
/*! @file zgstrf.c
 * \brief Computes an LU factorization of a general sparse matrix
 *
 * <pre>
 * -- SuperLU routine (version 3.0) --
 * Univ. of California Berkeley, Xerox Palo Alto Research Center,
 * and Lawrence Berkeley National Lab.
 * October 15, 2003
 * 
 * Copyright (c) 1994 by Xerox Corporation.  All rights reserved.
 *
 * THIS MATERIAL IS PROVIDED AS IS, WITH ABSOLUTELY NO WARRANTY
 * EXPRESSED OR IMPLIED.  ANY USE IS AT YOUR OWN RISK.
 * 
 * Permission is hereby granted to use or copy this program for any
 * purpose, provided the above notices are retained on all copies.
 * Permission to modify the code and to distribute modified code is
 * granted, provided the above notices are retained, and a notice that
 * the code was modified is included with the above copyright notice.
 * </pre>
 */


#include "slu_zdefs.h"

/*! \brief
 *
 * <pre>
 * Purpose
 * =======
 *
 * ZGSTRF computes an LU factorization of a general sparse m-by-n
 * matrix A using partial pivoting with row interchanges.
 * The factorization has the form
 *     Pr * A = L * U
 * where Pr is a row permutation matrix, L is lower triangular with unit
 * diagonal elements (lower trapezoidal if A->nrow > A->ncol), and U is upper 
 * triangular (upper trapezoidal if A->nrow < A->ncol).
 *
 * See supermatrix.h for the definition of 'SuperMatrix' structure.
 *
 * Arguments
 * =========
 *
 * options (input) superlu_options_t*
 *         The structure defines the input parameters to control
 *         how the LU decomposition will be performed.
 *
 * A        (input) SuperMatrix*
 *	    Original matrix A, permuted by columns, of dimension
 *          (A->nrow, A->ncol). The type of A can be:
 *          Stype = SLU_NCP; Dtype = SLU_Z; Mtype = SLU_GE.
 *
 * relax    (input) int
 *          To control degree of relaxing supernodes. If the number
 *          of nodes (columns) in a subtree of the elimination tree is less
 *          than relax, this subtree is considered as one supernode,
 *          regardless of the row structures of those columns.
 *
 * panel_size (input) int
 *          A panel consists of at most panel_size consecutive columns.
 *
 * etree    (input) int*, dimension (A->ncol)
 *          Elimination tree of A'*A.
 *          Note: etree is a vector of parent pointers for a forest whose
 *          vertices are the integers 0 to A->ncol-1; etree[root]==A->ncol.
 *          On input, the columns of A should be permuted so that the
 *          etree is in a certain postorder.
 *
 * work     (input/output) void*, size (lwork) (in bytes)
 *          User-supplied work space and space for the output data structures.
 *          Not referenced if lwork = 0;
 *
 * lwork   (input) int
 *         Specifies the size of work array in bytes.
 *         = 0:  allocate space internally by system malloc;
 *         > 0:  use user-supplied work array of length lwork in bytes,
 *               returns error if space runs out.
 *         = -1: the routine guesses the amount of space needed without
 *               performing the factorization, and returns it in
 *               *info; no other side effects.
 *
 * perm_c   (input) int*, dimension (A->ncol)
 *	    Column permutation vector, which defines the 
 *          permutation matrix Pc; perm_c[i] = j means column i of A is 
 *          in position j in A*Pc.
 *          When searching for diagonal, perm_c[*] is applied to the
 *          row subscripts of A, so that diagonal threshold pivoting
 *          can find the diagonal of A, rather than that of A*Pc.
 *
 * perm_r   (input/output) int*, dimension (A->nrow)
 *          Row permutation vector which defines the permutation matrix Pr,
 *          perm_r[i] = j means row i of A is in position j in Pr*A.
 *          If options->Fact == SamePattern_SameRowPerm, the pivoting routine
 *             will try to use the input perm_r, unless a certain threshold
 *             criterion is violated. In that case, perm_r is overwritten by
 *             a new permutation determined by partial pivoting or diagonal
 *             threshold pivoting.
