (double)) : NULL ;
C = cs_spalloc (m, n, anz + bnz, values, 0) ;
if (!C || !w || (values && !x)) return (cs_done (C, w, x, 0)) ;
Cp = C->p ; Ci = C->i ; Cx = C->x ;
for (j = 0 ; j < n ; j++)
{
Cp [j] = nz ; /* column j of C starts here */
nz = cs_scatter (A, j, alpha, w, x, j+1, C, nz) ; /* alpha*A(:,j)*/
nz = cs_scatter (B, j, beta, w, x, j+1, C, nz) ; /* beta*B(:,j) */
if (values) for (p = Cp [j] ; p < nz ; p++) Cx [p] = x [Ci [p]] ;
}
Cp [n] = nz ; /* finalize the last column of C */
cs_sprealloc (C, 0) ; /* remove extra space from C */
return (cs_done (C, w, x, 1)) ; /* success; free workspace, return C */
}
static int cs_wclear (int mark, int lemax, int *w, int n)
/*
Purpose:
CS_WCLEAR clears W.
Reference:
Timothy Davis,
Direct Methods for Sparse Linear Systems,
SIAM, Philadelphia, 2006.
*/
{
int k ;
if (mark < 2 || (mark + lemax < 0))
{
for (k = 0 ; k < n ; k++) if (w [k] != 0) w [k] = 1 ;
mark = 2 ;
}
return (mark) ; /* at this point, w [0..n-1] < mark holds */
}
/* keep off-diagonal entries; drop diagonal entries */
static int cs_diag (int i, int j, double aij, void *other)
{
return (i != j);
}
/* p = amd(A+A') if symmetric is true, or amd(A'A) otherwise */
int *cs_amd ( const cs *A, int order )
/*
Purpose:
CS_AMD carries out the approximate minimum degree algorithm.
Reference:
Timothy Davis,
Direct Methods for Sparse Linear Systems,
SIAM, Philadelphia, 2006.
Parameters:
Input, int ORDER:
-1:natural,
0:Cholesky,
1:LU,
2:QR
*/
{
cs *C, *A2, *AT ;
int *Cp, *Ci, *last, *ww, *len, *nv, *next, *P, *head, *elen, *degree, *w,
*hhead, *ATp, *ATi, d, dk, dext, lemax = 0, e, elenk, eln, i, j, k, k1,
k2, k3, jlast, ln, dense, nzmax, mindeg = 0, nvi, nvj, nvk, mark, wnvi,
ok, cnz, nel = 0, p, p1, p2, p3, p4, pj, pk, pk1, pk2, pn, q, n, m ;
unsigned int h ;
/* --- Construct matrix C ----------------------------------------------- */
if (!A || order < 0) return (NULL) ; /* check inputs; quick return */
AT = cs_transpose (A, 0) ; /* compute A' */
if (!AT) return (NULL) ;
m = A->m ; n = A->n ;
dense = (int)CS_MAX (16, 10 * sqrt ((double) n)) ; /* find dense threshold */
dense = CS_MIN (n-2, dense) ;
if (order == 0 && n == m)
{
C = cs_add (A, AT, 0, 0) ; /* C = A+A' */
}
else if (order == 1)
{
ATp = AT->p ; /* drop dense columns from AT */
ATi = AT->i ;
for (p2 = 0, j = 0 ; j < m ; j++)
{
p = ATp [j] ; /* column j of AT starts here */
ATp [j] = p2 ; /* new column j starts here */
if (ATp [j+1] - p > dense) continue ; /* skip dense col j */
for ( ; p < ATp [j+1] ; p++) ATi [p2++] = ATi [p] ;
}
ATp [m] = p2 ; /* finalize AT */
A2 = cs_transpose (AT, 0) ; /* A2 = AT' */
C = A2 ? cs_multiply (AT, A2) : NULL ; /* C=A'*A with no dense rows */
cs_spfree (A2) ;
}
else
{
C = cs_multiply (AT, A) ; /* C=A'*A */
}
cs_spfree (AT) ;
if (!C) return (NULL) ;
P = (int*)cs_malloc (n+1, sizeof (int)) ; /* allocate result */
ww = (int*)cs_malloc (8*(n+1), sizeof (int)) ;/* get workspace */
len = ww ; nv = ww + (n+1) ; next = ww + 2*(n+1) ;
head = ww + 3*(n+1) ; elen = ww + 4*(n+1) ; degree = ww + 5*(n+1) ;
w = ww + 6*(n+1) ; hhead = ww + 7*(n+1) ;
last = P ; /* use P as workspace for last */
cs_fkeep (C, &cs_diag, NULL) ; /* drop diagonal entries */
Cp = C->p ;
cnz = Cp [n] ;
if (!cs_sprealloc (C, cnz+cnz/5+2*n)) return (cs_idone (P, C, ww, 0)) ;
/* --- Initialize quotient graph ---------------------------------------- */
for (k = 0 ; k < n ; k++) len [k] = Cp [k+1] - Cp [k] ;
len [n] = 0 ;
nzmax = C->nzmax ;
Ci = C->i ;
for (i = 0 ; i <= n ; i++)
{
head [i] = -1 ; /* degree list i is empty */
last [i] = -1 ;
next [i] = -1 ;
hhead [i] = -1 ; /* hash list i is empty */
nv [i] = 1 ; /* node i is just one node */
w [i] = 1 ; /* node i is alive */
elen [i] = 0 ; /* Ek of node i is empty */
degree [i] = len [i] ; /* degree of node i */
}
mark = cs_wclear (0, 0, w, n) ; /* clear w */
elen [n] = -2 ; /* n is a dead element */
Cp [n] = -1 ; /* n is a root of assembly tree */
w [n] = 0 ; /* n is a dead element */
/* --- Initialize degree lists ------------------------------------------ */
for (i = 0 ; i < n ; i++)
{
d = degree [i] ;
if (d == 0) /* node i is empty */
{
elen [i] = -2 ; /* element i is dead */
nel++ ;
Cp [i] = -1 ; /* i is a root of assemby tree */
w [i] = 0 ;
}
else if (d > dense) /* node i is dense */
{
nv [i] = 0 ; /* absorb i into element n */
elen [i] = -1 ; /* node i is dead */
nel++ ;
Cp [i] = CS_FLIP (n) ;
nv [n]++ ;
}
else
{
if (head [d] != -1) last [head [d]] = i ;
next [i] = head [d] ; /* put node i in degree list d */
head [d] = i ;
}
}
while (nel < n) /* while (selecting pivots) do */
{
/* --- Select node of minimum approximate degree -------------------- */
for (k = -1 ; mindeg < n && (k = head [mindeg]) == -1 ; mindeg++) ;
if (next [k] != -1) last [next [k]] = -1 ;
head [mindeg] = next [k] ; /* remove k from degree list */
elenk = elen [k] ; /* elenk = |Ek| */
nvk = nv [k] ; /* # of nodes k represents */
nel += nvk ; /* nv[k] nodes of A eliminated */
/* --- Garbage collection ------------------------------------------- */
if (elenk > 0 && cnz + mindeg >= nzmax)
{
for (j = 0 ; j < n ; j++)
{
if ((p = Cp [j]) >= 0) /* j is a live node or element */
{
Cp [j] = Ci [p] ; /* save first entry of object */
Ci [p] = CS_FLIP (j) ; /* first entry is now CS_FLIP(j) */
}
}
for (q = 0, p = 0 ; p < cnz ; ) /* scan all of memory */
{
if ((j = CS_FLIP (Ci [p++])) >= 0) /* found object j */
{
Ci [q] = Cp [j] ; /* restore first entry of object */
Cp [j] = q++ ; /* new pointer to object j */
for (k3 = 0 ; k3 < len [j]-1 ; k3++) Ci [q++] = Ci [p++] ;
}
}
cnz = q ; /* Ci [cnz...nzmax-1] now free */
}
/* --- Construct new element ---------------------------------------- */
dk = 0 ;
nv [k] = -nvk ; /* flag k as in Lk */
p = Cp [k] ;
pk1 = (elenk == 0) ? p : cnz ; /* do in place if elen[k] == 0 */
pk2 = pk1 ;
for (k1 = 1 ; k1 <= elenk + 1 ; k1++)
{
if (k1 > elenk)
{
e = k ; /* search the nodes in k */
pj = p ; /* list of nodes starts at Ci[pj]*/
ln = len [k] - elenk ; /* length of list of nodes in k */
}
else
{
e = Ci [p++] ; /* search the nodes in e */
pj = Cp [e] ;
ln = len [e] ; /* length of list of nodes in e */
}
for (k2 = 1 ; k2 <= ln ; k2++)
{
i = Ci [pj++] ;
if ((nvi = nv [i]) <= 0) continue ; /* node i dead, or seen */
dk += nvi ; /* degree[Lk] += size of node i */
nv [i] = -nvi ; /* negate nv[i] to denote i in Lk*/
Ci [pk2++] = i ; /* place i in Lk */
if (next [i] != -1) last [next [i]] = last [i] ;
if (last [i] != -1) /* remove i from degree list */
{
next [last [i]] = next [i] ;
}
else
{
head [degree [i]] = next [i] ;
}
}
if (e != k)
{
Cp [e] = CS_FLIP (k) ; /* absorb e into k */
w [e] = 0 ; /* e is now a dead element */
}
}
if (elenk != 0) cnz = pk2 ; /* Ci [cnz...nzmax] is free */
degree [k] = dk ; /* external degree of k - |Lk\i| */
Cp [k] = pk1 ; /* element k is in Ci[pk1..pk2-1] */
len [k] = pk2 - pk1 ;
elen [k] = -2 ; /* k is now an element */
/* --- Find set differences ----------------------------------------- */
mark = cs_wclear (mark, lemax, w, n) ; /* clear w if necessary */
for (pk = pk1 ; pk < pk2 ; pk++) /* scan 1: find |Le\Lk| */
{
i = Ci [pk] ;
if ((eln = elen [i]) <= 0) continue ;/* skip if elen[i] empty */
nvi = -nv [i] ; /* nv [i] was negated */
wnvi = mark - nvi ;
for (p = Cp [i] ; p <= Cp [i] + eln - 1 ; p++) /* scan Ei */
{
e = Ci [p] ;
if (w [e] >= mark)
{
w [e] -= nvi ; /* decrement |Le\Lk| */
}
else if (w [e] != 0) /* ensure e is a live element */
{
w [e] = degree [e] + wnvi ; /* 1st time e seen in scan 1 */
}
}
}
/* --- Degree update ------------------------------------------------ */
for (pk = pk1 ; pk < pk2 ; pk++) /* scan2: degree update */
{
i = Ci [pk] ; /* consider node i in Lk */
p1 = Cp [i] ;
p2 = p1 + elen [i] - 1 ;
pn = p1 ;
for (h = 0, d = 0, p = p1 ; p <= p2 ; p++) /* scan Ei */
{
e = Ci [p] ;
if (w [e] != 0) /* e is an unabsorbed element */
{
dext = w [e] - mark ; /* dext = |Le\Lk| */
if (dext > 0)
{
d += dext ; /* sum up the set differences */
Ci [pn++] = e ; /* keep e in Ei */
h += e ; /* compute the hash of node i */
}
else
{
Cp [e] = CS_FLIP (k) ; /* aggressive absorb. e->k */
w [e] = 0 ; /* e is a dead element */
}
}
}
elen [i] = pn - p1 + 1 ; /* elen[i] = |Ei| */
p3 = pn ;
p4 = p1 + len [i] ;
for (p = p2 + 1 ; p < p4 ; p++) /* prune edges in Ai */
{
j = Ci [p] ;
if ((nvj = nv [j]) <= 0) continue ; /* node j dead or in Lk */
d += nvj ; /* degree(i) += |j| */
Ci [pn++] = j ; /* place j in node list of i */
h += j ; /* compute hash for node i */
