实例介绍
【实例简介】SuiteSparse是世界上最优秀的系数矩阵处理工程之一。但是SuiteSparse提供的官方代码仅包含在matlab、linux环境下编译的生成文件，不能生成在windows操作系统下VS环境下的C 库函数。本文件包括一个库函数cs.cpp和一个头文件cs.h，其中的代码是移植自SuiteSparse官方代码中的Csparse原始代码，功能包括除了复数矩阵以外的所有功能，已成功在vs2010的c 环境下执行过，在毕业设计中用于求解超大型稀疏矩阵的线性方程组（也就是大型稀疏矩阵的除法）。以下是SuiteSparse的介绍。 SuiteSparse是一组C、Fortran和MATLAB函数集，用来生成空间稀疏矩阵数据。在SuiteSparse中几何多种稀疏 矩阵的处理方法，包括矩阵的LU分解，QR分解，Cholesky分解，提供了解非线性方程组、实现最小二乘法等多种函数代码。
【实例截图】
【实例截图】
【核心代码】
#include "cs.h"
#include <malloc.h>
#include <stdio.h>
/* C = alpha*A beta*B */
cs *cs_add (const cs *A, const cs *B, double alpha, double beta)
{
int p, j, nz = 0, anz, *Cp, *Ci, *Bp, m, n, bnz, *w, values ;
double *x, *Bx, *Cx ;
cs *C ;
if (!CS_CSC (A) || !CS_CSC (B)) return (NULL) ; /* check inputs */
if (A->m != B->m || A->n != B->n) return (NULL) ;
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)) ; /* get workspace */
values = (A->x != NULL) && (Bx != NULL) ;
x = values ? (double *)cs_malloc (m, sizeof (double)) : NULL ; /* get workspace */
C = cs_spalloc (m, n, anz bnz, values, 0) ; /* allocate result*/
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 */
}
/* clear w */
static int cs_wclear (int mark, int lemax, int *w, int n)
{
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 (int order, const cs *A) /* order 0:natural, 1:Chol, 2:LU, 3:QR */
{
cs *C, *A2, *AT ;
int *Cp, *Ci, *last, *W, *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, t ;
unsigned int h ;
/* --- Construct matrix C ----------------------------------------------- */
if (!CS_CSC (A) || order <= 0 || order > 3) return (NULL) ; /* check */
AT = cs_transpose (A, 0) ; /* compute A' */
if (!AT) return (NULL) ;
m = A->m ; n = A->n ;
dense = CS_MAX (16, 10 * sqrt ((double) n)) ; /* find dense threshold */
dense = CS_MIN (n-2, dense) ;
if (order == 1 && n == m)
{
C = cs_add (A, AT, 0, 0) ; /* C = A A' */
}
else if (order == 2)
{
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) ;
cs_fkeep (C, &cs_diag, NULL) ; /* drop diagonal entries */
Cp = C->p ;
cnz = Cp [n] ;
P = (int *)cs_malloc (n 1, sizeof (int)) ; /* allocate result */
W = (int *)cs_malloc (8*(n 1), sizeof (int)) ; /* get workspace */
t = cnz cnz/5 2*n ; /* add elbow room to C */
if (!P || !W || !cs_sprealloc (C, t)) return (cs_idone (P, C, W, 0)) ;
len = W ; nv = W (n 1) ; next = W 2*(n 1) ;
head = W 3*(n 1) ; elen = W 4*(n 1) ; degree = W 5*(n 1) ;
w = W 6*(n 1) ; hhead = W 7*(n 1) ;
last = P ; /* use P as workspace for last */
/* --- 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 assembly 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, W, 1)) ;
}
csn *cs_chol (const cs *A, const css *S)
{
double d, lki, *Lx, *x, *Cx ;
int top, i, p, k, n, *Li, *Lp, *cp, *pinv, *s, *c, *parent, *Cp, *Ci ;
cs *L, *C, *E ;
csn *N ;
if (!CS_CSC (A) || !S || !S->cp || !S->parent) return (NULL) ;
n = A->n ;
N = (csn *)cs_calloc (1, sizeof (csn)) ; /* allocate result */
c = (int *)cs_malloc (2*n, sizeof (int)) ; /* get int workspace */
x = (double *)cs_malloc (n, sizeof (double)) ; /* get double workspace */
cp = S->cp ; pinv = S->pinv ; parent = S->parent ;
C = pinv ? cs_symperm (A, pinv, 1) : ((cs *) A) ;
E = pinv ? C : NULL ; /* E is alias for A, or a copy E=A(p,p) */
if (!N || !c || !x || !C) return (cs_ndone (N, E, c, x, 0)) ;
s = c n ;
Cp = C->p ; Ci = C->i ; Cx = C->x ;
N->L = L = cs_spalloc (n, n, cp [n], 1, 0) ; /* allocate result */
if (!L) return (cs_ndone (N, E, c, x, 0)) ;
