klu2_kernel.hpp

/* ========================================================================== */
/* === KLU_kernel =========================================================== */
/* ========================================================================== */
// @HEADER
// ***********************************************************************
//
//                   KLU2: A Direct Linear Solver package
//                    Copyright 2011 Sandia Corporation
//
// Under terms of Contract DE-AC04-94AL85000, with Sandia Corporation, the 
// U.S. Government retains certain rights in this software.
//
// This library is free software; you can redistribute it and/or modify
// it under the terms of the GNU Lesser General Public License as
// published by the Free Software Foundation; either version 2.1 of the
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//
// This library is distributed in the hope that it will be useful, but
// WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Lesser General Public License for more details.
// 
// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301
// USA
// Questions? Contact Mike A. Heroux (maherou@sandia.gov)
//
// KLU2 is derived work from KLU, licensed under LGPL, and copyrighted by
// University of Florida. The Authors of KLU are Timothy A. Davis and
// Eka Palamadai. See Doc/KLU_README.txt for the licensing and copyright
// information for KLU.
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// @HEADER

/* Sparse left-looking LU factorization, with partial pivoting.  Based on
 * Gilbert & Peierl's method, with a non-recursive DFS and with Eisenstat &
 * Liu's symmetric pruning.  No user-callable routines are in this file.
 */

#ifndef KLU2_KERNEL_HPP
#define KLU2_KERNEL_HPP

#include "klu2_internal.h"
#include "klu2_memory.hpp"

/* ========================================================================== */
/* === dfs ================================================================== */
/* ========================================================================== */

/* Does a depth-first-search, starting at node j. */

template <typename Int>
static Int dfs
(
    /* input, not modified on output: */
    Int j,              /* node at which to start the DFS */
    Int k,              /* mark value, for the Flag array */
    Int Pinv [ ],       /* Pinv [i] = k if row i is kth pivot row, or EMPTY if
                         * row i is not yet pivotal.  */
    Int Llen [ ],       /* size n, Llen [k] = # nonzeros in column k of L */
    Int Lip [ ],        /* size n, Lip [k] is position in LU of column k of L */

    /* workspace, not defined on input or output */
    Int Stack [ ],      /* size n */

    /* input/output: */
    Int Flag [ ],       /* Flag [i] == k means i is marked */
    Int Lpend [ ],      /* for symmetric pruning */
    Int top,            /* top of stack on input*/
    Unit LU [],
    Int *Lik,           /* Li row index array of the kth column */
    Int *plength,

    /* other, not defined on input or output */
    Int Ap_pos [ ]      /* keeps track of position in adj list during DFS */
)
{
    Int i, pos, jnew, head, l_length ;
    Int *Li ;

    l_length = *plength ;

    head = 0 ;
    Stack [0] = j ;
    ASSERT (Flag [j] != k) ;

    while (head >= 0)
    {
        j = Stack [head] ;
        jnew = Pinv [j] ;
        ASSERT (jnew >= 0 && jnew < k) ;        /* j is pivotal */

        if (Flag [j] != k)          /* a node is not yet visited */
        {
            /* first time that j has been visited */
            Flag [j] = k ;
            PRINTF (("[ start dfs at %d : new %d\n", j, jnew)) ;
            /* set Ap_pos [head] to one past the last entry in col j to scan */
            Ap_pos [head] =
                (Lpend [jnew] == EMPTY) ?  Llen [jnew] : Lpend [jnew] ;
        }

        /* add the adjacent nodes to the recursive stack by iterating through
         * until finding another non-visited pivotal node */
        Li = (Int *) (LU + Lip [jnew]) ;
        for (pos = --Ap_pos [head] ; pos >= 0 ; --pos)
        {
            i = Li [pos] ;
            if (Flag [i] != k)
            {
                /* node i is not yet visited */
                if (Pinv [i] >= 0)
                {
                    /* keep track of where we left off in the scan of the
                     * adjacency list of node j so we can restart j where we
                     * left off. */
                    Ap_pos [head] = pos ;

                    /* node i is pivotal; push it onto the recursive stack
                     * and immediately break so we can recurse on node i. */
                    Stack [++head] = i ;
                    break ;
                }
                else
                {
                    /* node i is not pivotal (no outgoing edges). */
                    /* Flag as visited and store directly into L,
                     * and continue with current node j. */
                    Flag [i] = k ;
                    Lik [l_length] = i ;
                    l_length++ ;
                }
            }
        }

        if (pos == -1)
        {
            /* if all adjacent nodes of j are already visited, pop j from
             * recursive stack and push j onto output stack */
            head-- ;
            Stack[--top] = j ;
            PRINTF (("  end   dfs at %d ] head : %d\n", j, head)) ;
        }
    }