}
if (d == 0) /* check for mass elimination */
{
Cp [i] = CS_FLIP (k) ; /* absorb i into k */
nvi = -nv [i] ;
dk -= nvi ; /* |Lk| -= |i| */
nvk += nvi ; /* |k| += nv[i] */
nel += nvi ;
nv [i] = 0 ;
elen [i] = -1 ; /* node i is dead */
}
else
{
degree [i] = CS_MIN (degree [i], d) ; /* update degree(i) */
Ci [pn] = Ci [p3] ; /* move first node to end */
Ci [p3] = Ci [p1] ; /* move 1st el. to end of Ei */
Ci [p1] = k ; /* add k as 1st element in of Ei */
len [i] = pn - p1 + 1 ; /* new len of adj. list of node i */
h %= n ; /* finalize hash of i */
next [i] = hhead [h] ; /* place i in hash bucket */
hhead [h] = i ;
last [i] = h ; /* save hash of i in last[i] */
}
} /* scan2 is done */
degree [k] = dk ; /* finalize |Lk| */
lemax = CS_MAX (lemax, dk) ;
mark = cs_wclear (mark+lemax, lemax, w, n) ; /* clear w */
/* --- Supernode detection ------------------------------------------ */
for (pk = pk1 ; pk < pk2 ; pk++)
{
i = Ci [pk] ;
if (nv [i] >= 0) continue ; /* skip if i is dead */
h = last [i] ; /* scan hash bucket of node i */
i = hhead [h] ;
hhead [h] = -1 ; /* hash bucket will be empty */
for ( ; i != -1 && next [i] != -1 ; i = next [i], mark++)
{
ln = len [i] ;
eln = elen [i] ;
for (p = Cp[i]+1 ; p <= Cp[i]+ln-1 ; p++) w [Ci [p]] = mark ;
jlast = i ;
for (j = next [i] ; j != -1 ; ) /* compare i with all j */
{
ok = (len [j] == ln) && (elen [j] == eln) ;
for (p = Cp [j] + 1 ; ok && p <= Cp [j] + ln - 1 ; p++)
{
if (w [Ci [p]] != mark) ok = 0 ; /* compare i and j*/
}
if (ok) /* i and j are identical */
{
Cp [j] = CS_FLIP (i) ; /* absorb j into i */
nv [i] += nv [j] ;
nv [j] = 0 ;
elen [j] = -1 ; /* node j is dead */
j = next [j] ; /* delete j from hash bucket */
next [jlast] = j ;
}
else
{
jlast = j ; /* j and i are different */
j = next [j] ;
}
}
}
}
/* --- Finalize new element------------------------------------------ */
for (p = pk1, pk = pk1 ; pk < pk2 ; pk++) /* finalize Lk */
{
i = Ci [pk] ;
if ((nvi = -nv [i]) <= 0) continue ;/* skip if i is dead */
nv [i] = nvi ; /* restore nv[i] */
d = degree [i] + dk - nvi ; /* compute external degree(i) */
d = CS_MIN (d, n - nel - nvi) ;
if (head [d] != -1) last [head [d]] = i ;
next [i] = head [d] ; /* put i back in degree list */
last [i] = -1 ;
head [d] = i ;
mindeg = CS_MIN (mindeg, d) ; /* find new minimum degree */
degree [i] = d ;
Ci [p++] = i ; /* place i in Lk */
}
nv [k] = nvk ; /* # nodes absorbed into k */
if ((len [k] = p-pk1) == 0) /* length of adj list of element k*/
{
Cp [k] = -1 ; /* k is a root of the tree */
w [k] = 0 ; /* k is now a dead element */
}
if (elenk != 0) cnz = p ; /* free unused space in Lk */
}
/* --- Postordering ----------------------------------------------------- */
for (i = 0 ; i < n ; i++) Cp [i] = CS_FLIP (Cp [i]) ;/* fix assembly tree */
for (j = 0 ; j <= n ; j++) head [j] = -1 ;
for (j = n ; j >= 0 ; j--) /* place unordered nodes in lists */
{
if (nv [j] > 0) continue ; /* skip if j is an element */
next [j] = head [Cp [j]] ; /* place j in list of its parent */
head [Cp [j]] = j ;
}
for (e = n ; e >= 0 ; e--) /* place elements in lists */
{
if (nv [e] <= 0) continue ; /* skip unless e is an element */
if (Cp [e] != -1)
{
next [e] = head [Cp [e]] ; /* place e in list of its parent */
head [Cp [e]] = e ;
}
}
for (k = 0, i = 0 ; i <= n ; i++) /* postorder the assembly tree */
{
if (Cp [i] == -1) k = cs_tdfs (i, k, head, next, P, w) ;
}
return (cs_idone (P, C, ww, 1)) ;
}
/* compute nonzero pattern of L(k,:) */
static
int cs_ereach (const cs *A, int k, const int *parent, int *s, int *w,
double *x, int top)
{
int i, p, len, *Ap = A->p, *Ai = A->i ;
double *Ax = A->x ;
for (p = Ap [k] ; p < Ap [k+1] ; p++) /* get pattern of L(k,:) */
{
i = Ai [p] ; /* A(i,k) is nonzero */
if (i > k) continue ; /* only use upper triangular part of A */
x [i] = Ax [p] ; /* x(i) = A(i,k) */
for (len = 0 ; w [i] != k ; i = parent [i]) /* traverse up etree */
{
s [len++] = i ; /* L(k,i) is nonzero */
w [i] = k ; /* mark i as visited */
}
while (len > 0) s [--top] = s [--len] ; /* push path onto stack */
}
return (top) ; /* s [top..n-1] contains pattern of L(k,:)*/
}
/* L = chol (A, [Pinv parent cp]), Pinv is optional */
csn *cs_chol (const cs *A, const css *S)
{
double d, lki, *Lx, *x ;
int top, i, p, k, n, *Li, *Lp, *cp, *Pinv, *w, *s, *c, *parent ;
cs *L, *C, *E ;
csn *N ;
if (!A || !S || !S->cp || !S->parent) return (NULL) ; /* check inputs */
n = A->n ;
N = (csn*)cs_calloc (1, sizeof (csn)) ;
w = (int*)cs_malloc (3*n, sizeof (int)) ; s = w + n, c = w + 2*n ;
x = (double*)cs_malloc (n, sizeof (double)) ;
cp = S->cp ; Pinv = S->Pinv ; parent = S->parent ;
C = Pinv ? cs_symperm (A, Pinv, 1) : ((cs *) A) ;
E = Pinv ? C : NULL ;
if (!N || !w || !x || !C) return (cs_ndone (N, E, w, x, 0)) ;
N->L = L = cs_spalloc (n, n, cp [n], 1, 0) ;
if (!L) return (cs_ndone (N, E, w, x, 0)) ;
Lp = L->p ; Li = L->i ; Lx = L->x ;
for (k = 0 ; k < n ; k++)
{
/* --- Nonzero pattern of L(k,:) ------------------------------------ */
Lp [k] = c [k] = cp [k] ; /* column k of L starts here */
x [k] = 0 ; /* x (0:k) is now zero */
w [k] = k ; /* mark node k as visited */
top = cs_ereach (C, k, parent, s, w, x, n) ; /* find row k of L*/
d = x [k] ; /* d = C(k,k) */
x [k] = 0 ; /* clear workspace for k+1st iteration */
/* --- Triangular solve --------------------------------------------- */
for ( ; top < n ; top++) /* solve L(0:k-1,0:k-1) * x = C(:,k) */
{
i = s [top] ; /* s [top..n-1] is pattern of L(k,:) */
lki = x [i] / Lx [Lp [i]] ; /* L(k,i) = x (i) / L(i,i) */
x [i] = 0 ; /* clear workspace for k+1st iteration */
for (p = Lp [i] + 1 ; p < c [i] ; p++)
{
x [Li [p]] -= Lx [p] * lki ;
}
d -= lki * lki ; /* d = d - L(k,i)*L(k,i) */
p = c [i]++ ;
Li [p] = k ; /* store L(k,i) in column i */
Lx [p] = lki ;
}
/* --- Compute L(k,k) ----------------------------------------------- */
if (d <= 0) return (cs_ndone (N, E, w, x, 0)) ; /* not pos def */
p = c [k]++ ;
Li [p] = k ; /* store L(k,k) = sqrt (d) in column k */
Lx [p] = sqrt (d) ;
}
Lp [n] = cp [n] ; /* finalize L */
return (cs_ndone (N, E, w, x, 1)) ; /* success: free E,w,x; return N */
}
/* x=A\b where A is symmetric positive definite; b overwritten with solution */
int cs_cholsol (const cs *A, double *b, int order)
{
double *x ;
css *S ;
csn *N ;
int n, ok ;
if (!A || !b) return (0) ; /* check inputs */
n = A->n ;
S = cs_schol (A, order) ; /* ordering and symbolic analysis */
N = cs_chol (A, S) ; /* numeric Cholesky factorization */
x = (double*)cs_malloc (n, sizeof (double)) ;
ok = (S && N && x) ;
if (ok)
{
cs_ipvec (n, S->Pinv, b, x) ; /* x = P*b */
cs_lsolve (N->L, x) ; /* x = L\x */
cs_ltsolve (N->L, x) ; /* x = L'\x */
cs_pvec (n, S->Pinv, x, b) ; /* b = P'*x */
}
cs_free (x) ;
cs_sfree (S) ;
cs_nfree (N) ;
return (ok) ;
}
/* process edge (j,i) of the matrix */
static void cs_cedge (int j, int i, const int *first, int *maxfirst, int *delta,
int *prevleaf, int *ancestor)
{
int q, s, sparent, jprev ;
if (i <= j || first [j] <= maxfirst [i]) return ;
maxfirst [i] = first [j] ; /* update max first[j] seen so far */
jprev = prevleaf [i] ; /* j is a leaf of the ith subtree */
delta [j]++ ; /* A(i,j) is in the skeleton matrix */
if (jprev != -1)
{
/* q = least common ancestor of jprev and j */
for (q = jprev ; q != ancestor [q] ; q = ancestor [q]) ;
for (s = jprev ; s != q ; s = sparent)
{
sparent = ancestor [s] ; /* path compression */
ancestor [s] = q ;
}
delta [q]-- ; /* decrement to account for overlap in q */
}
prevleaf [i] = j ; /* j is now previous leaf of ith subtree */
}
/* colcount = column counts of LL'=A or LL'=A'A, given parent & post ordering */
int *cs_counts (const cs *A, const int *parent, const int *post, int ata)
{
int i, j, k, p, n, m, ii, s, *ATp, *ATi, *maxfirst, *prevleaf, *ancestor,
*head = NULL, *next = NULL, *colcount, *w, *first, *delta ;
cs *AT ;
if (!A || !parent || !post) return (NULL) ; /* check inputs */
m = A->m ; n = A->n ;
s = 4*n + (ata ? (n+m+1) : 0) ;
w = (int*)cs_malloc (s, sizeof (int)) ; first = w+3*n ; /* get workspace */
ancestor = w ; maxfirst = w+n ; prevleaf = w+2*n ;
delta = colcount = (int*)cs_malloc (n, sizeof (int)) ; /* allocate result */
AT = cs_transpose (A, 0) ;
if (!AT || !colcount || !w) return (cs_idone (colcount, AT, w, 1)) ;
for (k = 0 ; k < s ; k++) w [k] = -1 ; /* clear workspace w [0..s-1] */
for (k = 0 ; k < n ; k++) /* find first [j] */
{
j = post [k] ;
delta [j] = (first [j] == -1) ? 1 : 0 ; /* delta[j]=1 if j is a leaf */