 *          Otherwise, perm_r is output argument;
 *
 * L        (output) SuperMatrix*
 *          The factor L from the factorization Pr*A=L*U; use compressed row 
 *          subscripts storage for supernodes, i.e., L has type: 
 *          Stype = SLU_SC, Dtype = SLU_Z, Mtype = SLU_TRLU.
 *
 * U        (output) SuperMatrix*
 *	    The factor U from the factorization Pr*A*Pc=L*U. Use column-wise
 *          storage scheme, i.e., U has types: Stype = SLU_NC, 
 *          Dtype = SLU_Z, Mtype = SLU_TRU.
 *
 * Glu      (input/output) GlobalLU_t *
 *          If options->Fact == SamePattern_SameRowPerm, it is an input;
 *              The matrix A will be factorized assuming that a 
 *              factorization of a matrix with the same sparsity pattern
 *              and similar numerical values was performed prior to this one.
 *              Therefore, this factorization will reuse both row and column
 *		scaling factors R and C, both row and column permutation
 *		vectors perm_r and perm_c, and the L & U data structures
 *		set up from the previous factorization.
 *          Otherwise, it is an output.
 *
 * stat     (output) SuperLUStat_t*
 *          Record the statistics on runtime and floating-point operation count.
 *          See slu_util.h for the definition of 'SuperLUStat_t'.
 *
 * info     (output) int*
 *          = 0: successful exit
 *          < 0: if info = -i, the i-th argument had an illegal value
 *          > 0: if info = i, and i is
 *             <= A->ncol: U(i,i) is exactly zero. The factorization has
 *                been completed, but the factor U is exactly singular,
 *                and division by zero will occur if it is used to solve a
 *                system of equations.
 *             > A->ncol: number of bytes allocated when memory allocation
 *                failure occurred, plus A->ncol. If lwork = -1, it is
 *                the estimated amount of space needed, plus A->ncol.
 *
 * ======================================================================
 *
 * Local Working Arrays: 
 * ======================
 *   m = number of rows in the matrix
 *   n = number of columns in the matrix
 *
 *   xprune[0:n-1]: xprune[*] points to locations in subscript 
 *	vector lsub[*]. For column i, xprune[i] denotes the point where 
 *	structural pruning begins. I.e. only xlsub[i],..,xprune[i]-1 need 
 *	to be traversed for symbolic factorization.
 *
 *   marker[0:3*m-1]: marker[i] = j means that node i has been 
 *	reached when working on column j.
 *	Storage: relative to original row subscripts
 *	NOTE: There are 3 of them: marker/marker1 are used for panel dfs, 
 *	      see zpanel_dfs.c; marker2 is used for inner-factorization,
 *            see zcolumn_dfs.c.
 *
 *   parent[0:m-1]: parent vector used during dfs
 *      Storage: relative to new row subscripts
 *
 *   xplore[0:m-1]: xplore[i] gives the location of the next (dfs) 
 *	unexplored neighbor of i in lsub[*]
 *
 *   segrep[0:nseg-1]: contains the list of supernodal representatives
 *	in topological order of the dfs. A supernode representative is the 
 *	last column of a supernode.
 *      The maximum size of segrep[] is n.
 *
 *   repfnz[0:W*m-1]: for a nonzero segment U[*,j] that ends at a 
 *	supernodal representative r, repfnz[r] is the location of the first 
 *	nonzero in this segment.  It is also used during the dfs: repfnz[r]>0
 *	indicates the supernode r has been explored.
 *	NOTE: There are W of them, each used for one column of a panel. 
 *
 *   panel_lsub[0:W*m-1]: temporary for the nonzeros row indices below 
 *      the panel diagonal. These are filled in during zpanel_dfs(), and are
 *      used later in the inner LU factorization within the panel.
 *	panel_lsub[]/dense[] pair forms the SPA data structure.
 *	NOTE: There are W of them.
 *
 *   dense[0:W*m-1]: sparse accumulating (SPA) vector for intermediate values;
 *	    	   NOTE: there are W of them.
 *
 *   tempv[0:*]: real temporary used for dense numeric kernels;
 *	The size of this array is defined by NUM_TEMPV() in slu_zdefs.h.
 * </pre>
 */