    *plength = l_length ;
    return (top) ;
}


/* ========================================================================== */
/* === lsolve_symbolic ====================================================== */
/* ========================================================================== */

/* Finds the pattern of x, for the solution of Lx=b */

template <typename Int>
static Int lsolve_symbolic
(
    /* input, not modified on output: */
    Int n,              /* L is n-by-n, where n >= 0 */
    Int k,              /* also used as the mark value, for the Flag array */
    Int Ap [ ],
    Int Ai [ ],
    Int Q [ ],
    Int Pinv [ ],       /* Pinv [i] = k if i is kth pivot row, or EMPTY if row i
                         * is not yet pivotal.  */

    /* workspace, not defined on input or output */
    Int Stack [ ],      /* size n */

    /* workspace, defined on input and output */
    Int Flag [ ],       /* size n.  Initially, all of Flag [0..n-1] < k.  After
                         * lsolve_symbolic is done, Flag [i] == k if i is in
                         * the pattern of the output, and Flag [0..n-1] <= k. */

    /* other */
    Int Lpend [ ],      /* for symmetric pruning */
    Int Ap_pos [ ],     /* workspace used in dfs */

    Unit LU [ ],        /* LU factors (pattern and values) */
    Int lup,            /* pointer to free space in LU */
    Int Llen [ ],       /* size n, Llen [k] = # nonzeros in column k of L */
    Int Lip [ ],        /* size n, Lip [k] is position in LU of column k of L */

    /* ---- the following are only used in the BTF case --- */

    Int k1,             /* the block of A is from k1 to k2-1 */
    Int PSinv [ ]       /* inverse of P from symbolic factorization */
)
{
    Int *Lik ;
    Int i, p, pend, oldcol, kglobal, top, l_length ;

    top = n ;
    l_length = 0 ;
    Lik = (Int *) (LU + lup);

    /* ---------------------------------------------------------------------- */
    /* BTF factorization of A (k1:k2-1, k1:k2-1) */
    /* ---------------------------------------------------------------------- */

    kglobal = k + k1 ;  /* column k of the block is col kglobal of A */
    oldcol = Q [kglobal] ;      /* Q must be present for BTF case */
    pend = Ap [oldcol+1] ;
    for (p = Ap [oldcol] ; p < pend ; p++)
    {
        i = PSinv [Ai [p]] - k1 ;
        if (i < 0) continue ;   /* skip entry outside the block */

        /* (i,k) is an entry in the block.  start a DFS at node i */
        PRINTF (("\n ===== DFS at node %d in b, inew: %d\n", i, Pinv [i])) ;
        if (Flag [i] != k)
        {
            if (Pinv [i] >= 0)
            {
                top = dfs (i, k, Pinv, Llen, Lip, Stack, Flag,
                           Lpend, top, LU, Lik, &l_length, Ap_pos) ;
            }
            else
            {
                /* i is not pivotal, and not flagged. Flag and put in L */
                Flag [i] = k ;
                Lik [l_length] = i ;
                l_length++;
            }
        }
    }

    /* If Llen [k] is zero, the matrix is structurally singular */
    Llen [k] = l_length ;
    return (top) ;
}


/* ========================================================================== */
/* === construct_column ===================================================== */
/* ========================================================================== */

/* Construct the kth column of A, and the off-diagonal part, if requested.
 * Scatter the numerical values into the workspace X, and construct the
 * corresponding column of the off-diagonal matrix. */

template <typename Entry, typename Int>
static void construct_column
(
    /* inputs, not modified on output */
    Int k,          /* the column of A (or the column of the block) to get */
    Int Ap [ ],
    Int Ai [ ],
    Entry Ax [ ],
    Int Q [ ],      /* column pre-ordering */

    /* zero on input, modified on output */
    Entry X [ ],

    /* ---- the following are only used in the BTF case --- */

    /* inputs, not modified on output */
    Int k1,         /* the block of A is from k1 to k2-1 */
    Int PSinv [ ],  /* inverse of P from symbolic factorization */
    double Rs [ ],  /* scale factors for A */
    Int scale,      /* 0: no scaling, nonzero: scale the rows with Rs */

    /* inputs, modified on output */
    Int Offp [ ],   /* off-diagonal matrix (modified by this routine) */
    Int Offi [ ],
    Entry Offx [ ]
)
{
    Entry aik ;
    Int i, p, pend, oldcol, kglobal, poff, oldrow ;