for ( ; j != -1 && first [j] == -1 ; j = parent [j]) first [j] = k ;
}
ATp = AT->p ; ATi = AT->i ;
if (ata)
{
head = w+4*n ; next = w+5*n+1 ;
for (k = 0 ; k < n ; k++) w [post [k]] = k ; /* invert post */
for (i = 0 ; i < m ; i++)
{
k = n ; /* k = least postordered column in row i */
for (p = ATp [i] ; p < ATp [i+1] ; p++) k = CS_MIN (k, w [ATi [p]]);
next [i] = head [k] ; /* place row i in link list k */
head [k] = i ;
}
}
for (i = 0 ; i < n ; i++) ancestor [i] = i ; /* each node in its own set */
for (k = 0 ; k < n ; k++)
{
j = post [k] ; /* j is the kth node in postordered etree */
if (parent [j] != -1) delta [parent [j]]-- ; /* j is not a root */
if (ata)
{
for (ii = head [k] ; ii != -1 ; ii = next [ii])
{
for (p = ATp [ii] ; p < ATp [ii+1] ; p++)
cs_cedge (j, ATi [p], first, maxfirst, delta, prevleaf,
ancestor) ;
}
}
else
{
for (p = ATp [j] ; p < ATp [j+1] ; p++)
cs_cedge (j, ATi [p], first, maxfirst, delta, prevleaf,
ancestor) ;
}
if (parent [j] != -1) ancestor [j] = parent [j] ;
}
for (j = 0 ; j < n ; j++) /* sum up delta's of each child */
{
if (parent [j] != -1) colcount [parent [j]] += colcount [j] ;
}
return (cs_idone (colcount, AT, w, 1)) ; /* success: free workspace */
}
/* p [0..n] = cumulative sum of c [0..n-1], and then copy p [0..n-1] into c */
int cs_cumsum (int *p, int *c, int n)
{
int i, nz = 0 ;
if (!p || !c) return (-1) ; /* check inputs */
for (i = 0 ; i < n ; i++)
{
p [i] = nz ;
nz += c [i] ;
c [i] = p [i] ;
}
p [n] = nz ;
return (nz) ; /* return sum (c [0..n-1]) */
}
/* depth-first-search of the graph of a matrix, starting at node j */
int cs_dfs (int j, cs *L, int top, int *xi, int *pstack, const int *Pinv)
{
int i, p, p2, done, jnew, head = 0, *Lp, *Li ;
if (!L || !xi || !pstack) return (-1) ;
Lp = L->p ; Li = L->i ;
xi [0] = j ; /* initialize the recursion stack */
while (head >= 0)
{
j = xi [head] ; /* get j from the top of the recursion stack */
jnew = Pinv ? (Pinv [j]) : j ;
if (!CS_MARKED(Lp,j))
{
CS_MARK (Lp,j) ; /* mark node j as visited */
pstack [head] = (jnew < 0) ? 0 : CS_UNFLIP (Lp [jnew]) ;
}
done = 1 ; /* node j done if no unvisited neighbors */
p2 = (jnew < 0) ? 0 : CS_UNFLIP (Lp [jnew+1]) ;
for (p = pstack [head] ; p < p2 ; p++) /* examine all neighbors of j */
{
i = Li [p] ; /* consider neighbor node i */
if (CS_MARKED (Lp,i)) continue ; /* skip visited node i */
pstack [head] = p ; /* pause depth-first search of node j */
xi [++head] = i ; /* start dfs at node i */
done = 0 ; /* node j is not done */
break ; /* break, to start dfs (i) */
}
if (done) /* depth-first search at node j is done */
{
head-- ; /* remove j from the recursion stack */
xi [--top] = j ; /* and place in the output stack */
}
}
return (top) ;
}
/* breadth-first search for coarse decomposition (C0,C1,R1 or R0,R3,C3) */
static int cs_bfs (const cs *A, int n, int *wi, int *wj, int *queue,
const int *imatch, const int *jmatch, int mark)
{
int *Ap, *Ai, head = 0, tail = 0, j, i, p, j2 ;
cs *C ;
for (j = 0 ; j < n ; j++) /* place all unmatched nodes in queue */
{
if (imatch [j] >= 0) continue ; /* skip j if matched */
wj [j] = 0 ; /* j in set C0 (R0 if transpose) */
queue [tail++] = j ; /* place unmatched col j in queue */
}
if (tail == 0) return (1) ; /* quick return if no unmatched nodes */
C = (mark == 1) ? ((cs *) A) : cs_transpose (A, 0) ;
if (!C) return (0) ; /* bfs of C=A' to find R0,R3,C3 */
Ap = C->p ; Ai = C->i ;
while (head < tail) /* while queue is not empty */
{
j = queue [head++] ; /* get the head of the queue */
for (p = Ap [j] ; p < Ap [j+1] ; p++)
{
i = Ai [p] ;
if (wi [i] >= 0) continue ; /* skip if i is marked */
wi [i] = mark ; /* i in set R1 (C3 if transpose) */
j2 = jmatch [i] ; /* traverse alternating path to j2 */
if (wj [j2] >= 0) continue ;/* skip j2 if it is marked */
wj [j2] = mark ; /* j2 in set C1 (R3 if transpose) */
queue [tail++] = j2 ; /* add j2 to queue */
}
}
if (mark != 1) cs_spfree (C) ; /* free A' if it was created */
return (1) ;
}
/* collect matched rows and columns into P and Q */
static void cs_matched (int m, const int *wi, const int *jmatch, int *P, int *Q,
int *cc, int *rr, int set, int mark)
{
int kc = cc [set], i ;
int kr = rr [set-1] ;
for (i = 0 ; i < m ; i++)
{
if (wi [i] != mark) continue ; /* skip if i is not in R set */
P [kr++] = i ;
Q [kc++] = jmatch [i] ;
}
cc [set+1] = kc ;
rr [set] = kr ;
}
static void cs_unmatched (int m, const int *wi, int *P, int *rr, int set)
/*
Purpose:
CS_UNMATCHED collects unmatched rows into the permutation vector P.
*/
{
int i, kr = rr [set] ;
for (i = 0 ; i < m ; i++) if (wi [i] == 0) P [kr++] = i ;
rr [set+1] = kr ;
}
/* return 1 if row i is in R2 */
static int cs_rprune (int i, int j, double aij, void *other)
{
int *rr = (int *) other ;
return (i >= rr [1] && i < rr [2]) ;
}
/* Given A, find coarse dmperm */
csd *cs_dmperm (const cs *A)
{
int m, n, i, j, k, p, cnz, nc, *jmatch, *imatch, *wi, *wj, *Pinv, *Cp, *Ci,
*Ps, *Rs, nb1, nb2, *P, *Q, *cc, *rr, *R, *S, ok ;
cs *C ;
csd *D, *scc ;
/* --- Maximum matching ------------------------------------------------- */
if (!A) return (NULL) ; /* check inputs */
m = A->m ; n = A->n ;
D = cs_dalloc (m, n) ; /* allocate result */
if (!D) return (NULL) ;
P = D->P ; Q = D->Q ; R = D->R ; S = D->S ; cc = D->cc ; rr = D->rr ;
jmatch = cs_maxtrans (A) ; /* max transversal */
imatch = jmatch + m ; /* imatch = inverse of jmatch */
if (!jmatch) return (cs_ddone (D, NULL, jmatch, 0)) ;
/* --- Coarse decomposition --------------------------------------------- */
wi = R ; wj = S ; /* use R and S as workspace */
for (j = 0 ; j < n ; j++) wj [j] = -1 ; /* unmark all cols for bfs */
for (i = 0 ; i < m ; i++) wi [i] = -1 ; /* unmark all rows for bfs */
cs_bfs (A, n, wi, wj, Q, imatch, jmatch, 1) ; /* find C0, C1, R1 */
ok = cs_bfs (A, m, wj, wi, P, jmatch, imatch, 3) ; /* find R0, R3, C3 */
if (!ok) return (cs_ddone (D, NULL, jmatch, 0)) ;
cs_unmatched (n, wj, Q, cc, 0) ; /* unmatched set C0 */
cs_matched (m, wi, jmatch, P, Q, cc, rr, 1, 1) ; /* set R1 and C1 */
cs_matched (m, wi, jmatch, P, Q, cc, rr, 2, -1) ; /* set R2 and C2 */
cs_matched (m, wi, jmatch, P, Q, cc, rr, 3, 3) ; /* set R3 and C3 */
cs_unmatched (m, wi, P, rr, 3) ; /* unmatched set R0 */
cs_free (jmatch) ;
/* --- Fine decomposition ----------------------------------------------- */
Pinv = cs_pinv (P, m) ; /* Pinv=P' */
if (!Pinv) return (cs_ddone (D, NULL, NULL, 0)) ;
C = cs_permute (A, Pinv, Q, 0) ;/* C=A(P,Q) (it will hold A(R2,C2)) */
cs_free (Pinv) ;
if (!C) return (cs_ddone (D, NULL, NULL, 0)) ;
Cp = C->p ; Ci = C->i ;
nc = cc [3] - cc [2] ; /* delete cols C0, C1, and C3 from C */
if (cc [2] > 0) for (j = cc [2] ; j <= cc [3] ; j++) Cp [j-cc[2]] = Cp [j] ;
C->n = nc ;
if (rr [2] - rr [1] < m) /* delete rows R0, R1, and R3 from C */
{
cs_fkeep (C, cs_rprune, rr) ;
cnz = Cp [nc] ;
if (rr [1] > 0) for (p = 0 ; p < cnz ; p++) Ci [p] -= rr [1] ;
}
C->m = nc ;
scc = cs_scc (C) ; /* find strongly-connected components of C*/
if (!scc) return (cs_ddone (D, C, NULL, 0)) ;
/* --- Combine coarse and fine decompositions --------------------------- */
Ps = scc->P ; /* C(Ps,Ps) is the permuted matrix */
Rs = scc->R ; /* kth block is Rs[k]..Rs[k+1]-1 */
nb1 = scc->nb ; /* # of blocks of A(*/
for (k = 0 ; k < nc ; k++) wj [k] = Q [Ps [k] + cc [2]] ; /* combine */
for (k = 0 ; k < nc ; k++) Q [k + cc [2]] = wj [k] ;
for (k = 0 ; k < nc ; k++) wi [k] = P [Ps [k] + rr [1]] ;
for (k = 0 ; k < nc ; k++) P [k + rr [1]] = wi [k] ;
nb2 = 0 ; /* create the fine block partitions */
R [0] = 0 ;
S [0] = 0 ;
if (cc [2] > 0) nb2++ ; /* leading coarse block A (R1, [C0 C1]) */
for (k = 0 ; k < nb1 ; k++) /* coarse block A (R2,C2) */
{
R [nb2] = Rs [k] + rr [1] ; /* A (R2,C2) splits into nb1 fine blocks */
S [nb2] = Rs [k] + cc [2] ;
nb2++ ;
}
if (rr [2] < m)
{
R [nb2] = rr [2] ; /* trailing coarse block A ([R3 R0], C3) */
S [nb2] = cc [3] ;
nb2++ ;
}
R [nb2] = m ;
S [nb2] = n ;
D->nb = nb2 ;
cs_dfree (scc) ;
return (cs_ddone (D, C, NULL, 1)) ;
}
static int cs_tol (int i, int j, double aij, void *tol)
{
return (fabs (aij) > *((double *) tol)) ;
}
int cs_droptol (cs *A, double tol)
{
return (cs_fkeep (A, &cs_tol, &tol)) ; /* keep all large entries */
}
static int cs_nonzero (int i, int j, double aij, void *other)
{
return (aij != 0) ;
}
int cs_dropzeros (cs *A)
{
return (cs_fkeep (A, &cs_nonzero, NULL)) ; /* keep all nonzero entries */
}
int cs_dupl (cs *A)
/*
Purpose:
CS_DUPL removes duplicate entries from A.