Lp = L->p ; Li = L->i ; Lx = L->x ;
for (k = 0 ; k < n ; k ) Lp [k] = c [k] = cp [k] ;
for (k = 0 ; k < n ; k ) /* compute L(k,:) for L*L' = C */
{
/* --- Nonzero pattern of L(k,:) ------------------------------------ */
top = cs_ereach (C, k, parent, s, c) ; /* find pattern of L(k,:) */
x [k] = 0 ; /* x (0:k) is now zero */
for (p = Cp [k] ; p < Cp [k 1] ; p ) /* x = full(triu(C(:,k))) */
{
if (Ci [p] <= k) x [Ci [p]] = Cx [p] ;
}
d = x [k] ; /* d = C(k,k) */
x [k] = 0 ; /* clear x 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 x 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, c, 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, c, x, 1)) ; /* success: free E,s,x; return N */
}
/* x=A\b where A is symmetric positive definite; b overwritten with solution */
int cs_cholsol (int order, const cs *A, double *b)
{
double *x ;
css *S ;
csn *N ;
int n, ok ;
if (!CS_CSC (A) || !b) return (0) ; /* check inputs */
n = A->n ;
S = cs_schol (order, A) ; /* ordering and symbolic analysis */
N = cs_chol (A, S) ; /* numeric Cholesky factorization */
x = (double *)cs_malloc (n, sizeof (double)) ; /* get workspace */
ok = (S && N && x) ;
if (ok)
{
cs_ipvec (S->pinv, b, x, n) ; /* x = P*b */
cs_lsolve (N->L, x) ; /* x = L\x */
cs_ltsolve (N->L, x) ; /* x = L'\x */
cs_pvec (S->pinv, x, b, n) ; /* b = P'*x */
}
cs_free (x) ;
cs_sfree (S) ;
cs_nfree (N) ;
return (ok) ;
}
/* C = compressed-column form of a triplet matrix T */
cs *cs_compress (const cs *T)
{
int m, n, nz, p, k, *Cp, *Ci, *w, *Ti, *Tj ;
double *Cx, *Tx ;
cs *C ;
if (!CS_TRIPLET (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 */
}
/* column counts of LL'=A or LL'=A'A, given parent & post ordering */
#define HEAD(k,j) (ata ? head [k] : j)
#define NEXT(J) (ata ? next [J] : -1)
static void init_ata (cs *AT, const int *post, int *w, int **head, int **next)
{
int i, k, p, m = AT->n, n = AT->m, *ATp = AT->p, *ATi = AT->i ;
*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 )
{
for (k = n, p = ATp[i] ; p < ATp[i 1] ; p ) k = CS_MIN (k, w [ATi[p]]);
(*next) [i] = (*head) [k] ; /* place row i in linked list k */
(*head) [k] = i ;
}
}
int *cs_counts (const cs *A, const int *parent, const int *post, int ata)
{
int i, j, k, n, m, J, s, p, q, jleaf, *ATp, *ATi, *maxfirst, *prevleaf,
*ancestor, *head = NULL, *next = NULL, *colcount, *w, *first, *delta ;
cs *AT ;
if (!CS_CSC (A) || !parent || !post) return (NULL) ; /* check inputs */
m = A->m ; n = A->n ;
s = 4*n (ata ? (n m 1) : 0) ;
delta = colcount = (int *)cs_malloc (n, sizeof (int)) ; /* allocate result */
w = (int *)cs_malloc (s, sizeof (int)) ; /* get workspace */
AT = cs_transpose (A, 0) ; /* AT = A' */
if (!AT || !colcount || !w) return (cs_idone (colcount, AT, w, 0)) ;
ancestor = w ; maxfirst = w n ; prevleaf = w 2*n ; first = w 3*n ;
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) init_ata (AT, post, w, &head, &next) ;
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 */
for (J = HEAD (k,j) ; J != -1 ; J = NEXT (J)) /* J=j for LL'=A case */
{
for (p = ATp [J] ; p < ATp [J 1] ; p )
{
i = ATi [p] ;
q = cs_leaf (i, j, first, maxfirst, prevleaf, ancestor, &jleaf);
if (jleaf >= 1) delta [j] ; /* A(i,j) is in skeleton */
if (jleaf == 2) delta [q]-- ; /* account for overlap in q */
}
}
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 */
}
#include "cs.h"