    /* ---------------------------------------------------------------------- */
    /* Scale and scatter the column into X. */
    /* ---------------------------------------------------------------------- */

    kglobal = k + k1 ;          /* column k of the block is col kglobal of A */
    poff = Offp [kglobal] ;     /* start of off-diagonal column */
    oldcol = Q [kglobal] ;
    pend = Ap [oldcol+1] ;

    if (scale <= 0)
    {
        /* no scaling */
        for (p = Ap [oldcol] ; p < pend ; p++)
        {
            oldrow = Ai [p] ;
            i = PSinv [oldrow] - k1 ;
            aik = Ax [p] ;
            if (i < 0)
            {
                /* this is an entry in the off-diagonal part */
                Offi [poff] = oldrow ;
                Offx [poff] = aik ;
                poff++ ;
            }
            else
            {
                /* (i,k) is an entry in the block.  scatter into X */
                X [i] = aik ;
            }
        }
    }
    else
    {
        /* row scaling */
        for (p = Ap [oldcol] ; p < pend ; p++)
        {
            oldrow = Ai [p] ;
            i = PSinv [oldrow] - k1 ;
            aik = Ax [p] ;
            SCALE_DIV (aik, Rs [oldrow]) ;
            if (i < 0)
            {
                /* this is an entry in the off-diagonal part */
                Offi [poff] = oldrow ;
                Offx [poff] = aik ;
                poff++ ;
            }
            else
            {
                /* (i,k) is an entry in the block.  scatter into X */
                X [i] = aik ;
            }
        }
    }

    Offp [kglobal+1] = poff ;   /* start of the next col of off-diag part */
}


/* ========================================================================== */
/* === lsolve_numeric ======================================================= */
/* ========================================================================== */

/* Computes the numerical values of x, for the solution of Lx=b.  Note that x
 * may include explicit zeros if numerical cancelation occurs.  L is assumed
 * to be unit-diagonal, with possibly unsorted columns (but the first entry in
 * the column must always be the diagonal entry). */

template <typename Entry, typename Int>
static void lsolve_numeric
(
    /* input, not modified on output: */
    Int Pinv [ ],       /* Pinv [i] = k if i is kth pivot row, or EMPTY if row i
                         * is not yet pivotal.  */
    Unit *LU,           /* LU factors (pattern and values) */
    Int Stack [ ],      /* stack for dfs */
    Int Lip [ ],        /* size n, Lip [k] is position in LU of column k of L */
    Int top,            /* top of stack on input */
    Int n,              /* A is n-by-n */
    Int Llen [ ],       /* size n, Llen [k] = # nonzeros in column k of L */

    /* output, must be zero on input: */
    Entry X [ ] /* size n, initially zero.  On output,
                 * X [Ui [up1..up-1]] and X [Li [lp1..lp-1]]
                 * contains the solution. */

)
{
    Entry xj ;
    Entry *Lx ;
    Int *Li ;
    Int p, s, j, jnew, len ;

    /* solve Lx=b */
    for (s = top ; s < n ; s++)
    {
        /* forward solve with column j of L */
        j = Stack [s] ;
        jnew = Pinv [j] ;
        ASSERT (jnew >= 0) ;
        xj = X [j] ;
        GET_POINTER (LU, Lip, Llen, Li, Lx, jnew, len) ;
        ASSERT (Lip [jnew] <= Lip [jnew+1]) ;
        for (p = 0 ; p < len ; p++)
        {
            /*X [Li [p]] -= Lx [p] * xj ; */
            MULT_SUB (X [Li [p]], Lx [p], xj) ;
        }
    }
}


/* ========================================================================== */
/* === lpivot =============================================================== */
/* ========================================================================== */

/* Find a pivot via partial pivoting, and scale the column of L. */

template <typename Entry, typename Int>
static Int lpivot
(
    Int diagrow,
    Int *p_pivrow,
    Entry *p_pivot,
    double *p_abs_pivot,
    double tol,
    Entry X [ ],
    Unit *LU,           /* LU factors (pattern and values) */
    Int Lip [ ],
    Int Llen [ ],
    Int k,
    Int n,

    Int Pinv [ ],       /* Pinv [i] = k if row i is kth pivot row, or EMPTY if
                         * row i is not yet pivotal.  */