void
zgstrf (superlu_options_t *options, SuperMatrix *A,
        int relax, int panel_size, int *etree, void *work, int lwork,
        int *perm_c, int *perm_r, SuperMatrix *L, SuperMatrix *U,
    	GlobalLU_t *Glu, /* persistent to facilitate multiple factorizations */
        SuperLUStat_t *stat, int *info)
{
    /* Local working arrays */
    NCPformat *Astore;
    int       *iperm_r = NULL; /* inverse of perm_r; used when 
                                  options->Fact == SamePattern_SameRowPerm */
    int       *iperm_c; /* inverse of perm_c */
    int       *iwork;
    doublecomplex    *zwork;
    int	      *segrep, *repfnz, *parent, *xplore;
    int	      *panel_lsub; /* dense[]/panel_lsub[] pair forms a w-wide SPA */
    int	      *xprune;
    int	      *marker;
    doublecomplex    *dense, *tempv;
    int       *relax_end;
    doublecomplex    *a;
    int       *asub;
    int       *xa_begin, *xa_end;
    int       *xsup, *supno;
    int       *xlsub, *xlusup, *xusub;
    int       nzlumax;
    double fill_ratio = sp_ienv(6);  /* estimated fill ratio */

    /* Local scalars */
    fact_t    fact = options->Fact;
    double    diag_pivot_thresh = options->DiagPivotThresh;
    int       pivrow;   /* pivotal row number in the original matrix A */
    int       nseg1;	/* no of segments in U-column above panel row jcol */
    int       nseg;	/* no of segments in each U-column */
    register int jcol;	
    register int kcol;	/* end column of a relaxed snode */
    register int icol;
    register int i, k, jj, new_next, iinfo;
    int       m, n, min_mn, jsupno, fsupc, nextlu, nextu;
    int       w_def;	/* upper bound on panel width */
    int       usepr, iperm_r_allocated = 0;
    int       nnzL, nnzU;
    int       *panel_histo = stat->panel_histo;
    flops_t   *ops = stat->ops;

    iinfo    = 0;
    m        = A->nrow;
    n        = A->ncol;
    min_mn   = SUPERLU_MIN(m, n);
    Astore   = A->Store;
    a        = Astore->nzval;
    asub     = Astore->rowind;
    xa_begin = Astore->colbeg;
    xa_end   = Astore->colend;

    /* Allocate storage common to the factor routines */
    *info = zLUMemInit(fact, work, lwork, m, n, Astore->nnz,
                       panel_size, fill_ratio, L, U, Glu, &iwork, &zwork);
    if ( *info ) return;
    
    xsup    = Glu->xsup;
    supno   = Glu->supno;
    xlsub   = Glu->xlsub;
    xlusup  = Glu->xlusup;
    xusub   = Glu->xusub;
    
    SetIWork(m, n, panel_size, iwork, &segrep, &parent, &xplore,
	     &repfnz, &panel_lsub, &xprune, &marker);
    zSetRWork(m, panel_size, zwork, &dense, &tempv);
    
    usepr = (fact == SamePattern_SameRowPerm);
    if ( usepr ) {
	/* Compute the inverse of perm_r */
	iperm_r = (int *) intMalloc(m);
	for (k = 0; k < m; ++k) iperm_r[perm_r[k]] = k;
	iperm_r_allocated = 1;
    }
    iperm_c = (int *) intMalloc(n);
    for (k = 0; k < n; ++k) iperm_c[perm_c[k]] = k;

