/* solve L'x=b where x and b are dense. x=b on input, solution on output. */
CS_INT cs_ltsolve (const cs *L, CS_ENTRY *x)
{
CS_INT p, j, n, *Lp, *Li ;
CS_ENTRY *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] -= CS_CONJ (Lx [p]) * x [Li [p]] ;
}
x [j] /= CS_CONJ (Lx [Lp [j]]) ;
}
return (1) ;
}
/* sparse Cholesky update/downdate, L*L' + sigma*w*w' (sigma = +1 or -1) */
CS_INT cs_updown (cs *L, CS_INT sigma, const cs *C, const CS_INT *parent)
{
CS_INT n, p, f, j, *Lp, *Li, *Cp, *Ci ;
CS_ENTRY *Lx, *Cx, alpha, gamma, w1, w2, *w ;
double beta = 1, beta2 = 1, delta ;
#ifdef CS_COMPLEX
cs_complex_t phase ;
#endif
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 = cs_malloc (n, sizeof (CS_ENTRY)) ; /* 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*CS_CONJ(alpha) ;
if (beta2 <= 0) break ; /* not positive definite */
beta2 = sqrt (beta2) ;
delta = (sigma > 0) ? (beta / beta2) : (beta2 / beta) ;
gamma = sigma * CS_CONJ(alpha) / (beta2 * beta) ;
Lx [p] = delta * Lx [p] + ((sigma > 0) ? (gamma * w [j]) : 0) ;
beta = beta2 ;
#ifdef CS_COMPLEX
phase = CS_ABS (Lx [p]) / Lx [p] ; /* phase = abs(L(j,j))/L(j,j)*/
Lx [p] *= phase ; /* L(j,j) = L(j,j) * phase */
#endif
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) ;
#ifdef CS_COMPLEX
Lx [p] *= phase ; /* L(i,j) = L(i,j) * phase */
#endif
}
}
cs_free (w) ;
return (beta2 > 0) ;
}
/* apply the ith Householder vector to x */
CS_INT cs_happly (const cs *V, CS_INT i, double beta, CS_ENTRY *x)
{
CS_INT p, *Vp, *Vi ;
CS_ENTRY *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 += CS_CONJ (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) ;
}
/* C = A(p,p) where A and C are symmetric the upper part stored; pinv not p */
cs *cs_symperm (const cs *A, const CS_INT *pinv, CS_INT values)
{
CS_INT i, j, p, q, i2, j2, n, *Ap, *Ai, *Cp, *Ci, *w ;
CS_ENTRY *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 = cs_calloc (n, sizeof (CS_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] = (i2 <= j2) ? Ax [p] : CS_CONJ (Ax [p]) ;
}
}
return (cs_done (C, w, NULL, 1)) ; /* success; free workspace, return C */
}