    Int *p_firstrow,
    KLU_common<Entry, Int> *Common
)
{
    Entry x, pivot, *Lx ;
    double abs_pivot, xabs ;
    Int p, i, ppivrow, pdiag, pivrow, *Li, last_row_index, firstrow, len ;

    pivrow = EMPTY ;
    if (Llen [k] == 0)
    {
        /* matrix is structurally singular */
        if (Common->halt_if_singular)
        {
            return (FALSE) ;
        }
        for (firstrow = *p_firstrow ; firstrow < n ; firstrow++)
        {
            PRINTF (("check %d\n", firstrow)) ;
            if (Pinv [firstrow] < 0)
            {
                /* found the lowest-numbered non-pivotal row.  Pick it. */
                pivrow = firstrow ;
                PRINTF (("Got pivotal row: %d\n", pivrow)) ;
                break ;
            }
        }
        ASSERT (pivrow >= 0 && pivrow < n) ;
        CLEAR (pivot) ;
        *p_pivrow = pivrow ;
        *p_pivot = pivot ;
        *p_abs_pivot = 0 ;
        *p_firstrow = firstrow ;
        return (FALSE) ;
    }

    pdiag = EMPTY ;
    ppivrow = EMPTY ;
    abs_pivot = EMPTY ;
    i = Llen [k] - 1 ;
    GET_POINTER (LU, Lip, Llen, Li, Lx, k, len) ;
    last_row_index = Li [i] ;

    /* decrement the length by 1 */
    Llen [k] = i ;
    GET_POINTER (LU, Lip, Llen, Li, Lx, k, len) ;

    /* look in Li [0 ..Llen [k] - 1 ] for a pivot row */
    for (p = 0 ; p < len ; p++)
    {
        /* gather the entry from X and store in L */
        i = Li [p] ;
        x = X [i] ;
        CLEAR (X [i]) ;

        Lx [p] = x ;
        /* xabs = ABS (x) ; */
        KLU2_ABS (xabs, x) ;

        /* find the diagonal */
        if (i == diagrow)
        {
            pdiag = p ;
        }

        /* find the partial-pivoting choice */
        if (xabs > abs_pivot)
        {
            abs_pivot = xabs ;
            ppivrow = p ;
        }
    }

    /* xabs = ABS (X [last_row_index]) ;*/
    KLU2_ABS (xabs, X [last_row_index]) ;
    if (xabs > abs_pivot)
    {
        abs_pivot = xabs ;
        ppivrow = EMPTY ;
    }

    /* compare the diagonal with the largest entry */
    if (last_row_index == diagrow)
    {
        if (xabs >= tol * abs_pivot)
        {
            abs_pivot = xabs ;
            ppivrow = EMPTY ;
        }
    }
    else if (pdiag != EMPTY)
    {
        /* xabs = ABS (Lx [pdiag]) ;*/
        KLU2_ABS (xabs, Lx [pdiag]) ;
        if (xabs >= tol * abs_pivot)
        {
            /* the diagonal is large enough */
            abs_pivot = xabs ;
            ppivrow = pdiag ;
        }
    }

    if (ppivrow != EMPTY)
    {
        pivrow = Li [ppivrow] ;
        pivot  = Lx [ppivrow] ;
        /* overwrite the ppivrow values with last index values */
        Li [ppivrow] = last_row_index ;
        Lx [ppivrow] = X [last_row_index] ;
    }
    else
    {
        pivrow = last_row_index ;
        pivot = X [last_row_index] ;
    }
    CLEAR (X [last_row_index]) ;

    *p_pivrow = pivrow ;
    *p_pivot = pivot ;
    *p_abs_pivot = abs_pivot ;
    ASSERT (pivrow >= 0 && pivrow < n) ;

    if (IS_ZERO (pivot) && Common->halt_if_singular)
    {
        /* numerically singular case */
        return (FALSE) ;
    }

    /* divide L by the pivot value */
    for (p = 0 ; p < Llen [k] ; p++)
    {
        /* Lx [p] /= pivot ; */
        DIV (Lx [p], Lx [p], pivot) ;
    }

    return (TRUE) ;
}


/* ========================================================================== */
/* === prune ================================================================ */
/* ========================================================================== */

/* Prune the columns of L to reduce work in subsequent depth-first searches */
template <typename Entry, typename Int>
static void prune
(
    /* input/output: */
    Int Lpend [ ],      /* Lpend [j] marks symmetric pruning point for L(:,j) */

    /* input: */
    Int Pinv [ ],       /* Pinv [i] = k if row i is kth pivot row, or EMPTY if
                         * row i is not yet pivotal.  */
    Int k,              /* prune using column k of U */
    Int pivrow,         /* current pivot row */

    /* input/output: */
    Unit *LU,           /* LU factors (pattern and values) */