Reference:
Timothy Davis,
Direct Methods for Sparse Linear Systems,
SIAM, Philadelphia, 2006.
*/
{
int i, j, p, q, nz = 0, n, m, *Ap, *Ai, *w ;
double *Ax ;
if (!A) return (0) ; /* check inputs */
m = A->m ; n = A->n ; Ap = A->p ; Ai = A->i ; Ax = A->x ;
w = (int*)cs_malloc (m, sizeof (int)) ; /* get workspace */
if (!w) return (0) ; /* out of memory */
for (i = 0 ; i < m ; i++) w [i] = -1 ; /* row i not yet seen */
for (j = 0 ; j < n ; j++)
{
q = nz ; /* column j will start at q */
for (p = Ap [j] ; p < Ap [j+1] ; p++)
{
i = Ai [p] ; /* A(i,j) is nonzero */
if (w [i] >= q)
{
Ax [w [i]] += Ax [p] ; /* A(i,j) is a duplicate */
}
else
{
w [i] = nz ; /* record where row i occurs */
Ai [nz] = i ; /* keep A(i,j) */
Ax [nz++] = Ax [p] ;
}
}
Ap [j] = q ; /* record start of column j */
}
Ap [n] = nz ; /* finalize A */
cs_free (w) ; /* free workspace */
return (cs_sprealloc (A, 0)) ; /* remove extra space from A */
}
/* add an entry to a triplet matrix; return 1 if ok, 0 otherwise */
int cs_entry (cs *T, int i, int j, double x)
{
if (!T || (T->nz >= T->nzmax && !cs_sprealloc (T, 2*(T->nzmax)))) return(0);
if (T->x) T->x [T->nz] = x ;
T->i [T->nz] = i ;
T->p [T->nz++] = j ;
T->m = CS_MAX (T->m, i+1) ;
T->n = CS_MAX (T->n, j+1) ;
return (1) ;
}
/* compute the etree of A (using triu(A), or A'A without forming A'A */
int *cs_etree (const cs *A, int ata)
{
int i, k, p, m, n, inext, *Ap, *Ai, *w, *parent, *ancestor, *prev ;
if (!A) return (NULL) ; /* check inputs */
m = A->m ; n = A->n ; Ap = A->p ; Ai = A->i ;
parent = (int*)cs_malloc (n, sizeof (int)) ;
w = (int*)cs_malloc (n + (ata ? m : 0), sizeof (int)) ;
ancestor = w ; prev = w + n ;
if (!w || !parent) return (cs_idone (parent, NULL, w, 0)) ;
if (ata) for (i = 0 ; i < m ; i++) prev [i] = -1 ;
for (k = 0 ; k < n ; k++)
{
parent [k] = -1 ; /* node k has no parent yet */
ancestor [k] = -1 ; /* nor does k have an ancestor */
for (p = Ap [k] ; p < Ap [k+1] ; p++)
{
i = ata ? (prev [Ai [p]]) : (Ai [p]) ;
for ( ; i != -1 && i < k ; i = inext) /* traverse from i to k */
{
inext = ancestor [i] ; /* inext = ancestor of i */
ancestor [i] = k ; /* path compression */
if (inext == -1) parent [i] = k ; /* no anc., parent is k */
}
if (ata) prev [Ai [p]] = k ;
}
}
return (cs_idone (parent, NULL, w, 1)) ;
}
/* drop entries for which fkeep(A(i,j)) is false; return nz if OK, else -1 */
int cs_fkeep (cs *A, int (*fkeep) (int, int, double, void *), void *other)
{
int j, p, nz = 0, n, *Ap, *Ai ;
double *Ax ;
if (!A || !fkeep) return (-1) ; /* check inputs */
n = A->n ; Ap = A->p ; Ai = A->i ; Ax = A->x ;
for (j = 0 ; j < n ; j++)
{
p = Ap [j] ; /* get current location of col j */
Ap [j] = nz ; /* record new location of col j */
for ( ; p < Ap [j+1] ; p++)
{
if (fkeep (Ai [p], j, Ax ? Ax [p] : 1, other))
{
if (Ax) Ax [nz] = Ax [p] ; /* keep A(i,j) */
Ai [nz++] = Ai [p] ;
}
}
}
return (Ap [n] = nz) ; /* finalize A and return nnz(A) */
}
/* y = A*x+y */
int cs_gaxpy (const cs *A, const double *x, double *y)
{
int p, j, n, *Ap, *Ai ;
double *Ax ;
if (!A || !x || !y) return (0) ; /* check inputs */
n = A->n ; Ap = A->p ; Ai = A->i ; Ax = A->x ;
for (j = 0 ; j < n ; j++)
{
for (p = Ap [j] ; p < Ap [j+1] ; p++)
{
y [Ai [p]] += Ax [p] * x [j] ;
}
}
return (1) ;
}
/* apply the ith Householder vector to x */
int cs_happly (const cs *V, int i, double beta, double *x)
{
int p, *Vp, *Vi ;
double *Vx, tau = 0 ;
if (!V || !x) return (0) ; /* check inputs */
Vp = V->p ; Vi = V->i ; Vx = V->x ;
for (p = Vp [i] ; p < Vp [i+1] ; p++) /* tau = v'*x */
{
tau += Vx [p] * x [Vi [p]] ;
}
tau *= beta ; /* tau = beta*(v'*x) */
for (p = Vp [i] ; p < Vp [i+1] ; p++) /* x = x - v*tau */
{
x [Vi [p]] -= Vx [p] * tau ;
}
return (1) ;
}
/* create a Householder reflection [v,beta,s]=house(x), overwrite x with v,
* where (I-beta*v*v')*x = s*x. See Algo 5.1.1, Golub & Van Loan, 3rd ed. */
double cs_house (double *x, double *beta, int n)
{
double s, sigma = 0 ;
int i ;
if (!x || !beta) return (-1) ; /* check inputs */
for (i = 1 ; i < n ; i++) sigma += x [i] * x [i] ;
if (sigma == 0)
{
s = fabs (x [0]) ; /* s = |x(0)| */
(*beta) = (x [0] <= 0) ? 2 : 0 ;
x [0] = 1 ;
}
else
{
s = sqrt (x [0] * x [0] + sigma) ; /* s = norm (x) */
x [0] = (x [0] <= 0) ? (x [0] - s) : (-sigma / (x [0] + s)) ;
(*beta) = -1. / (s * x [0]) ;
}
return (s) ;
}
/* x(P) = b, for dense vectors x and b; P=NULL denotes identity */
int cs_ipvec (int n, const int *P, const double *b, double *x)
{
int k ;
if (!x || !b) return (0) ; /* check inputs */
for (k = 0 ; k < n ; k++) x [P ? P [k] : k] = b [k] ;
return (1) ;
}
cs *cs_load ( FILE *f )
/*
Purpose:
CS_LOAD loads a triplet matrix from a file.
Reference:
Timothy Davis,
Direct Methods for Sparse Linear Systems,
SIAM, Philadelphia, 2006.
*/
{
int i, j ;
double x ;
cs *T ;
if (!f) return (NULL) ;
T = cs_spalloc (0, 0, 1, 1, 1) ;
while (fscanf (f, "%d %d %lg\n", &i, &j, &x) == 3)
{
if (!cs_entry (T, i, j, x)) return (cs_spfree (T)) ;
}
return (T) ;
}
int cs_lsolve ( const cs *L, double *x )
/*
Purpose:
CS_LSOLVE solves L*x=b.
Discussion:
On input, X contains the right hand side, and on output, the solution.
Reference:
Timothy Davis,
Direct Methods for Sparse Linear Systems,
SIAM, Philadelphia, 2006.
*/
{
int p, j, n, *Lp, *Li ;
double *Lx ;
if (!L || !x) return (0) ; /* check inputs */
n = L->n ; Lp = L->p ; Li = L->i ; Lx = L->x ;
for (j = 0 ; j < n ; j++)
{
x [j] /= Lx [Lp [j]] ;
for (p = Lp [j]+1 ; p < Lp [j+1] ; p++)
{
x [Li [p]] -= Lx [p] * x [j] ;
}
}
return (1) ;
}
int cs_ltsolve ( const cs *L, double *x )
/*
Purpose:
CS_LTSOLVE solves L'*x=b.
Discussion:
On input, X contains the right hand side, and on output, the solution.
Reference:
Timothy Davis,
Direct Methods for Sparse Linear Systems,
SIAM, Philadelphia, 2006.