/* p [0..n] = cumulative sum of c [0..n-1], and then copy p [0..n-1] into c */
double cs_cumsum (int *p, int *c, int n)
{
int i, nz = 0 ;
double nz2 = 0 ;
if (!p || !c) return (-1) ; /* check inputs */
for (i = 0 ; i < n ; i )
{
p [i] = nz ;
nz = c [i] ;
nz2 = c [i] ; /* also in double to avoid int overflow */
c [i] = p [i] ; /* also copy p[0..n-1] back into c[0..n-1]*/
}
p [n] = nz ;
return (nz2) ; /* return sum (c [0..n-1]) */
}
int cs_dfs (int j, cs *G, int top, int *xi, int *pstack, const int *pinv)
{
int i, p, p2, done, jnew, head = 0, *Gp, *Gi ;
if (!CS_CSC (G) || !xi || !pstack) return (-1) ; /* check inputs */
Gp = G->p ; Gi = G->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 (Gp, j))
{
CS_MARK (Gp, j) ; /* mark node j as visited */
pstack [head] = (jnew < 0) ? 0 : CS_UNFLIP (Gp [jnew]) ;
}
done = 1 ; /* node j done if no unvisited neighbors */
p2 = (jnew < 0) ? 0 : CS_UNFLIP (Gp [jnew 1]) ;
for (p = pstack [head] ; p < p2 ; p ) /* examine all neighbors of j */
{
i = Gi [p] ; /* consider neighbor node i */
if (CS_MARKED (Gp, 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 R3,C3 from R0 */
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 n, const int *wj, const int *imatch, int *p, int *q,
int *cc, int *rr, int set, int mark)
{
int kc = cc [set], j ;
int kr = rr [set-1] ;
for (j = 0 ; j < n ; j )
{
if (wj [j] != mark) continue ; /* skip if j is not in C set */
p [kr ] = imatch [j] ;
q [kc ] = j ;
}
cc [set 1] = kc ;
rr [set] = kr ;
}
/* collect unmatched rows into the permutation vector p */
static void cs_unmatched (int m, const int *wi, int *p, int *rr, int set)
{
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, compute coarse and then fine dmperm */
csd *cs_dmperm (const cs *A, int seed)
{
int m, n, i, j, k, 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 (!CS_CSC (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, seed) ; /* 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 C1, R1 from C0*/
ok = cs_bfs (A, m, wj, wi, p, jmatch, imatch, 3) ; /* find R3, C3 from R0*/
if (!ok) return (cs_ddone (D, NULL, jmatch, 0)) ;
cs_unmatched (n, wj, q, cc, 0) ; /* unmatched set C0 */
cs_matched (n, wj, imatch, p, q, cc, rr, 1, 1) ; /* set R1 and C1 */
cs_matched (n, wj, imatch, p, q, cc, rr, 2, -1) ; /* set R2 and C2 */
cs_matched (n, wj, imatch, 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 ;
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] ;
Ci = C->i ;
if (rr [1] > 0) for (k = 0 ; k < cnz ; k ) Ci [k] -= 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(R2,C2) */
for (k = 0 ; k < nc ; k ) wj [k] = q [ps [k] cc [2]] ;
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] = 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 */
}
#include "cs.h"
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 */
}
/* remove duplicate entries from A */
int cs_dupl (cs *A)
{
int i, j, p, q, nz = 0, n, m, *Ap, *Ai, *w ;
double *Ax ;
if (!CS_CSC (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 */
}
int cs_entry (cs *T, int i, int j, double x)
{
if (!CS_TRIPLET (T) || i < 0 || j < 0) return (0) ; /* check inputs */
if (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) ;
}
/* find nonzero pattern of Cholesky L(k,1:k-1) using etree and triu(A(:,k)) */
int cs_ereach (const cs *A, int k, const int *parent, int *s, int *w)
{
int i, p, n, len, top, *Ap, *Ai ;
if (!CS_CSC (A) || !parent || !s || !w) return (-1) ; /* check inputs */