    /* input */
    Int Uip [ ],        /* size n, column pointers for U */
    Int Lip [ ],        /* size n, column pointers for L */
    Int Ulen [ ],       /* size n, column length of U */
    Int Llen [ ]        /* size n, column length of L */
)
{
    Entry x ;
    Entry *Lx, *Ux ;
    Int *Li, *Ui ;
    Int p, i, j, p2, phead, ptail, llen, ulen ;

    /* check to see if any column of L can be pruned */
    GET_POINTER (LU, Uip, Ulen, Ui, Ux, k, ulen) ;

    // Try not to warn about Ux never being used
    (void) Ux;
    for (p = 0 ; p < ulen ; p++)
    {
        j = Ui [p] ;
        ASSERT (j < k) ;
        PRINTF (("%d is pruned: %d. Lpend[j] %d Lip[j+1] %d\n",
            j, Lpend [j] != EMPTY, Lpend [j], Lip [j+1])) ;
        if (Lpend [j] == EMPTY)
        {
            /* scan column j of L for the pivot row */
            GET_POINTER (LU, Lip, Llen, Li, Lx, j, llen) ;
            for (p2 = 0 ; p2 < llen ; p2++)
            {
                if (pivrow == Li [p2])
                {
                    /* found it!  This column can be pruned */
#ifndef NDEBUGKLU2
                    PRINTF (("==== PRUNE: col j %d of L\n", j)) ;
                    {
                        Int p3 ;
                        for (p3 = 0 ; p3 < Llen [j] ; p3++)
                        {
                            PRINTF (("before: %i  pivotal: %d\n", Li [p3],
                                        Pinv [Li [p3]] >= 0)) ;
                        }
                    }
#endif

                    /* partition column j of L.  The unit diagonal of L
                     * is not stored in the column of L. */
                    phead = 0 ;
                    ptail = Llen [j] ;
                    while (phead < ptail)
                    {
                        i = Li [phead] ;
                        if (Pinv [i] >= 0)
                        {
                            /* leave at the head */
                            phead++ ;
                        }
                        else
                        {
                            /* swap with the tail */
                            ptail-- ;
                            Li [phead] = Li [ptail] ;
                            Li [ptail] = i ;
                            x = Lx [phead] ;
                            Lx [phead] = Lx [ptail] ;
                            Lx [ptail] = x ;
                        }
                    }

                    /* set Lpend to one past the last entry in the
                     * first part of the column of L.  Entries in
                     * Li [0 ... Lpend [j]-1] are the only part of
                     * column j of L that needs to be scanned in the DFS.
                     * Lpend [j] was EMPTY; setting it >= 0 also flags
                     * column j as pruned. */
                    Lpend [j] = ptail ;

#ifndef NDEBUGKLU2
                    {
                        Int p3 ;
                        for (p3 = 0 ; p3 < Llen [j] ; p3++)
                        {
                            if (p3 == Lpend [j]) PRINTF (("----\n")) ;
                            PRINTF (("after: %i  pivotal: %d\n", Li [p3],
                                        Pinv [Li [p3]] >= 0)) ;
                        }
                    }
#endif

                    break ;
                }
            }
        }
    }
}


/* ========================================================================== */
/* === KLU_kernel =========================================================== */
/* ========================================================================== */

template <typename Entry, typename Int>
size_t KLU_kernel   /* final size of LU on output */
(
    /* input, not modified */
    Int n,          /* A is n-by-n */
    Int Ap [ ],     /* size n+1, column pointers for A */
    Int Ai [ ],     /* size nz = Ap [n], row indices for A */
    Entry Ax [ ],   /* size nz, values of A */
    Int Q [ ],      /* size n, optional input permutation */
    size_t lusize,  /* initial size of LU on input */

    /* output, not defined on input */
    Int Pinv [ ],   /* size n, inverse row permutation, where Pinv [i] = k if
                     * row i is the kth pivot row */
    Int P [ ],      /* size n, row permutation, where P [k] = i if row i is the
                     * kth pivot row. */
    Unit **p_LU,        /* LU array, size lusize on input */
    Entry Udiag [ ],    /* size n, diagonal of U */
    Int Llen [ ],       /* size n, column length of L */
    Int Ulen [ ],       /* size n, column length of U */
    Int Lip [ ],        /* size n, column pointers for L */
    Int Uip [ ],        /* size n, column pointers for U */
    Int *lnz,           /* size of L*/
    Int *unz,           /* size of U*/
    /* workspace, not defined on input */
    Entry X [ ],    /* size n, undefined on input, zero on output */