*/
{
int p, j, n, *Lp, *Li ;
double *Lx ;
if (!L || !x) return (0) ; /* check inputs */
n = L->n ; Lp = L->p ; Li = L->i ; Lx = L->x ;
for (j = n-1 ; j >= 0 ; j--)
{
for (p = Lp [j]+1 ; p < Lp [j+1] ; p++)
{
x [j] -= Lx [p] * x [Li [p]] ;
}
x [j] /= Lx [Lp [j]] ;
}
return (1) ;
}
/* [L,U,Pinv]=lu(A, [Q lnz unz]). lnz and unz can be guess */
csn *cs_lu (const cs *A, const css *S, double tol)
{
cs *L, *U ;
csn *N ;
double pivot, *Lx, *Ux, *x, a, t ;
int *Lp, *Li, *Up, *Ui, *Pinv, *xi, *Q, n, ipiv, k, top, p, i, col, lnz,unz;
if (!A || !S) return (NULL) ; /* check inputs */
n = A->n ;
Q = S->Q ; lnz = S->lnz ; unz = S->unz ;
x = (double*)cs_malloc (n, sizeof (double)) ;
xi = (int*)cs_malloc (2*n, sizeof (int)) ;
N = (csn*)cs_calloc (1, sizeof (csn)) ;
if (!x || !xi || !N) return (cs_ndone (N, NULL, xi, x, 0)) ;
N->L = L = cs_spalloc (n, n, lnz, 1, 0) ; /* initial L and U */
N->U = U = cs_spalloc (n, n, unz, 1, 0) ;
N->Pinv = Pinv = (int*)cs_malloc (n, sizeof (int)) ;
if (!L || !U || !Pinv) return (cs_ndone (N, NULL, xi, x, 0)) ;
Lp = L->p ; Up = U->p ;
for (i = 0 ; i < n ; i++) x [i] = 0 ; /* clear workspace */
for (i = 0 ; i < n ; i++) Pinv [i] = -1 ; /* no rows pivotal yet */
for (k = 0 ; k <= n ; k++) Lp [k] = 0 ; /* no cols of L yet */
lnz = unz = 0 ;
for (k = 0 ; k < n ; k++) /* compute L(:,k) and U(:,k) */
{
/* --- Triangular solve --------------------------------------------- */
Lp [k] = lnz ; /* L(:,k) starts here */
Up [k] = unz ; /* U(:,k) starts here */
if ((lnz + n > L->nzmax && !cs_sprealloc (L, 2*L->nzmax + n)) ||
(unz + n > U->nzmax && !cs_sprealloc (U, 2*U->nzmax + n)))
{
return (cs_ndone (N, NULL, xi, x, 0)) ;
}
Li = L->i ; Lx = L->x ; Ui = U->i ; Ux = U->x ;
col = Q ? (Q [k]) : k ;
top = cs_splsolve (L, A, col, xi, x, Pinv) ; /* x = L\A(:,col) */
/* --- Find pivot --------------------------------------------------- */
ipiv = -1 ;
a = -1 ;
for (p = top ; p < n ; p++)
{
i = xi [p] ; /* x(i) is nonzero */
if (Pinv [i] < 0) /* row i is not pivotal */
{
if ((t = fabs (x [i])) > a)
{
a = t ; /* largest pivot candidate so far */
ipiv = i ;
}
}
else /* x(i) is the entry U(Pinv[i],k) */
{
Ui [unz] = Pinv [i] ;
Ux [unz++] = x [i] ;
}
}
if (ipiv == -1 || a <= 0) return (cs_ndone (N, NULL, xi, x, 0)) ;
if (Pinv [col] < 0 && fabs (x [col]) >= a*tol) ipiv = col ;
/* --- Divide by pivot ---------------------------------------------- */
pivot = x [ipiv] ; /* the chosen pivot */
Ui [unz] = k ; /* last entry in U(:,k) is U(k,k) */
Ux [unz++] = pivot ;
Pinv [ipiv] = k ; /* ipiv is the kth pivot row */
Li [lnz] = ipiv ; /* first entry in L(:,k) is L(k,k) = 1 */
Lx [lnz++] = 1 ;
for (p = top ; p < n ; p++) /* L(k+1:n,k) = x / pivot */
{
i = xi [p] ;
if (Pinv [i] < 0) /* x(i) is an entry in L(:,k) */
{
Li [lnz] = i ; /* save unpermuted row in L */
Lx [lnz++] = x [i] / pivot ; /* scale pivot column */
}
x [i] = 0 ; /* x [0..n-1] = 0 for next k */
}
}
/* --- Finalize L and U ------------------------------------------------- */
Lp [n] = lnz ;
Up [n] = unz ;
Li = L->i ; /* fix row indices of L for final Pinv */
for (p = 0 ; p < lnz ; p++) Li [p] = Pinv [Li [p]] ;
cs_sprealloc (L, 0) ; /* remove extra space from L and U */
cs_sprealloc (U, 0) ;
return (cs_ndone (N, NULL, xi, x, 1)) ; /* success */
}
/* x=A\b where A is unsymmetric; b overwritten with solution */
int cs_lusol (const cs *A, double *b, int order, double tol)
{
double *x ;
css *S ;
csn *N ;
int n, ok ;
if (!A || !b) return (0) ; /* check inputs */
n = A->n ;
S = cs_sqr (A, order, 0) ; /* ordering and symbolic analysis */
N = cs_lu (A, S, tol) ; /* numeric LU factorization */
x = (double*)cs_malloc (n, sizeof (double)) ;
ok = (S && N && x) ;
if (ok)
{
cs_ipvec (n, N->Pinv, b, x) ; /* x = P*b */
cs_lsolve (N->L, x) ; /* x = L\x */
cs_usolve (N->U, x) ; /* x = U\x */
cs_ipvec (n, S->Q, x, b) ; /* b = Q*x */
}
cs_free (x) ;
cs_sfree (S) ;
cs_nfree (N) ;
return (ok) ;
}
#ifdef MATLAB_MEX_FILE
#define malloc mxMalloc
#define free mxFree
#define realloc mxRealloc
#define calloc mxCalloc
#endif
/* wrapper for malloc */
void *cs_malloc (int n, size_t size)
{
return (CS_OVERFLOW (n,size) ? NULL : malloc (CS_MAX (n,1) * size)) ;
}
/* wrapper for calloc */
void *cs_calloc (int n, size_t size)
{
return (CS_OVERFLOW (n,size) ? NULL : calloc (CS_MAX (n,1), size)) ;
}
/* wrapper for free */
void *cs_free (void *p)
{
if (p) free (p) ; /* free p if it is not already NULL */
return (NULL) ; /* return NULL to simplify the use of cs_free */
}
/* wrapper for realloc */
void *cs_realloc (void *p, int n, size_t size, int *ok)
{
void *p2 ;
*ok = !CS_OVERFLOW (n,size) ; /* guard against int overflow */
if (!(*ok)) return (p) ; /* p unchanged if n too large */
p2 = realloc (p, CS_MAX (n,1) * size) ; /* realloc the block */
*ok = (p2 != NULL) ;
return ((*ok) ? p2 : p) ; /* return original p if failure */
}
/* find an augmenting path starting at column k and extend the match if found */
static void cs_augment (int k, const cs *A, int *jmatch, int *cheap, int *w,
int *js, int *is, int *ps)
{
int found = 0, p, i = -1, *Ap = A->p, *Ai = A->i, head = 0, j ;
js [0] = k ; /* start with just node k in jstack */
while (head >= 0)
{
/* --- Start (or continue) depth-first-search at node j ------------- */
j = js [head] ; /* get j from top of jstack */
if (w [j] != k) /* 1st time j visited for kth path */
{
w [j] = k ; /* mark j as visited for kth path */
for (p = cheap [j] ; p < Ap [j+1] && !found ; p++)
{
i = Ai [p] ; /* try a cheap assignment (i,j) */
found = (jmatch [i] == -1) ;
}
cheap [j] = p ; /* start here next time j is traversed*/
if (found)
{
is [head] = i ; /* column j matched with row i */
break ; /* end of augmenting path */
}
ps [head] = Ap [j] ; /* no cheap match: start dfs for j */
}
/* --- Depth-first-search of neighbors of j ------------------------- */
for (p = ps [head] ; p < Ap [j+1] ; p++)
{
i = Ai [p] ; /* consider row i */
if (w [jmatch [i]] == k) continue ; /* skip jmatch [i] if marked */
ps [head] = p + 1 ; /* pause dfs of node j */
is [head] = i ; /* i will be matched with j if found */
js [++head] = jmatch [i] ; /* start dfs at column jmatch [i] */
break ;
}
if (p == Ap [j+1]) head-- ; /* node j is done; pop from stack */
} /* augment the match if path found: */
if (found) for (p = head ; p >= 0 ; p--) jmatch [is [p]] = js [p] ;
}
/* find a maximum transveral */
int *cs_maxtrans (const cs *A) /* returns jmatch [0..m-1]; imatch [0..n-1] */
{
int i, j, k, n, m, p, n2 = 0, m2 = 0, *Ap, *jimatch, *w, *cheap, *js, *is,
*ps, *Ai, *Cp, *jmatch, *imatch ;
cs *C ;
if (!A) return (NULL) ; /* check inputs */
n = A->n ; m = A->m ; Ap = A->p ; Ai = A->i ;
w = jimatch = (int*)cs_calloc (m+n, sizeof (int)) ; /* allocate result */
if (!jimatch) return (NULL) ;
for (j = 0 ; j < n ; j++) /* count non-empty rows and columns */
{
n2 += (Ap [j] < Ap [j+1]) ;
for (p = Ap [j] ; p < Ap [j+1] ; p++) w [Ai [p]] = 1 ;
}
for (i = 0 ; i < m ; i++) m2 += w [i] ;
C = (m2 < n2) ? cs_transpose (A,0) : ((cs *) A) ; /* transpose if needed */
if (!C) return (cs_idone (jimatch, (m2 < n2) ? C : NULL, NULL, 0)) ;
n = C->n ; m = C->m ; Cp = C->p ;
jmatch = (m2 < n2) ? jimatch + n : jimatch ;
imatch = (m2 < n2) ? jimatch : jimatch + m ;
w = (int*)cs_malloc (5*n, sizeof (int)) ; /* allocate workspace */
if (!w) return (cs_idone (jimatch, (m2 < n2) ? C : NULL, w, 0)) ;
cheap = w + n ; js = w + 2*n ; is = w + 3*n ; ps = w + 4*n ;
for (j = 0 ; j < n ; j++) cheap [j] = Cp [j] ; /* for cheap assignment */
for (j = 0 ; j < n ; j++) w [j] = -1 ; /* all columns unflagged */
for (i = 0 ; i < m ; i++) jmatch [i] = -1 ; /* nothing matched yet */
for (k = 0 ; k < n ; k++) cs_augment (k, C, jmatch, cheap, w, js, is, ps) ;
for (j = 0 ; j < n ; j++) imatch [j] = -1 ; /* find row match */
for (i = 0 ; i < m ; i++) if (jmatch [i] >= 0) imatch [jmatch [i]] = i ;
return (cs_idone (jimatch, (m2 < n2) ? C : NULL, w, 1)) ;
}
/* C = A*B */
cs *cs_multiply (const cs *A, const cs *B)
{
int p, j, nz = 0, anz, *Cp, *Ci, *Bp, m, n, bnz, *w, values, *Bi ;
double *x, *Bx, *Cx ;
cs *C ;
if (!A || !B) return (NULL) ; /* check inputs */
m = A->m ; anz = A->p [A->n] ;
n = B->n ; Bp = B->p ; Bi = B->i ; Bx = B->x ; bnz = Bp [n] ;
w = (int*)cs_calloc (m, sizeof (int)) ;
values = (A->x != NULL) && (Bx != NULL) ;
x = values ? (double*)cs_malloc (m, sizeof (double)) : NULL ;
C = cs_spalloc (m, n, anz + bnz, values, 0) ;