top = n = A->n ; Ap = A->p ; Ai = A->i ;
CS_MARK (w, k) ; /* mark node k as visited */
for (p = Ap [k] ; p < Ap [k 1] ; p )
{
i = Ai [p] ; /* A(i,k) is nonzero */
if (i > k) continue ; /* only use upper triangular part of A */
for (len = 0 ; !CS_MARKED (w,i) ; i = parent [i]) /* traverse up etree*/
{
s [len ] = i ; /* L(k,i) is nonzero */
CS_MARK (w, i) ; /* mark i as visited */
}
while (len > 0) s [--top] = s [--len] ; /* push path onto stack */
}
for (p = top ; p < n ; p ) CS_MARK (w, s [p]) ; /* unmark all nodes */
CS_MARK (w, k) ; /* unmark node k */
return (top) ; /* s [top..n-1] contains pattern of L(k,:)*/
}
/* 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 (!CS_CSC (A)) return (NULL) ; /* check inputs */
m = A->m ; n = A->n ; Ap = A->p ; Ai = A->i ;
parent = (int *)cs_malloc (n, sizeof (int)) ; /* allocate result */
w = (int *)cs_malloc (n (ata ? m : 0), sizeof (int)) ; /* get workspace */
if (!w || !parent) return (cs_idone (parent, NULL, w, 0)) ;
ancestor = w ; prev = w n ;
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 (!CS_CSC (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] ;
}
}
}
Ap [n] = nz ; /* finalize A */
cs_sprealloc (A, 0) ; /* remove extra space from A */
return (nz) ;
}
/* y = A*x y */
int cs_gaxpy (const cs *A, const double *x, double *y)
{
int p, j, n, *Ap, *Ai ;
double *Ax ;
if (!CS_CSC (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 (!CS_CSC (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*e1. 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 (const int *p, const double *b, double *x, int n)
{
int k ;
if (!x || !b) return (0) ; /* check inputs */
for (k = 0 ; k < n ; k ) x [p ? p [k] : k] = b [k] ;
return (1) ;
}
/* consider A(i,j), node j in ith row subtree and return lca(jprev,j) */
int cs_leaf (int i, int j, const int *first, int *maxfirst, int *prevleaf,
int *ancestor, int *jleaf)
{
int q, s, sparent, jprev ;
if (!first || !maxfirst || !prevleaf || !ancestor || !jleaf) return (-1) ;
*jleaf = 0 ;
if (i <= j || first [j] <= maxfirst [i]) return (-1) ; /* j not a leaf */
maxfirst [i] = first [j] ; /* update max first[j] seen so far */
jprev = prevleaf [i] ; /* jprev = previous leaf of ith subtree */
prevleaf [i] = j ;
*jleaf = (jprev == -1) ? 1: 2 ; /* j is first or subsequent leaf */
if (*jleaf == 1) return (i) ; /* if 1st leaf, q = root of ith subtree */
for (q = jprev ; q != ancestor [q] ; q = ancestor [q]) ;
for (s = jprev ; s != q ; s = sparent)
{
sparent = ancestor [s] ; /* path compression */
ancestor [s] = q ;
}
return (q) ; /* q = least common ancester (jprev,j) */
}
/* load a triplet matrix from a file */
cs *cs_load (FILE *f)
{
int i, j ;
double x ;
cs *T ;
if (!f) return (NULL) ; /* check inputs */
T = cs_spalloc (0, 0, 1, 1, 1) ; /* allocate result */
while (fscanf (f, "%d %d %lg\n", &i, &j, &x) == 3)
{
if (!cs_entry (T, i, j, x)) return (cs_spfree (T)) ;
}
return (T) ;
}
/* solve Lx=b where x and b are dense. x=b on input, solution on output. */
int cs_lsolve (const cs *L, double *x)
{
int p, j, n, *Lp, *Li ;
double *Lx ;
if (!CS_CSC (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) ;
}
/* solve L'x=b where x and b are dense. x=b on input, solution on output. */
int cs_ltsolve (const cs *L, double *x)
{
int p, j, n, *Lp, *Li ;
double *Lx ;