    /* workspace, not defined on input or output */
    Int Stack [ ],  /* size n */
    Int Flag [ ],   /* size n */
    Int Ap_pos [ ],     /* size n */

    /* other workspace: */
    Int Lpend [ ],                  /* size n workspace, for pruning only */

    /* inputs, not modified on output */
    Int k1,             /* the block of A is from k1 to k2-1 */
    Int PSinv [ ],      /* inverse of P from symbolic factorization */
    double Rs [ ],      /* scale factors for A */

    /* inputs, modified on output */
    Int Offp [ ],   /* off-diagonal matrix (modified by this routine) */
    Int Offi [ ],
    Entry Offx [ ],
    /* --------------- */
    KLU_common<Entry, Int> *Common
)
{
    Entry pivot ;
    double abs_pivot, xsize, nunits, tol, memgrow ;
    Entry *Ux ;
    Int *Li, *Ui ;
    Unit *LU ;          /* LU factors (pattern and values) */
    Int k, p, i, j, pivrow = 0, kbar, diagrow, firstrow, lup, top, scale, len ;
    size_t newlusize ;

#ifndef NDEBUGKLU2
    Entry *Lx ;
#endif

    ASSERT (Common != NULL) ;
    scale = Common->scale ;
    tol = Common->tol ;
    memgrow = Common->memgrow ;
    *lnz = 0 ;
    *unz = 0 ;
    CLEAR (pivot) ;

    /* ---------------------------------------------------------------------- */
    /* get initial Li, Lx, Ui, and Ux */
    /* ---------------------------------------------------------------------- */

    PRINTF (("input: lusize %d \n", lusize)) ;
    ASSERT (lusize > 0) ;
    LU = *p_LU ;

    /* ---------------------------------------------------------------------- */
    /* initializations */
    /* ---------------------------------------------------------------------- */

    firstrow = 0 ;
    lup = 0 ;

    for (k = 0 ; k < n ; k++)
    {
        /* X [k] = 0 ; */
        CLEAR (X [k]) ;
        Flag [k] = EMPTY ;
        Lpend [k] = EMPTY ;     /* flag k as not pruned */
    }

    /* ---------------------------------------------------------------------- */
    /* mark all rows as non-pivotal and determine initial diagonal mapping */
    /* ---------------------------------------------------------------------- */

    /* PSinv does the symmetric permutation, so don't do it here */
    for (k = 0 ; k < n ; k++)
    {
        P [k] = k ;
        Pinv [k] = FLIP (k) ;   /* mark all rows as non-pivotal */
    }
    /* initialize the construction of the off-diagonal matrix */
    Offp [0] = 0 ;

    /* P [k] = row means that UNFLIP (Pinv [row]) = k, and visa versa.
     * If row is pivotal, then Pinv [row] >= 0.  A row is initially "flipped"
     * (Pinv [k] < EMPTY), and then marked "unflipped" when it becomes
     * pivotal. */

#ifndef NDEBUGKLU2
    for (k = 0 ; k < n ; k++)
    {
        PRINTF (("Initial P [%d] = %d\n", k, P [k])) ;
    }
#endif

    /* ---------------------------------------------------------------------- */
    /* factorize */
    /* ---------------------------------------------------------------------- */

    for (k = 0 ; k < n ; k++)
    {

        PRINTF (("\n\n==================================== k: %d\n", k)) ;

        /* ------------------------------------------------------------------ */
        /* determine if LU factors have grown too big */
        /* ------------------------------------------------------------------ */

        /* (n - k) entries for L and k entries for U */
        nunits = DUNITS (Int, n - k) + DUNITS (Int, k) +
                 DUNITS (Entry, n - k) + DUNITS (Entry, k) ;

        /* LU can grow by at most 'nunits' entries if the column is dense */
        PRINTF (("lup %d lusize %g lup+nunits: %g\n", lup, (double) lusize,
            lup+nunits));
        xsize = ((double) lup) + nunits ;
        if (xsize > (double) lusize)
        {
            /* check here how much to grow */
            xsize = (memgrow * ((double) lusize) + 4*n + 1) ;
            if (INT_OVERFLOW (xsize))
            {
                PRINTF (("Matrix is too large (Int overflow)\n")) ;
                Common->status = KLU_TOO_LARGE ;
                return (lusize) ;
            }
            newlusize = (size_t) (memgrow * lusize + 2*n + 1) ;
            /* Future work: retry mechanism in case of malloc failure */
            LU = (Unit *) KLU_realloc (newlusize, lusize, sizeof (Unit), LU, Common) ;
            Common->nrealloc++ ;
            *p_LU = LU ;
            if (Common->status == KLU_OUT_OF_MEMORY)
            {
                PRINTF (("Matrix is too large (LU)\n")) ;
                return (lusize) ;
            }
            lusize = newlusize ;
            PRINTF (("inc LU to %d done\n", lusize)) ;
        }