    /* Identify relaxed snodes */
    relax_end = (int *) intMalloc(n);
    if ( options->SymmetricMode == YES ) {
        heap_relax_snode(n, etree, relax, marker, relax_end); 
    } else {
        relax_snode(n, etree, relax, marker, relax_end); 
    }
    
    ifill (perm_r, m, EMPTY);
    ifill (marker, m * NO_MARKER, EMPTY);
    supno[0] = -1;
    xsup[0]  = xlsub[0] = xusub[0] = xlusup[0] = 0;
    w_def    = panel_size;

    /* 
     * Work on one "panel" at a time. A panel is one of the following: 
     *	   (a) a relaxed supernode at the bottom of the etree, or
     *	   (b) panel_size contiguous columns, defined by the user
     */
    for (jcol = 0; jcol < min_mn; ) {

	if ( relax_end[jcol] != EMPTY ) { /* start of a relaxed snode */
   	    kcol = relax_end[jcol];	  /* end of the relaxed snode */
	    panel_histo[kcol-jcol+1]++;

	    /* --------------------------------------
	     * Factorize the relaxed supernode(jcol:kcol) 
	     * -------------------------------------- */
	    /* Determine the union of the row structure of the snode */
	    if ( (*info = zsnode_dfs(jcol, kcol, asub, xa_begin, xa_end,
				    xprune, marker, Glu)) != 0 )
		return;

            nextu    = xusub[jcol];
	    nextlu   = xlusup[jcol];
	    jsupno   = supno[jcol];
	    fsupc    = xsup[jsupno];
	    new_next = nextlu + (xlsub[fsupc+1]-xlsub[fsupc])*(kcol-jcol+1);
	    nzlumax = Glu->nzlumax;
	    while ( new_next > nzlumax ) {
		if ( (*info = zLUMemXpand(jcol, nextlu, LUSUP, &nzlumax, Glu)) )
		    return;
	    }
    
	    for (icol = jcol; icol<= kcol; icol++) {
		xusub[icol+1] = nextu;
		
    		/* Scatter into SPA dense[*] */
    		for (k = xa_begin[icol]; k < xa_end[icol]; k++)
        	    dense[asub[k]] = a[k];

	       	/* Numeric update within the snode */
	        zsnode_bmod(icol, jsupno, fsupc, dense, tempv, Glu, stat);

		if ( (*info = zpivotL(icol, diag_pivot_thresh, &usepr, perm_r,
				      iperm_r, iperm_c, &pivrow, Glu, stat)) )
		    if ( iinfo == 0 ) iinfo = *info;
		
#ifdef DEBUG
		zprint_lu_col("[1]: ", icol, pivrow, xprune, Glu);
#endif

	    }

	    jcol = icol;

	} else { /* Work on one panel of panel_size columns */
	    
	    /* Adjust panel_size so that a panel won't overlap with the next 
	     * relaxed snode.
	     */
	    panel_size = w_def;
	    for (k = jcol + 1; k < SUPERLU_MIN(jcol+panel_size, min_mn); k++) 
		if ( relax_end[k] != EMPTY ) {
		    panel_size = k - jcol;
		    break;
		}
	    if ( k == min_mn ) panel_size = min_mn - jcol;
	    panel_histo[panel_size]++;

	    /* symbolic factor on a panel of columns */
	    zpanel_dfs(m, panel_size, jcol, A, perm_r, &nseg1,
		      dense, panel_lsub, segrep, repfnz, xprune,
		      marker, parent, xplore, Glu);
	    
	    /* numeric sup-panel updates in topological order */
	    zpanel_bmod(m, panel_size, jcol, nseg1, dense,
		        tempv, segrep, repfnz, Glu, stat);
	    