if (!C || !w || (values && !x)) return (cs_done (C, w, x, 0)) ;
Cp = C->p ;
for (j = 0 ; j < n ; j++)
{
if (nz + m > C->nzmax && !cs_sprealloc (C, 2*(C->nzmax)+m))
{
return (cs_done (C, w, x, 0)) ; /* out of memory */
}
Ci = C->i ; Cx = C->x ; /* C may have been reallocated */
Cp [j] = nz ; /* column j of C starts here */
for (p = Bp [j] ; p < Bp [j+1] ; p++)
{
nz = cs_scatter (A, Bi [p], Bx ? Bx [p] : 1, w, x, j+1, C, nz) ;
}
if (values) for (p = Cp [j] ; p < nz ; p++) Cx [p] = x [Ci [p]] ;
}
Cp [n] = nz ; /* finalize the last column of C */
cs_sprealloc (C, 0) ; /* remove extra space from C */
return (cs_done (C, w, x, 1)) ; /* success; free workspace, return C */
}
/* 1-norm of a sparse matrix = max (sum (abs (A))), largest column sum */
double cs_norm (const cs *A)
{
int p, j, n, *Ap ;
double *Ax, norm = 0, s ;
if (!A || !A->x) return (-1) ; /* check inputs */
n = A->n ; Ap = A->p ; Ax = A->x ;
for (j = 0 ; j < n ; j++)
{
for (s = 0, p = Ap [j] ; p < Ap [j+1] ; p++) s += fabs (Ax [p]) ;
norm = CS_MAX (norm, s) ;
}
return (norm) ;
}
/* C = A(P,Q) where P and Q are permutations of 0..m-1 and 0..n-1. */
cs *cs_permute (const cs *A, const int *Pinv, const int *Q, int values)
{
int p, j, k, nz = 0, m, n, *Ap, *Ai, *Cp, *Ci ;
double *Cx, *Ax ;
cs *C ;
if (!A) return (NULL) ; /* check inputs */
m = A->m ; n = A->n ; Ap = A->p ; Ai = A->i ; Ax = A->x ;
C = cs_spalloc (m, n, Ap [n], values && Ax != NULL, 0) ;
if (!C) return (cs_done (C, NULL, NULL, 0)) ; /* out of memory */
Cp = C->p ; Ci = C->i ; Cx = C->x ;
for (k = 0 ; k < n ; k++)
{
Cp [k] = nz ; /* column k of C is column Q[k] of A */
j = Q ? (Q [k]) : k ;
for (p = Ap [j] ; p < Ap [j+1] ; p++)
{
if (Cx) Cx [nz] = Ax [p] ; /* row i of A is row Pinv[i] of C */
Ci [nz++] = Pinv ? (Pinv [Ai [p]]) : Ai [p] ;
}
}
Cp [n] = nz ; /* finalize the last column of C */
return (cs_done (C, NULL, NULL, 1)) ;
}
/* Pinv = P', or P = Pinv' */
int *cs_pinv (int const *P, int n)
{
int k, *Pinv ;
if (!P) return (NULL) ; /* P = NULL denotes identity */
Pinv = (int*)cs_malloc (n, sizeof (int)) ; /* allocate resuult */
if (!Pinv) return (NULL) ; /* out of memory */
for (k = 0 ; k < n ; k++) Pinv [P [k]] = k ;/* invert the permutation */
return (Pinv) ; /* return result */
}
/* post order a forest */
int *cs_post (int n, const int *parent)
{
int j, k = 0, *post, *w, *head, *next, *stack ;
if (!parent) return (NULL) ; /* check inputs */
post = (int*)cs_malloc (n, sizeof (int)) ; /* allocate result */
w = (int*)cs_malloc (3*n, sizeof (int)) ; /* 3*n workspace */
head = w ; next = w + n ; stack = w + 2*n ;
if (!w || !post) return (cs_idone (post, NULL, w, 0)) ;
for (j = 0 ; j < n ; j++) head [j] = -1 ; /* empty link lists */
for (j = n-1 ; j >= 0 ; j--) /* traverse nodes in reverse order*/
{
if (parent [j] == -1) continue ; /* j is a root */
next [j] = head [parent [j]] ; /* add j to list of its parent */
head [parent [j]] = j ;
}
for (j = 0 ; j < n ; j++)
{
if (parent [j] != -1) continue ; /* skip j if it is not a root */
k = cs_tdfs (j, k, head, next, post, stack) ;
}
return (cs_idone (post, NULL, w, 1)) ; /* success; free w, return post */
}
/* print a sparse matrix */
int cs_print (const cs *A, int brief)
{
int p, j, m, n, nzmax, nz, *Ap, *Ai ;
double *Ax ;
if (!A) { printf ("(null)\n") ; return (0) ; }
m = A->m ; n = A->n ; Ap = A->p ; Ai = A->i ; Ax = A->x ;
nzmax = A->nzmax ; nz = A->nz ;
printf ("CSparse Version %d.%d.%d, %s. %s\n", CS_VER, CS_SUBVER,
CS_SUBSUB, CS_DATE, CS_COPYRIGHT) ;
if (nz < 0)
{
printf ("%d-by-%d, nzmax: %d nnz: %d, 1-norm: %g\n", m, n, nzmax,
Ap [n], cs_norm (A)) ;
for (j = 0 ; j < n ; j++)
{
printf (" col %d : locations %d to %d\n", j, Ap [j], Ap [j+1]-1);
for (p = Ap [j] ; p < Ap [j+1] ; p++)
{
printf (" %d : %g\n", Ai [p], Ax ? Ax [p] : 1) ;
if (brief && p > 20) { printf (" ...\n") ; return (1) ; }
}
}
}
else
{
printf ("triplet: %d-by-%d, nzmax: %d nnz: %d\n", m, n, nzmax, nz) ;
for (p = 0 ; p < nz ; p++)
{
printf (" %d %d : %g\n", Ai [p], Ap [p], Ax ? Ax [p] : 1) ;
if (brief && p > 20) { printf (" ...\n") ; return (1) ; }
}
}
return (1) ;
}
/* x = b(P), for dense vectors x and b; P=NULL denotes identity */
int cs_pvec (int n, const int *P, const double *b, double *x)
{
int k ;
if (!x || !b) return (0) ; /* check inputs */
for (k = 0 ; k < n ; k++) x [k] = b [P ? P [k] : k] ;
return (1) ;
}
/* sparse QR factorization [V,beta,p,R] = qr (A) */
csn *cs_qr (const cs *A, const css *S)
{
double *Rx, *Vx, *Ax, *Beta, *x ;
int i, k, p, m, n, vnz, p1, top, m2, len, col, rnz, *s, *leftmost, *Ap,
*Ai, *parent, *Rp, *Ri, *Vp, *Vi, *w, *Pinv, *Q ;
cs *R, *V ;
csn *N ;
if (!A || !S || !S->parent || !S->Pinv) return (NULL) ; /* check inputs */
m = A->m ; n = A->n ; Ap = A->p ; Ai = A->i ; Ax = A->x ;
Q = S->Q ; parent = S->parent ; Pinv = S->Pinv ; m2 = S->m2 ;
vnz = S->lnz ; rnz = S->unz ;
leftmost = Pinv + m + n ;
w = (int*)cs_malloc (m2+n, sizeof (int)) ;
x = (double*)cs_malloc (m2, sizeof (double)) ;
N = (csn*)cs_calloc (1, sizeof (csn)) ;
if (!w || !x || !N) return (cs_ndone (N, NULL, w, x, 0)) ;
s = w + m2 ; /* size n */
for (k = 0 ; k < m2 ; k++) x [k] = 0 ; /* clear workspace x */
N->L = V = cs_spalloc (m2, n, vnz, 1, 0) ; /* allocate V */
N->U = R = cs_spalloc (m2, n, rnz, 1, 0) ; /* allocate R, m2-by-n */
N->B = Beta = (double*)cs_malloc (n, sizeof (double)) ;
if (!R || !V || !Beta) return (cs_ndone (N, NULL, w, x, 0)) ;
Rp = R->p ; Ri = R->i ; Rx = R->x ;
Vp = V->p ; Vi = V->i ; Vx = V->x ;
for (i = 0 ; i < m2 ; i++) w [i] = -1 ; /* clear w, to mark nodes */
rnz = 0 ; vnz = 0 ;
for (k = 0 ; k < n ; k++) /* compute V and R */
{
Rp [k] = rnz ; /* R(:,k) starts here */
Vp [k] = p1 = vnz ; /* V(:,k) starts here */
w [k] = k ; /* add V(k,k) to pattern of V */
Vi [vnz++] = k ;
top = n ;
col = Q ? Q [k] : k ;
for (p = Ap [col] ; p < Ap [col+1] ; p++) /* find R(:,k) pattern */
{
i = leftmost [Ai [p]] ; /* i = min(find(A(i,Q))) */
for (len = 0 ; w [i] != k ; i = parent [i]) /* traverse up to k */
{
s [len++] = i ;
w [i] = k ;
}
while (len > 0) s [--top] = s [--len] ; /* push path on stack */
i = Pinv [Ai [p]] ; /* i = permuted row of A(:,col) */
x [i] = Ax [p] ; /* x (i) = A(.,col) */
if (i > k && w [i] < k) /* pattern of V(:,k) = x (k+1:m) */
{
Vi [vnz++] = i ; /* add i to pattern of V(:,k) */
w [i] = k ;
}
}
for (p = top ; p < n ; p++) /* for each i in pattern of R(:,k) */
{
i = s [p] ; /* R(i,k) is nonzero */
cs_happly (V, i, Beta [i], x) ; /* apply (V(i),Beta(i)) to x */
Ri [rnz] = i ; /* R(i,k) = x(i) */
Rx [rnz++] = x [i] ;
x [i] = 0 ;
if (parent [i] == k) vnz = cs_scatter (V, i, 0, w, NULL, k, V, vnz);
}
for (p = p1 ; p < vnz ; p++) /* gather V(:,k) = x */
{
Vx [p] = x [Vi [p]] ;
x [Vi [p]] = 0 ;
}
Ri [rnz] = k ; /* R(k,k) = norm (x) */
Rx [rnz++] = cs_house (Vx+p1, Beta+k, vnz-p1) ; /* [v,beta]=house(x) */
}
Rp [n] = rnz ; /* finalize R */
Vp [n] = vnz ; /* finalize V */
return (cs_ndone (N, NULL, w, x, 1)) ; /* success */
}
/* x=A\b where A can be rectangular; b overwritten with solution */
int cs_qrsol (const cs *A, double *b, int order)
{
double *x ;
css *S ;
csn *N ;
cs *AT = NULL ;
int k, m, n, ok ;
if (!A || !b) return (0) ; /* check inputs */
n = A->n ;
m = A->m ;
if (m >= n)
{
S = cs_sqr (A, order, 1) ; /* ordering and symbolic analysis */
N = cs_qr (A, S) ; /* numeric QR factorization */
x = (double*)cs_calloc (S ? S->m2 : 1, sizeof (double)) ;
ok = (S && N && x) ;
if (ok)
{
cs_ipvec (m, S->Pinv, b, x) ; /* x(0:m-1) = P*b(0:m-1) */
for (k = 0 ; k < n ; k++) /* apply Householder refl. to x */
{
cs_happly (N->L, k, N->B [k], x) ;
}
cs_usolve (N->U, x) ; /* x = R\x */
cs_ipvec (n, S->Q, x, b) ; /* b(0:n-1) = Q*x (permutation) */
}
}
else
{
AT = cs_transpose (A, 1) ; /* Ax=b is underdetermined */
S = cs_sqr (AT, order, 1) ; /* ordering and symbolic analysis */
N = cs_qr (AT, S) ; /* numeric QR factorization of A' */
x = (double*)cs_calloc (S ? S->m2 : 1, sizeof (double)) ;
ok = (AT && S && N && x) ;
if (ok)
{
cs_pvec (m, S->Q, b, x) ; /* x(0:m-1) = Q'*b (permutation) */
cs_utsolve (N->U, x) ; /* x = R'\x */
for (k = m-1 ; k >= 0 ; k--) /* apply Householder refl. to x */
{
cs_happly (N->L, k, N->B [k], x) ;
}
cs_pvec (n, S->Pinv, x, b) ; /* b (0:n-1) = P'*x */
}
}
cs_free (x) ;
cs_sfree (S) ;
cs_nfree (N) ;
cs_spfree (AT) ;
return (ok) ;
}
/* xi [top...n-1] = nodes reachable from graph of L*P' via nodes in B(:,k).