        /* ------------------------------------------------------------------ */
        /* start the kth column of L and U */
        /* ------------------------------------------------------------------ */

        Lip [k] = lup ;

        /* ------------------------------------------------------------------ */
        /* compute the nonzero pattern of the kth column of L and U */
        /* ------------------------------------------------------------------ */

#ifndef NDEBUGKLU2
        for (i = 0 ; i < n ; i++)
        {
            ASSERT (Flag [i] < k) ;
            /* ASSERT (X [i] == 0) ; */
            ASSERT (IS_ZERO (X [i])) ;
        }
#endif

        top = lsolve_symbolic (n, k, Ap, Ai, Q, Pinv, Stack, Flag,
                    Lpend, Ap_pos, LU, lup, Llen, Lip, k1, PSinv) ;

#ifndef NDEBUGKLU2
        PRINTF (("--- in U:\n")) ;
        for (p = top ; p < n ; p++)
        {
            PRINTF (("pattern of X for U: %d : %d pivot row: %d\n",
                p, Stack [p], Pinv [Stack [p]])) ;
            ASSERT (Flag [Stack [p]] == k) ;
        }
        PRINTF (("--- in L:\n")) ;
        Li = (Int *) (LU + Lip [k]);
        for (p = 0 ; p < Llen [k] ; p++)
        {
            PRINTF (("pattern of X in L: %d : %d pivot row: %d\n",
                p, Li [p], Pinv [Li [p]])) ;
            ASSERT (Flag [Li [p]] == k) ;
        }
        p = 0 ;
        for (i = 0 ; i < n ; i++)
        {
            ASSERT (Flag [i] <= k) ;
            if (Flag [i] == k) p++ ;
        }
#endif

        /* ------------------------------------------------------------------ */
        /* get the column of the matrix to factorize and scatter into X */
        /* ------------------------------------------------------------------ */

        construct_column <Entry> (k, Ap, Ai, Ax, Q, X,
            k1, PSinv, Rs, scale, Offp, Offi, Offx) ;

        /* ------------------------------------------------------------------ */
        /* compute the numerical values of the kth column (s = L \ A (:,k)) */
        /* ------------------------------------------------------------------ */

        lsolve_numeric <Entry> (Pinv, LU, Stack, Lip, top, n, Llen, X) ;

#ifndef NDEBUGKLU2
        for (p = top ; p < n ; p++)
        {
            PRINTF (("X for U %d : ",  Stack [p])) ;
            PRINT_ENTRY (X [Stack [p]]) ;
        }
        Li = (Int *) (LU + Lip [k]) ;
        for (p = 0 ; p < Llen [k] ; p++)
        {
            PRINTF (("X for L %d : ", Li [p])) ;
            PRINT_ENTRY (X [Li [p]]) ;
        }
#endif

        /* ------------------------------------------------------------------ */
        /* partial pivoting with diagonal preference */
        /* ------------------------------------------------------------------ */

        /* determine what the "diagonal" is */
        diagrow = P [k] ;   /* might already be pivotal */
        PRINTF (("k %d, diagrow = %d, UNFLIP (diagrow) = %d\n",
            k, diagrow, UNFLIP (diagrow))) ;

        /* find a pivot and scale the pivot column */
        if (!lpivot <Entry> (diagrow, &pivrow, &pivot, &abs_pivot, tol, X, LU, Lip,
                    Llen, k, n, Pinv, &firstrow, Common))
        {
            /* matrix is structurally or numerically singular */
            Common->status = KLU_SINGULAR ;
            if (Common->numerical_rank == EMPTY)
            {
                Common->numerical_rank = k+k1 ;
                Common->singular_col = Q [k+k1] ;
            }
            if (Common->halt_if_singular)
            {
                /* do not continue the factorization */
                return (lusize) ;
            }
        }

        /* we now have a valid pivot row, even if the column has NaN's or
         * has no entries on or below the diagonal at all. */
        PRINTF (("\nk %d : Pivot row %d : ", k, pivrow)) ;
        PRINT_ENTRY (pivot) ;
        ASSERT (pivrow >= 0 && pivrow < n) ;
        ASSERT (Pinv [pivrow] < 0) ;

        /* set the Uip pointer */
        Uip [k] = Lip [k] + UNITS (Int, Llen [k]) + UNITS (Entry, Llen [k]) ;