if (!CS_CSC (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 (!CS_CSC (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)) ; /* get double workspace */
xi = (int *)cs_malloc (2*n, sizeof (int)) ; /* get int workspace */
N = (csn *)cs_calloc (1, sizeof (csn)) ; /* allocate result */
if (!x || !xi || !N) return (cs_ndone (N, NULL, xi, x, 0)) ;
N->L = L = cs_spalloc (n, n, lnz, 1, 0) ; /* allocate result L */
N->U = U = cs_spalloc (n, n, unz, 1, 0) ; /* allocate result U */
N->pinv = pinv = (int *)cs_malloc (n, sizeof (int)) ; /* allocate result pinv */
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_spsolve (L, A, col, xi, x, pinv, 1) ; /* 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 yet 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 (int order, const cs *A, double *b, double tol)
{
double *x ;
css *S ;
csn *N ;
int n, ok ;
if (!CS_CSC (A) || !b) return (0) ; /* check inputs */
n = A->n ;
S = cs_sqr (order, A, 0) ; /* ordering and symbolic analysis */
N = cs_lu (A, S, tol) ; /* numeric LU factorization */
x = (double *)cs_malloc (n, sizeof (double)) ; /* get workspace */
ok = (S && N && x) ;
if (ok)
{
cs_ipvec (N->pinv, b, x, n) ; /* x = b(p) */
cs_lsolve (N->L, x) ; /* x = L\x */
cs_usolve (N->U, x) ; /* x = U\x */
cs_ipvec (S->q, x, b, n) ; /* 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 (malloc (CS_MAX (n,1) * size)) ;
}
/* wrapper for calloc */
void *cs_calloc (int n, size_t size)
{
return (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 *pnew ;
pnew = realloc (p, CS_MAX (n,1) * size) ; /* realloc the block */
*ok = (pnew != NULL) ; /* realloc fails if pnew is NULL */
return ((*ok) ? pnew : 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, int seed) /*[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, *q ;
cs *C ;
if (!CS_CSC (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 (k = 0, j = 0 ; j < n ; j ) /* count nonempty rows and columns */
{
n2 = (Ap [j] < Ap [j 1]) ;
for (p = Ap [j] ; p < Ap [j 1] ; p )
{
w [Ai [p]] = 1 ;
k = (j == Ai [p]) ; /* count entries already on diagonal */
}
}
if (k == CS_MIN (m,n)) /* quick return if diagonal zero-free */
{
jmatch = jimatch ; imatch = jimatch m ;
for (i = 0 ; i < k ; i ) jmatch [i] = i ;
for ( ; i < m ; i ) jmatch [i] = -1 ;
for (j = 0 ; j < k ; j ) imatch [j] = j ;
for ( ; j < n ; j ) imatch [j] = -1 ;
return (cs_idone (jimatch, NULL, NULL, 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)) ; /* get 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 */
q = cs_randperm (n, seed) ; /* q = random permutation */
for (k = 0 ; k < n ; k ) /* augment, starting at column q[k] */
{
cs_augment (q ? q [k]: k, C, jmatch, cheap, w, js, is, ps) ;
}
cs_free (q) ;
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 (!CS_CSC (A) || !CS_CSC (B)) return (NULL) ; /* check inputs */
if (A->n != B->m) return (NULL) ;
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)) ; /* get workspace */
values = (A->x != NULL) && (Bx != NULL) ;
x = values ? (double *)cs_malloc (m, sizeof (double)) : NULL ; /* get workspace */
C = cs_spalloc (m, n, anz bnz, values, 0) ; /* allocate result */
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->i and C->x may be 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 (!CS_CSC (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 t, j, k, nz = 0, m, n, *Ap, *Ai, *Cp, *Ci ;
double *Cx, *Ax ;
cs *C ;
if (!CS_CSC (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) ; /* alloc result */