	    /* Sparse LU within the panel, and below panel diagonal */
    	    for ( jj = jcol; jj < jcol + panel_size; jj++) {
 		k = (jj - jcol) * m; /* column index for w-wide arrays */

		nseg = nseg1;	/* Begin after all the panel segments */

	    	if ((*info = zcolumn_dfs(m, jj, perm_r, &nseg, &panel_lsub[k],
					segrep, &repfnz[k], xprune, marker,
					parent, xplore, Glu)) != 0) return;

	      	/* Numeric updates */
	    	if ((*info = zcolumn_bmod(jj, (nseg - nseg1), &dense[k],
					 tempv, &segrep[nseg1], &repfnz[k],
					 jcol, Glu, stat)) != 0) return;
		
	        /* Copy the U-segments to ucol[*] */
		if ((*info = zcopy_to_ucol(jj, nseg, segrep, &repfnz[k],
					  perm_r, &dense[k], Glu)) != 0)
		    return;

	    	if ( (*info = zpivotL(jj, diag_pivot_thresh, &usepr, perm_r,
				      iperm_r, iperm_c, &pivrow, Glu, stat)) )
		    if ( iinfo == 0 ) iinfo = *info;

		/* Prune columns (0:jj-1) using column jj */
	    	zpruneL(jj, perm_r, pivrow, nseg, segrep,
                        &repfnz[k], xprune, Glu);

		/* Reset repfnz[] for this column */
	    	resetrep_col (nseg, segrep, &repfnz[k]);
		
#ifdef DEBUG
		zprint_lu_col("[2]: ", jj, pivrow, xprune, Glu);
#endif

	    }

   	    jcol += panel_size;	/* Move to the next panel */

	} /* else */

    } /* for */

    *info = iinfo;
    
    if ( m > n ) {
	k = 0;
        for (i = 0; i < m; ++i) 
            if ( perm_r[i] == EMPTY ) {
    		perm_r[i] = n + k;
		++k;
	    }
    }

    countnz(min_mn, xprune, &nnzL, &nnzU, Glu);
    fixupL(min_mn, perm_r, Glu);

    zLUWorkFree(iwork, zwork, Glu); /* Free work space and compress storage */

    if ( fact == SamePattern_SameRowPerm ) {
        /* L and U structures may have changed due to possibly different
	   pivoting, even though the storage is available.
	   There could also be memory expansions, so the array locations
           may have changed, */
        ((SCformat *)L->Store)->nnz = nnzL;
	((SCformat *)L->Store)->nsuper = Glu->supno[n];
	((SCformat *)L->Store)->nzval = (doublecomplex *) Glu->lusup;
	((SCformat *)L->Store)->nzval_colptr = Glu->xlusup;
	((SCformat *)L->Store)->rowind = Glu->lsub;
	((SCformat *)L->Store)->rowind_colptr = Glu->xlsub;
	((NCformat *)U->Store)->nnz = nnzU;
	((NCformat *)U->Store)->nzval = (doublecomplex *) Glu->ucol;
	((NCformat *)U->Store)->rowind = Glu->usub;
	((NCformat *)U->Store)->colptr = Glu->xusub;
    } else {
        zCreate_SuperNode_Matrix(L, A->nrow, min_mn, nnzL, 
	      (doublecomplex *) Glu->lusup, Glu->xlusup, 
              Glu->lsub, Glu->xlsub, Glu->supno, Glu->xsup,
              SLU_SC, SLU_Z, SLU_TRLU);
    	zCreate_CompCol_Matrix(U, min_mn, min_mn, nnzU, 
	      (doublecomplex *) Glu->ucol, Glu->usub, Glu->xusub,
              SLU_NC, SLU_Z, SLU_TRU);
    }
    
    ops[FACT] += ops[TRSV] + ops[GEMV];	
    stat->expansions = --(Glu->num_expansions);
    
    if ( iperm_r_allocated ) SUPERLU_FREE (iperm_r);
    SUPERLU_FREE (iperm_c);
    SUPERLU_FREE (relax_end);

}