        /* move the lup pointer to the position where indices of U
         * should be stored */
        lup += UNITS (Int, Llen [k]) + UNITS (Entry, Llen [k]) ;

        Ulen [k] = n - top ;

        /* extract Stack [top..n-1] to Ui and the values to Ux and clear X */
        GET_POINTER (LU, Uip, Ulen, Ui, Ux, k, len) ;
        for (p = top, i = 0 ; p < n ; p++, i++)
        {
            j = Stack [p] ;
            Ui [i] = Pinv [j] ;
            Ux [i] = X [j] ;
            CLEAR (X [j]) ;
        }

        /* position the lu index at the starting point for next column */
        lup += UNITS (Int, Ulen [k]) + UNITS (Entry, Ulen [k]) ;

        /* U(k,k) = pivot */
        Udiag [k] = pivot ;

        /* ------------------------------------------------------------------ */
        /* log the pivot permutation */
        /* ------------------------------------------------------------------ */

        ASSERT (UNFLIP (Pinv [diagrow]) < n) ;
        ASSERT (P [UNFLIP (Pinv [diagrow])] == diagrow) ;

        if (pivrow != diagrow)
        {
            /* an off-diagonal pivot has been chosen */
            Common->noffdiag++ ;
            PRINTF ((">>>>>>>>>>>>>>>>> pivrow %d k %d off-diagonal\n",
                        pivrow, k)) ;
            if (Pinv [diagrow] < 0)
            {
                /* the former diagonal row index, diagrow, has not yet been
                 * chosen as a pivot row.  Log this diagrow as the "diagonal"
                 * entry in the column kbar for which the chosen pivot row,
                 * pivrow, was originally logged as the "diagonal" */
                kbar = FLIP (Pinv [pivrow]) ;
                P [kbar] = diagrow ;
                Pinv [diagrow] = FLIP (kbar) ;
            }
        }
        P [k] = pivrow ;
        Pinv [pivrow] = k ;

#ifndef NDEBUGKLU2
        for (i = 0 ; i < n ; i++) { ASSERT (IS_ZERO (X [i])) ;}
        GET_POINTER (LU, Uip, Ulen, Ui, Ux, k, len) ;
        for (p = 0 ; p < len ; p++)
        {
            PRINTF (("Column %d of U: %d : ", k, Ui [p])) ;
            PRINT_ENTRY (Ux [p]) ;
        }
        GET_POINTER (LU, Lip, Llen, Li, Lx, k, len) ;
        for (p = 0 ; p < len ; p++)
        {
            PRINTF (("Column %d of L: %d : ", k, Li [p])) ;
            PRINT_ENTRY (Lx [p]) ;
        }
#endif

        /* ------------------------------------------------------------------ */
        /* symmetric pruning */
        /* ------------------------------------------------------------------ */

        prune<Entry> (Lpend, Pinv, k, pivrow, LU, Uip, Lip, Ulen, Llen) ;

        *lnz += Llen [k] + 1 ; /* 1 added to lnz for diagonal */
        *unz += Ulen [k] + 1 ; /* 1 added to unz for diagonal */
    }

    /* ---------------------------------------------------------------------- */
    /* finalize column pointers for L and U, and put L in the pivotal order */
    /* ---------------------------------------------------------------------- */

    for (p = 0 ; p < n ; p++)
    {
        Li = (Int *) (LU + Lip [p]) ;
        for (i = 0 ; i < Llen [p] ; i++)
        {
            Li [i] = Pinv [Li [i]] ;
        }
    }

#ifndef NDEBUGKLU2
    for (i = 0 ; i < n ; i++)
    {
        PRINTF (("P [%d] = %d   Pinv [%d] = %d\n", i, P [i], i, Pinv [i])) ;
    }
    for (i = 0 ; i < n ; i++)
    {
        ASSERT (Pinv [i] >= 0 && Pinv [i] < n) ;
        ASSERT (P [i] >= 0 && P [i] < n) ;
        ASSERT (P [Pinv [i]] == i) ;
        ASSERT (IS_ZERO (X [i])) ;
    }
#endif

    /* ---------------------------------------------------------------------- */
    /* shrink the LU factors to just the required size */
    /* ---------------------------------------------------------------------- */

    newlusize = lup ;
    ASSERT ((size_t) newlusize <= lusize) ;

    /* this cannot fail, since the block is descreasing in size */
    LU = (Unit *) KLU_realloc (newlusize, lusize, sizeof (Unit), LU, Common) ;
    *p_LU = LU ;
    return (newlusize) ;
}

#endif