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 (t = Ap [j] ; t < Ap [j 1] ; t )
{
if (Cx) Cx [nz] = Ax [t] ; /* row i of A is row pinv[i] of C */
Ci [nz ] = pinv ? (pinv [Ai [t]]) : Ai [t] ;
}
}
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 result */
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 (const int *parent, int n)
{
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)) ; /* get workspace */
if (!w || !post) return (cs_idone (post, NULL, w, 0)) ;
head = w ; next = w n ; stack = w 2*n ;
for (j = 0 ; j < n ; j ) head [j] = -1 ; /* empty linked 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 (const int *p, const double *b, double *x, int n)
{
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,pinv,R] = qr (A) */
csn *cs_qr (const cs *A, const css *S)
{
double *Rx, *Vx, *Ax, *x, *Beta ;
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 (!CS_CSC (A) || !S) return (NULL) ;
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 = S->leftmost ;
w = (int *)cs_malloc (m2 n, sizeof (int)) ; /* get int workspace */
x = (double *)cs_malloc (m2, sizeof (double)) ; /* get double workspace */
N = (csn *)cs_calloc (1, sizeof (csn)) ; /* allocate result */
if (!w || !x || !N) return (cs_ndone (N, NULL, w, x, 0)) ;
s = w m2 ; /* s is 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 result V */
N->U = R = cs_spalloc (m2, n, rnz, 1, 0) ; /* allocate result R */
N->B = Beta = (double *)cs_malloc (n, sizeof (double)) ; /* allocate result Beta */
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 (int order, const cs *A, double *b)
{
double *x ;
css *S ;
csn *N ;
cs *AT = NULL ;
int k, m, n, ok ;
if (!CS_CSC (A) || !b) return (0) ; /* check inputs */
n = A->n ;
m = A->m ;
if (m >= n)
{
S = cs_sqr (order, A, 1) ; /* ordering and symbolic analysis */
N = cs_qr (A, S) ; /* numeric QR factorization */
x = (double *)cs_calloc (S ? S->m2 : 1, sizeof (double)) ; /* get workspace */
ok = (S && N && x) ;
if (ok)
{
cs_ipvec (S->pinv, b, x, m) ; /* x(0:m-1) = b(p(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 (S->q, x, b, n) ; /* b(q(0:n-1)) = x(0:n-1) */
}
}
else
{
AT = cs_transpose (A, 1) ; /* Ax=b is underdetermined */
S = cs_sqr (order, AT, 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)) ; /* get workspace */
ok = (AT && S && N && x) ;
if (ok)
{
cs_pvec (S->q, b, x, m) ; /* x(q(0:m-1)) = b(0:m-1) */
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 (S->pinv, x, b, n) ; /* b(0:n-1) = x(p(0:n-1)) */
}
}
cs_free (x) ;
cs_sfree (S) ;
cs_nfree (N) ;
cs_spfree (AT) ;
return (ok) ;
}
/* return a random permutation vector, the identity perm, or p = n-1:-1:0.
* seed = -1 means p = n-1:-1:0. seed = 0 means p = identity. otherwise
* p = random permutation. */
int *cs_randperm (int n, int seed)
{
int *p, k, j, t ;
if (seed == 0) return (NULL) ; /* return p = NULL (identity) */
p = (int *)cs_malloc (n, sizeof (int)) ; /* allocate result */
if (!p) return (NULL) ; /* out of memory */
for (k = 0 ; k < n ; k ) p [k] = n-k-1 ;
if (seed == -1) return (p) ; /* return reverse permutation */
srand (seed) ; /* get new random number seed */
for (k = 0 ; k < n ; k )
{
j = k (rand ( ) % (n-k)) ; /* j = rand int in range k to n-1 */
t = p [j] ; /* swap p[k] and p[j] */
p [j] = p [k] ;
p [k] = t ;
}
return (p) ;
}
/* xi [top...n-1] = nodes reachable from graph of G*P' via nodes in B(:,k).