* xi [n...2n-1] used as workspace */
int cs_reach (cs *L, const cs *B, int k, int *xi, const int *Pinv)
{
int p, n, top, *Bp, *Bi, *Lp ;
if (!L || !B || !xi) return (-1) ;
n = L->n ; Bp = B->p ; Bi = B->i ; Lp = L->p ;
top = n ;
for (p = Bp [k] ; p < Bp [k+1] ; p++)
{
if (!CS_MARKED (Lp, Bi [p])) /* start a dfs at unmarked node i */
{
top = cs_dfs (Bi [p], L, top, xi, xi+n, Pinv) ;
}
}
for (p = top ; p < n ; p++) CS_MARK (Lp, xi [p]) ; /* restore L */
return (top) ;
}
/* x = x + beta * A(:,j), where x is a dense vector and A(:,j) is sparse */
int cs_scatter (const cs *A, int j, double beta, int *w, double *x, int mark,
cs *C, int nz)
{
int i, p, *Ap, *Ai, *Ci ;
double *Ax ;
if (!A || !w || !C) return (-1) ; /* ensure inputs are valid */
Ap = A->p ; Ai = A->i ; Ax = A->x ; Ci = C->i ;
for (p = Ap [j] ; p < Ap [j+1] ; p++)
{
i = Ai [p] ; /* A(i,j) is nonzero */
if (w [i] < mark)
{
w [i] = mark ; /* i is new entry in column j */
Ci [nz++] = i ; /* add i to pattern of C(:,j) */
if (x) x [i] = beta * Ax [p] ; /* x(i) = beta*A(i,j) */
}
else if (x) x [i] += beta * Ax [p] ; /* i exists in C(:,j) already */
}
return (nz) ;
}
/* find the strongly connected components of a square matrix */
csd *cs_scc (cs *A) /* matrix A temporarily modified, then restored */
{
int n, i, k, b = 0, top, *xi, *pstack, *P, *R, *Ap, *ATp ;
cs *AT ;
csd *D ;
if (!A) return (NULL) ;
n = A->n ; Ap = A->p ;
D = cs_dalloc (n, 0) ;
AT = cs_transpose (A, 0) ; /* AT = A' */
xi = (int*)cs_malloc (2*n, sizeof (int)) ; /* allocate workspace */
pstack = xi + n ;
if (!D || !AT || !xi) return (cs_ddone (D, AT, xi, 0)) ;
P = D->P ; R = D->R ; ATp = AT->p ;
top = n ;
for (i = 0 ; i < n ; i++) /* first dfs(A) to find finish times (xi) */
{
if (!CS_MARKED (Ap,i)) top = cs_dfs (i, A, top, xi, pstack, NULL) ;
}
for (i = 0 ; i < n ; i++) CS_MARK (Ap, i) ; /* restore A; unmark all nodes*/
top = n ;
b = n ;
for (k = 0 ; k < n ; k++) /* dfs(A') to find strongly connnected comp. */
{
i = xi [k] ; /* get i in reverse order of finish times */
if (CS_MARKED (ATp,i)) continue ; /* skip node i if already ordered */
R [b--] = top ; /* node i is the start of a component in P */
top = cs_dfs (i, AT, top, P, pstack, NULL) ;
}
R [b] = 0 ; /* first block starts at zero; shift R up */
for (k = b ; k <= n ; k++) R [k-b] = R [k] ;
D->nb = R [n+1] = b = n-b ; /* b = # of strongly connected components */
return (cs_ddone (D, AT, xi, 1)) ;
}
/* ordering and symbolic analysis for a Cholesky factorization */
css *cs_schol (const cs *A, int order)
{
int n, *c, *post, *P ;
cs *C ;
css *S ;
if (!A) return (NULL) ; /* check inputs */
n = A->n ;
S = (css*)cs_calloc (1, sizeof (css)) ; /* allocate symbolic analysis */
if (!S) return (NULL) ; /* out of memory */
P = cs_amd (A, order) ; /* P = amd(A+A'), or natural */
S->Pinv = cs_pinv (P, n) ; /* find inverse permutation */
cs_free (P) ;
if (order >= 0 && !S->Pinv) return (cs_sfree (S)) ;
C = cs_symperm (A, S->Pinv, 0) ; /* C = spones(triu(A(P,P))) */
S->parent = cs_etree (C, 0) ; /* find etree of C */
post = cs_post (n, S->parent) ; /* postorder the etree */
c = cs_counts (C, S->parent, post, 0) ; /* find column counts of chol(C) */
cs_free (post) ;
cs_spfree (C) ;
S->cp = (int*)cs_malloc (n+1, sizeof (int)) ; /* find column pointers for L */
S->unz = S->lnz = cs_cumsum (S->cp, c, n) ;
cs_free (c) ;
return ((S->lnz >= 0) ? S : cs_sfree (S)) ;
}
/* solve Lx=b(:,k), leaving pattern in xi[top..n-1], values scattered in x. */
int cs_splsolve (cs *L, const cs *B, int k, int *xi, double *x, const int *Pinv)
{
int j, jnew, p, px, top, n, *Lp, *Li, *Bp, *Bi ;
double *Lx, *Bx ;
if (!L || !B || !xi || !x) return (-1) ;
Lp = L->p ; Li = L->i ; Lx = L->x ; n = L->n ;
Bp = B->p ; Bi = B->i ; Bx = B->x ;
top = cs_reach (L, B, k, xi, Pinv) ; /* xi[top..n-1]=Reach(B(:,k)) */
for (p = top ; p < n ; p++) x [xi [p]] = 0 ;/* clear x */
for (p = Bp [k] ; p < Bp [k+1] ; p++) x [Bi [p]] = Bx [p] ; /* scatter B */
for (px = top ; px < n ; px++)
{
j = xi [px] ; /* x(j) is nonzero */
jnew = Pinv ? (Pinv [j]) : j ; /* j is column jnew of L */
if (jnew < 0) continue ; /* column jnew is empty */
for (p = Lp [jnew]+1 ; p < Lp [jnew+1] ; p++)
{
x [Li [p]] -= Lx [p] * x [j] ; /* x(i) -= L(i,j) * x(j) */
}
}
return (top) ; /* return top of stack */
}
/* compute vnz, Pinv, leftmost, m2 from A and parent */
static int *cs_vcount (const cs *A, const int *parent, int *m2, int *vnz)
{
int i, k, p, pa, n = A->n, m = A->m, *Ap = A->p, *Ai = A->i ;
int *Pinv = (int*)cs_malloc (2*m+n, sizeof (int)), *leftmost = Pinv + m + n ;
int *w = (int*)cs_malloc (m+3*n, sizeof (int)) ;
int *next = w, *head = w + m, *tail = w + m + n, *nque = w + m + 2*n ;
if (!Pinv || !w) return (cs_idone (Pinv, NULL, w, 0)) ;
for (k = 0 ; k < n ; k++) head [k] = -1 ; /* queue k is empty */
for (k = 0 ; k < n ; k++) tail [k] = -1 ;
for (k = 0 ; k < n ; k++) nque [k] = 0 ;
for (i = 0 ; i < m ; i++) leftmost [i] = -1 ;
for (k = n-1 ; k >= 0 ; k--)
{
for (p = Ap [k] ; p < Ap [k+1] ; p++)
{
leftmost [Ai [p]] = k ; /* leftmost[i] = min(find(A(i,:)))*/
}
}
for (i = m-1 ; i >= 0 ; i--) /* scan rows in reverse order */
{
Pinv [i] = -1 ; /* row i is not yet ordered */
k = leftmost [i] ;
if (k == -1) continue ; /* row i is empty */
if (nque [k]++ == 0) tail [k] = i ; /* first row in queue k */
next [i] = head [k] ; /* put i at head of queue k */
head [k] = i ;
}
(*vnz) = 0 ;
(*m2) = m ;
for (k = 0 ; k < n ; k++) /* find row permutation and nnz(V)*/
{
i = head [k] ; /* remove row i from queue k */
(*vnz)++ ; /* count V(k,k) as nonzero */
if (i < 0) i = (*m2)++ ; /* add a fictitious row */
Pinv [i] = k ; /* associate row i with V(:,k) */
if (--nque [k] <= 0) continue ; /* skip if V(k+1:m,k) is empty */
(*vnz) += nque [k] ; /* nque [k] = nnz (V(k+1:m,k)) */
if ((pa = parent [k]) != -1) /* move all rows to parent of k */
{
if (nque [pa] == 0) tail [pa] = tail [k] ;
next [tail [k]] = head [pa] ;
head [pa] = next [i] ;
nque [pa] += nque [k] ;
}
}
for (i = 0 ; i < m ; i++) if (Pinv [i] < 0) Pinv [i] = k++ ;
return (cs_idone (Pinv, NULL, w, 1)) ;
}
/* symbolic analysis for QR or LU */
css *cs_sqr (const cs *A, int order, int qr)
{
int n, k, ok = 1, *post ;
css *S ;
if (!A) return (NULL) ; /* check inputs */
n = A->n ;
S = (css*)cs_calloc (1, sizeof (css)) ; /* allocate symbolic analysis */
if (!S) return (NULL) ; /* out of memory */
S->Q = cs_amd (A, order) ; /* fill-reducing ordering */
if (order >= 0 && !S->Q) return (cs_sfree (S)) ;
if (qr) /* QR symbolic analysis */
{
cs *C = (order >= 0) ? cs_permute (A, NULL, S->Q, 0) : ((cs *) A) ;
S->parent = cs_etree (C, 1) ; /* etree of C'*C, where C=A(:,Q) */
post = cs_post (n, S->parent) ;
S->cp = cs_counts (C, S->parent, post, 1) ; /* col counts chol(C'*C) */
cs_free (post) ;
ok = C && S->parent && S->cp ;
if (ok) S->Pinv = cs_vcount (C, S->parent, &(S->m2), &(S->lnz)) ;
ok = ok && S->Pinv ;
if (ok) for (S->unz = 0, k = 0 ; k < n ; k++) S->unz += S->cp [k] ;
if (order >= 0) cs_spfree (C) ;
}
else
{
S->unz = 4*(A->p [n]) + n ; /* for LU factorization only, */
S->lnz = S->unz ; /* guess nnz(L) and nnz(U) */
}
return (ok ? S : cs_sfree (S)) ;
}
/* C = A(p,p) where A and C are symmetric the upper part stored, Pinv not P */
cs *cs_symperm (const cs *A, const int *Pinv, int values)
{
int i, j, p, q, i2, j2, n, *Ap, *Ai, *Cp, *Ci, *w ;
double *Cx, *Ax ;
cs *C ;
if (!A) return (NULL) ;
n = A->n ; Ap = A->p ; Ai = A->i ; Ax = A->x ;
C = cs_spalloc (n, n, Ap [n], values && (Ax != NULL), 0) ;
w = (int*)cs_calloc (n, sizeof (int)) ;
if (!C || !w) return (cs_done (C, w, NULL, 0)) ; /* out of memory */
Cp = C->p ; Ci = C->i ; Cx = C->x ;
for (j = 0 ; j < n ; j++) /* count entries in each column of C */
{
j2 = Pinv ? Pinv [j] : j ; /* column j of A is column j2 of C */
for (p = Ap [j] ; p < Ap [j+1] ; p++)
{
i = Ai [p] ;
if (i > j) continue ; /* skip lower triangular part of A */
i2 = Pinv ? Pinv [i] : i ; /* row i of A is row i2 of C */
w [CS_MAX (i2, j2)]++ ; /* column count of C */
}
}
cs_cumsum (Cp, w, n) ; /* compute column pointers of C */
for (j = 0 ; j < n ; j++)
{
j2 = Pinv ? Pinv [j] : j ; /* column j of A is column j2 of C */
for (p = Ap [j] ; p < Ap [j+1] ; p++)
{
i = Ai [p] ;
if (i > j) continue ; /* skip lower triangular part of A*/
