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#include "cs.h"
/* 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) ;
}

/* 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_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 = cs_malloc (triplet ? nzmax : n+1, sizeof (int)) ;
    A->i = cs_malloc (nzmax, sizeof (int)) ;
    A->x = values ? 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 = cs_realloc (A->i, nzmax, sizeof (int), &oki) ;
    if (A->nz >= 0) A->p = cs_realloc (A->p, nzmax, sizeof (int), &okj) ;
    if (A->x) A->x = 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_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 (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 (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 = cs_calloc (1, sizeof (csd)) ;
    if (!D) return (NULL) ;
    D->P = cs_malloc (m, sizeof (int)) ;
    D->R = cs_malloc (m+6, sizeof (int)) ;
    D->Q = cs_malloc (n, sizeof (int)) ;
    D->S = 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 (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 : 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 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) ;
}

/* 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 = cs_malloc (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) ;
}

/* 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 = 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 */
}

/* 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 = 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 */
}

/* 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(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 = 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 */
}

/* 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 = cs_malloc (2*m+n, sizeof (int)), *leftmost = Pinv + m + n ;
    int *w = 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 = 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)) ;
}

/* 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 */
}

/* 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 = 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 = 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)) ;
}

/* 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 = 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)) ;
}

/* 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) ;
}

/* 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=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 = 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 = 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) ;
}

/* 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 = cs_malloc (m2+n, sizeof (int)) ;
    x = cs_malloc (m2, sizeof (double)) ;
    N = 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 = 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 = 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) ;
}

/* 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) ;
}

/* 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 = cs_malloc (n, sizeof (int)) ;        /* allocate result */
    w = 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 */
}

/* 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 = 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 */
}

/* 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)) ;
}

/* 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*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 = cs_calloc (m, sizeof (int)) ;
    values = (A->x != NULL) && (Bx != NULL) ;
    x = values ? 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 */
}

/* 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 = 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 = 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)) ;
}

#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 */
}

/* 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 = 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) ;
}

/* [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 = cs_malloc (n, sizeof (double)) ;
    xi = cs_malloc (2*n, sizeof (int)) ;
    N = 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 = 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 */
}

/* 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 (!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) ;
}

/* 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 (!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) ;
}
#include "cs.h"
/* load a triplet matrix from a file */
cs *cs_load (FILE *f)
{
    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) ;
}

/* 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) ;
}

/* 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) ;
}

/* 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) ;
}

/* 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) ;
}

/* 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) */
}

/* 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 = cs_malloc (n, sizeof (int)) ;
    w = 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)) ;
}

/* 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) ;
}

/* 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 (!A) return (0) ;            /* check inputs */
    m = A->m ; n = A->n ; Ap = A->p ; Ai = A->i ; Ax = A->x ;
    w = 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 */
}

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 */
}

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 */
}

/* 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 ;
}

/* 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, 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)) ;
}

/* 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) ;
}

/* 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]) */
}

/* 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 = cs_malloc (s, sizeof (int)) ; first = w+3*n ;   /* get workspace */
    ancestor = w ; maxfirst = w+n ; prevleaf = w+2*n ;
    delta = colcount = 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 */
}

/* 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 = 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) ;
}

/* 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 = cs_calloc (1, sizeof (csn)) ;
    w = cs_malloc (3*n, sizeof (int)) ; s = w + n, c = w + 2*n ;
    x = 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 */
}

/* 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 (const cs *A, int order)  /* order -1:natural, 0:Chol, 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 = 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 = cs_malloc (n+1, sizeof (int)) ;     /* allocate result */
    ww = 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)) ;
}
#include "cs.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 (!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 = cs_calloc (m, sizeof (int)) ;
    values = (A->x != NULL) && (Bx != NULL) ;
    x = values ? 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 */
}