* xi [n...2n-1] used as workspace */
int cs_reach (cs *G, const cs *B, int k, int *xi, const int *pinv)
{
int p, n, top, *Bp, *Bi, *Gp ;
if (!CS_CSC (G) || !CS_CSC (B) || !xi) return (-1) ; /* check inputs */
n = G->n ; Bp = B->p ; Bi = B->i ; Gp = G->p ;
top = n ;
for (p = Bp [k] ; p < Bp [k 1] ; p )
{
if (!CS_MARKED (Gp, Bi [p])) /* start a dfs at unmarked node i */
{
top = cs_dfs (Bi [p], G, top, xi, xi n, pinv) ;
}
}
for (p = top ; p < n ; p ) CS_MARK (Gp, xi [p]) ; /* restore G */
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 (!CS_CSC (A) || !w || !CS_CSC (C)) return (-1) ; /* check inputs */
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, nb = 0, top, *xi, *pstack, *p, *r, *Ap, *ATp, *rcopy, *Blk ;
cs *AT ;
csd *D ;
if (!CS_CSC (A)) return (NULL) ; /* check inputs */
n = A->n ; Ap = A->p ;
D = cs_dalloc (n, 0) ; /* allocate result */
AT = cs_transpose (A, 0) ; /* AT = A' */
xi = (int *)cs_malloc (2*n 1, sizeof (int)) ; /* get workspace */
if (!D || !AT || !xi) return (cs_ddone (D, AT, xi, 0)) ;
Blk = xi ; rcopy = pstack = xi n ;
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 ;
nb = 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 [nb--] = top ; /* node i is the start of a component in p */
top = cs_dfs (i, AT, top, p, pstack, NULL) ;
}
r [nb] = 0 ; /* first block starts at zero; shift r up */
for (k = nb ; k <= n ; k ) r [k-nb] = r [k] ;
D->nb = nb = n-nb ; /* nb = # of strongly connected components */
for (b = 0 ; b < nb ; b ) /* sort each block in natural order */
{
for (k = r [b] ; k < r [b 1] ; k ) Blk [p [k]] = b ;
}
for (b = 0 ; b <= nb ; b ) rcopy [b] = r [b] ;
for (i = 0 ; i < n ; i ) p [rcopy [Blk [i]] ] = i ;
return (cs_ddone (D, AT, xi, 1)) ;
}
/* ordering and symbolic analysis for a Cholesky factorization */
css *cs_schol (int order, const cs *A)
{
int n, *c, *post, *P ;
cs *C ;
css *S ;
if (!CS_CSC (A)) return (NULL) ; /* check inputs */
n = A->n ;
S = (css *)cs_calloc (1, sizeof (css)) ; /* allocate result S */
if (!S) return (NULL) ; /* out of memory */
P = cs_amd (order, A) ; /* P = amd(A A'), or natural */
S->pinv = cs_pinv (P, n) ; /* find inverse permutation */
cs_free (P) ;
if (order && !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 (S->parent, n) ; /* 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)) ; /* allocate result S->cp */
S->unz = S->lnz = cs_cumsum (S->cp, c, n) ; /* find column pointers for L */
cs_free (c) ;
return ((S->lnz >= 0) ? S : cs_sfree (S)) ;
}
/* solve Gx=b(:,k), where G is either upper (lo=0) or lower (lo=1) triangular */
int cs_spsolve (cs *G, const cs *B, int k, int *xi, double *x, const int *pinv,
int lo)
{
int j, J, p, q, px, top, n, *Gp, *Gi, *Bp, *Bi ;
double *Gx, *Bx ;
if (!CS_CSC (G) || !CS_CSC (B) || !xi || !x) return (-1) ;
Gp = G->p ; Gi = G->i ; Gx = G->x ; n = G->n ;
Bp = B->p ; Bi = B->i ; Bx = B->x ;
top = cs_reach (G, 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 */
J = pinv ? (pinv [j]) : j ; /* j maps to col J of G */
if (J < 0) continue ; /* column J is empty */
x [j] /= Gx [lo ? (Gp [J]) : (Gp [J 1]-1)] ;/* x(j) /= G(j,j) */
p = lo ? (Gp [J] 1) : (Gp [J]) ; /* lo: L(j,j) 1st entry */
q = lo ? (Gp [J 1]) : (Gp [J 1]-1) ; /* up: U(j,j) last entry */
for ( ; p < q ; p )
{
x [Gi [p]] -= Gx [p] * x [j] ; /* x(i) -= G(i,j) * x(j) */
}
}
return (top) ; /* return top of stack */
}
/* compute nnz(V) = S->lnz, S->pinv, S->leftmost, S->m2 from A and S->parent */
static int cs_vcount (const cs *A, css *S)