i2 = Pinv ? Pinv [i] : i ; /* row i of A is row i2 of C */
Ci [q = w [CS_MAX (i2, j2)]++] = CS_MIN (i2, j2) ;
if (Cx) Cx [q] = Ax [p] ;
}
}
return (cs_done (C, w, NULL, 1)) ; /* success; free workspace, return C */
}
/* depth-first search and postorder of a tree rooted at node j */
int cs_tdfs (int j, int k, int *head, const int *next, int *post, int *stack)
{
int i, p, top = 0 ;
if (!head || !next || !post || !stack) return (-1) ; /* check inputs */
stack [0] = j ; /* place j on the stack */
while (top >= 0) /* while (stack is not empty) */
{
p = stack [top] ; /* p = top of stack */
i = head [p] ; /* i = youngest child of p */
if (i == -1)
{
top-- ; /* p has no unordered children left */
post [k++] = p ; /* node p is the kth postordered node */
}
else
{
head [p] = next [i] ; /* remove i from children of p */
stack [++top] = i ; /* start dfs on child node i */
}
}
return (k) ;
}
/* C = A' */
cs *cs_transpose (const cs *A, int values)
{
int p, q, j, *Cp, *Ci, n, m, *Ap, *Ai, *w ;
double *Cx, *Ax ;
cs *C ;
if (!A) return (NULL) ;
m = A->m ; n = A->n ; Ap = A->p ; Ai = A->i ; Ax = A->x ;
C = cs_spalloc (n, m, Ap [n], values && Ax, 0) ; /* allocate result */
w = (int*)cs_calloc (m, sizeof (int)) ;
if (!C || !w) return (cs_done (C, w, NULL, 0)) ; /* out of memory */
Cp = C->p ; Ci = C->i ; Cx = C->x ;
for (p = 0 ; p < Ap [n] ; p++) w [Ai [p]]++ ; /* row counts */
cs_cumsum (Cp, w, m) ; /* row pointers */
for (j = 0 ; j < n ; j++)
{
for (p = Ap [j] ; p < Ap [j+1] ; p++)
{
Ci [q = w [Ai [p]]++] = j ; /* place A(i,j) as entry C(j,i) */
if (Cx) Cx [q] = Ax [p] ;
}
}
return (cs_done (C, w, NULL, 1)) ; /* success; free w and return C */
}
/* C = compressed-column form of a triplet matrix T */
cs *cs_triplet (const cs *T)
{
int m, n, nz, p, k, *Cp, *Ci, *w, *Ti, *Tj ;
double *Cx, *Tx ;
cs *C ;
if (!T) return (NULL) ; /* check inputs */
m = T->m ; n = T->n ; Ti = T->i ; Tj = T->p ; Tx = T->x ; nz = T->nz ;
C = cs_spalloc (m, n, nz, Tx != NULL, 0) ; /* allocate result */
w = (int*)cs_calloc (n, sizeof (int)) ; /* get workspace */
if (!C || !w) return (cs_done (C, w, NULL, 0)) ; /* out of memory */
Cp = C->p ; Ci = C->i ; Cx = C->x ;
for (k = 0 ; k < nz ; k++) w [Tj [k]]++ ; /* column counts */
cs_cumsum (Cp, w, n) ; /* column pointers */
for (k = 0 ; k < nz ; k++)
{
Ci [p = w [Tj [k]]++] = Ti [k] ; /* A(i,j) is the pth entry in C */
if (Cx) Cx [p] = Tx [k] ;
}
return (cs_done (C, w, NULL, 1)) ; /* success; free w and return C */
}
/* sparse Cholesky update/downdate, L*L' + sigma*w*w' (sigma = +1 or -1) */
int cs_updown (cs *L, int sigma, const cs *C, const int *parent)
{
int p, f, j, *Lp, *Li, *Cp, *Ci ;
double *Lx, *Cx, alpha, beta = 1, delta, gamma, w1, w2, *w, n, beta2 = 1 ;
if (!L || !C || !parent) return (0) ;
Lp = L->p ; Li = L->i ; Lx = L->x ; n = L->n ;
Cp = C->p ; Ci = C->i ; Cx = C->x ;
if ((p = Cp [0]) >= Cp [1]) return (1) ; /* return if C empty */
w = (double*)cs_malloc ((int)n, sizeof (double)) ;
if (!w) return (0) ;
f = Ci [p] ;
for ( ; p < Cp [1] ; p++) f = CS_MIN (f, Ci [p]) ; /* f = min (find (C)) */
for (j = f ; j != -1 ; j = parent [j]) w [j] = 0 ; /* clear workspace w */
for (p = Cp [0] ; p < Cp [1] ; p++) w [Ci [p]] = Cx [p] ; /* w = C */
for (j = f ; j != -1 ; j = parent [j]) /* walk path f up to root */
{
p = Lp [j] ;
alpha = w [j] / Lx [p] ; /* alpha = w(j) / L(j,j) */
beta2 = beta*beta + sigma*alpha*alpha ;
if (beta2 <= 0) break ; /* not positive definite */
beta2 = sqrt (beta2) ;
delta = (sigma > 0) ? (beta / beta2) : (beta2 / beta) ;
gamma = sigma * alpha / (beta2 * beta) ;
Lx [p] = delta * Lx [p] + ((sigma > 0) ? (gamma * w [j]) : 0) ;
beta = beta2 ;
for (p++ ; p < Lp [j+1] ; p++)
{
w1 = w [Li [p]] ;
w [Li [p]] = w2 = w1 - alpha * Lx [p] ;
Lx [p] = delta * Lx [p] + gamma * ((sigma > 0) ? w1 : w2) ;
}
}
cs_free (w) ;
return (beta2 > 0) ;
}
/* solve Ux=b where x and b are dense. x=b on input, solution on output. */
int cs_usolve (const cs *U, double *x)
{
int p, j, n, *Up, *Ui ;
double *Ux ;
if (!U || !x) return (0) ; /* check inputs */
n = U->n ; Up = U->p ; Ui = U->i ; Ux = U->x ;
for (j = n-1 ; j >= 0 ; j--)
{
x [j] /= Ux [Up [j+1]-1] ;
for (p = Up [j] ; p < Up [j+1]-1 ; p++)
{
x [Ui [p]] -= Ux [p] * x [j] ;
}
}
return (1) ;
}
/* allocate a sparse matrix (triplet form or compressed-column form) */
cs *cs_spalloc (int m, int n, int nzmax, int values, int triplet)
{
cs *A = (cs*)cs_calloc (1, sizeof (cs)) ; /* allocate the cs struct */
if (!A) return (NULL) ; /* out of memory */
A->m = m ; /* define dimensions and nzmax */
A->n = n ;
A->nzmax = nzmax = CS_MAX (nzmax, 1) ;
A->nz = triplet ? 0 : -1 ; /* allocate triplet or comp.col */
A->p = (int*)cs_malloc (triplet ? nzmax : n+1, sizeof (int)) ;
A->i = (int*)cs_malloc (nzmax, sizeof (int)) ;
A->x = values ? (double*)cs_malloc (nzmax, sizeof (double)) : NULL ;
return ((!A->p || !A->i || (values && !A->x)) ? cs_spfree (A) : A) ;
}
/* change the max # of entries sparse matrix */
int cs_sprealloc (cs *A, int nzmax)
{
int ok, oki, okj = 1, okx = 1 ;
if (!A) return (0) ;
nzmax = (nzmax <= 0) ? (A->p [A->n]) : nzmax ;
A->i = (int*)cs_realloc (A->i, nzmax, sizeof (int), &oki) ;
if (A->nz >= 0) A->p = (int*)cs_realloc (A->p, nzmax, sizeof (int), &okj) ;
if (A->x) A->x = (double*)cs_realloc (A->x, nzmax, sizeof (double), &okx) ;
ok = (oki && okj && okx) ;
if (ok) A->nzmax = nzmax ;
return (ok) ;
}
/* free a sparse matrix */
cs *cs_spfree (cs *A)
{
if (!A) return (NULL) ; /* do nothing if A already NULL */
cs_free (A->p) ;
cs_free (A->i) ;
cs_free (A->x) ;
return (cs*)(cs_free (A)) ; /* free the cs struct and return NULL */
}
/* free a numeric factorization */
csn *cs_nfree (csn *N)
{
if (!N) return (NULL) ; /* do nothing if N already NULL */
cs_spfree (N->L) ;
cs_spfree (N->U) ;
cs_free (N->Pinv) ;
cs_free (N->B) ;
return (csn*)(cs_free (N)) ; /* free the csn struct and return NULL */
}
/* free a symbolic factorization */
css *cs_sfree (css *S)
{
if (!S) return (NULL) ; /* do nothing if S already NULL */
cs_free (S->Pinv) ;
cs_free (S->Q) ;
cs_free (S->parent) ;
cs_free (S->cp) ;
return (css*)(cs_free (S)) ; /* free the css struct and return NULL */
}
/* allocate a cs_dmperm or cs_scc result */
csd *cs_dalloc (int m, int n)
{
csd *D ;
D = (csd*)cs_calloc (1, sizeof (csd)) ;
if (!D) return (NULL) ;
D->P = (int*)cs_malloc (m, sizeof (int)) ;
D->R = (int*)cs_malloc (m+6, sizeof (int)) ;
D->Q = (int*)cs_malloc (n, sizeof (int)) ;
D->S = (int*)cs_malloc (n+6, sizeof (int)) ;
return ((!D->P || !D->R || !D->Q || !D->S) ? cs_dfree (D) : D) ;
}
/* free a cs_dmperm or cs_scc result */
csd *cs_dfree (csd *D)
{
if (!D) return (NULL) ; /* do nothing if D already NULL */
cs_free (D->P) ;
cs_free (D->Q) ;
cs_free (D->R) ;
cs_free (D->S) ;
return (csd*)(cs_free (D)) ;
}
/* free workspace and return a sparse matrix result */
cs *cs_done (cs *C, void *w, void *x, int ok)
{
cs_free (w) ; /* free workspace */
cs_free (x) ;
return (ok ? C : cs_spfree (C)) ; /* return result if OK, else free it */
}
/* free workspace and return int array result */
int *cs_idone (int *p, cs *C, void *w, int ok)
{
cs_spfree (C) ; /* free temporary matrix */
cs_free (w) ; /* free workspace */
return (ok ? p : (int*)cs_free (p)) ; /* return result if OK, else free it */
}
/* free workspace and return a numeric factorization (Cholesky, LU, or QR) */
csn *cs_ndone (csn *N, cs *C, void *w, void *x, int ok)
{
cs_spfree (C) ; /* free temporary matrix */
cs_free (w) ; /* free workspace */
cs_free (x) ;
return (ok ? N : cs_nfree (N)) ; /* return result if OK, else free it */
}
/* free workspace and return a csd result */
csd *cs_ddone (csd *D, cs *C, void *w, int ok)
{
cs_spfree (C) ; /* free temporary matrix */
cs_free (w) ; /* free workspace */
return (ok ? D : cs_dfree (D)) ; /* return result if OK, else free it */
}
/* solve U'x=b where x and b are dense. x=b on input, solution on output. */
int cs_utsolve (const cs *U, double *x)
{
int p, j, n, *Up, *Ui ;
double *Ux ;
if (!U || !x) return (0) ; /* check inputs */
n = U->n ; Up = U->p ; Ui = U->i ; Ux = U->x ;
for (j = 0 ; j < n ; j++)
{
for (p = Up [j] ; p < Up [j+1]-1 ; p++)
{
x [j] -= Ux [p] * x [Ui [p]] ;
}
x [j] /= Ux [p] ;
}
return (1) ;
}
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