{
int i, k, p, pa, n = A->n, m = A->m, *Ap = A->p, *Ai = A->i, *next, *head,
*tail, *nque, *pinv, *leftmost, *w, *parent = S->parent ;
S->pinv = pinv = (int *)cs_malloc (m n, sizeof (int)) ; /* allocate pinv, */
S->leftmost = leftmost = (int *)cs_malloc (m, sizeof (int)) ; /* and leftmost */
w = (int *)cs_malloc (m 3*n, sizeof (int)) ; /* get workspace */
if (!pinv || !w || !leftmost)
{
cs_free (w) ; /* pinv and leftmost freed later */
return (0) ; /* out of memory */
}
next = w ; head = w m ; tail = w m n ; nque = w m 2*n ;
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 ;
}
S->lnz = 0 ;
S->m2 = m ;
for (k = 0 ; k < n ; k ) /* find row permutation and nnz(V)*/
{
i = head [k] ; /* remove row i from queue k */
S->lnz ; /* count V(k,k) as nonzero */
if (i < 0) i = S->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 */
S->lnz = nque [k] ; /* nque [k] is 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 ;
cs_free (w) ;
return (1) ;
}
/* symbolic ordering and analysis for QR or LU */
css *cs_sqr (int order, const cs *A, int qr)
{
int n, k, ok = 1, *post ;
css *S ;
if (!CS_CSC (A)) return (NULL) ; /* check inputs */
n = A->n ;
S = (css *)cs_calloc (1, sizeof (css)) ; /* allocate result S */
if (!S) return (NULL) ; /* out of memory */
S->q = cs_amd (order, A) ; /* fill-reducing ordering */
if (order && !S->q) return (cs_sfree (S)) ;
if (qr) /* QR symbolic analysis */
{
cs *C = order ? 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 (S->parent, n) ;
S->cp = cs_counts (C, S->parent, post, 1) ; /* col counts chol(C'*C) */
cs_free (post) ;
ok = C && S->parent && S->cp && cs_vcount (C, S) ;
if (ok) for (S->unz = 0, k = 0 ; k < n ; k ) S->unz = S->cp [k] ;
ok = ok && S->lnz >= 0 && S->unz >= 0 ; /* int overflow guard */
if (order) 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)) ; /* return result 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 (!CS_CSC (A)) return (NULL) ; /* check inputs */
n = A->n ; Ap = A->p ; Ai = A->i ; Ax = A->x ;
C = cs_spalloc (n, n, Ap [n], values && (Ax != NULL), 0) ; /* alloc 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 (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 (!CS_CSC (A)) return (NULL) ; /* check inputs */
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)) ; /* 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 (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 */
}
/* 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 n, p, f, j, *Lp, *Li, *Cp, *Ci ;
double *Lx, *Cx, alpha, beta = 1, delta, gamma, w1, w2, *w, beta2 = 1 ;
if (!CS_CSC (L) || !CS_CSC (C) || !parent) return (0) ; /* check inputs */
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 (n, sizeof (double)) ; /* get workspace */
if (!w) return (0) ; /* out of memory */
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 (!CS_CSC (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) ;
if (nzmax <= 0) nzmax = (CS_CSC (A)) ? (A->p [A->n]) : A->nz ;
A->i = (int *)cs_realloc (A->i, nzmax, sizeof (int), &oki) ;
if (CS_TRIPLET (A)) 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) ;
cs_free (S->leftmost) ;
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 the csd struct and return NULL */
}
/* 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, or 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 (!CS_CSC (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 [Up [j 1]-1] ;
}
return (1) ;
}
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