lin_alg.c

/*----------------------------------------------------------------------
  SerialReax - Reax Force Field Simulator

  Copyright (2010) Purdue University
  Hasan Metin Aktulga, haktulga@cs.purdue.edu
  Joseph Fogarty, jcfogart@mail.usf.edu
  Sagar Pandit, pandit@usf.edu
  Ananth Y Grama, ayg@cs.purdue.edu

  This program is free software; you can redistribute it and/or
  modify it under the terms of the GNU General Public License as
  published by the Free Software Foundation; either version 2 of
  the License, or (at your option) any later version.

  This program 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 General Public License for more details:
  <http://www.gnu.org/licenses/>.
  ----------------------------------------------------------------------*/

#include "lin_alg.h"
#include "allocate.h"
#include "tool_box.h"
#include "vector.h"

#include "print_utils.h"

/* Intel MKL */
#if defined(HAVE_LAPACKE_MKL)
#include "mkl.h"
/* reference LAPACK */
#elif defined(HAVE_LAPACKE)
#include "lapacke.h"
#endif

typedef struct
{
    unsigned int j;
    real val;
} sparse_matrix_entry;


/* global to make OpenMP shared (Sparse_MatVec) */
#ifdef _OPENMP
real *b_local = NULL;
#endif
/* global to make OpenMP shared (apply_preconditioner) */
real *Dinv_L = NULL, *Dinv_U = NULL;
/* global to make OpenMP shared (tri_solve_level_sched) */
int levels = 1;
int levels_L = 1, levels_U = 1;
unsigned int *row_levels_L = NULL, *level_rows_L = NULL, *level_rows_cnt_L = NULL;
unsigned int *row_levels_U = NULL, *level_rows_U = NULL, *level_rows_cnt_U = NULL;
unsigned int *row_levels, *level_rows, *level_rows_cnt;
unsigned int *top = NULL;
/* global to make OpenMP shared (graph_coloring) */
unsigned int *color = NULL;
unsigned int *to_color = NULL;
unsigned int *conflict = NULL;
unsigned int *conflict_cnt = NULL;
unsigned int *temp_ptr;
unsigned int *recolor = NULL;
unsigned int recolor_cnt;
unsigned int *color_top = NULL;
/* global to make OpenMP shared (sort_colors) */
unsigned int *permuted_row_col = NULL;
unsigned int *permuted_row_col_inv = NULL;
real *y_p = NULL;
/* global to make OpenMP shared (permute_vector) */
real *x_p = NULL;
unsigned int *mapping = NULL;
sparse_matrix *H_full;
sparse_matrix *H_p;
/* global to make OpenMP shared (jacobi_iter) */
real *Dinv_b = NULL, *rp = NULL, *rp2 = NULL, *rp3 = NULL;


#if defined(TEST_MAT)
static sparse_matrix * create_test_mat( void )
{
    unsigned int i, n;
    sparse_matrix *H_test;

    if ( Allocate_Matrix( &H_test, 3, 6 ) == FAILURE )
    {
        fprintf( stderr, "not enough memory for test matrices. terminating.\n" );
        exit( INSUFFICIENT_MEMORY );
    }

    //3x3, SPD, store lower half
    i = 0;
    n = 0;
    H_test->start[n] = i;
    H_test->j[i] = 0;
    H_test->val[i] = 4.;
    ++i;
    ++n;
    H_test->start[n] = i;
    H_test->j[i] = 0;
    H_test->val[i] = 12.;
    ++i;
    H_test->j[i] = 1;
    H_test->val[i] = 37.;
    ++i;
    ++n;
    H_test->start[n] = i;
    H_test->j[i] = 0;
    H_test->val[i] = -16.;
    ++i;
    H_test->j[i] = 1;
    H_test->val[i] = -43.;
    ++i;
    H_test->j[i] = 2;
    H_test->val[i] = 98.;
    ++i;
    ++n;
    H_test->start[n] = i;

    return H_test;
}
#endif


/* Routine used with qsort for sorting nonzeros within a sparse matrix row
 *
 * v1/v2: pointers to column indices of nonzeros within a row (unsigned int)
 */
static int compare_matrix_entry(const void *v1, const void *v2)
{
    /* larger element has larger column index */
    return ((sparse_matrix_entry *)v1)->j - ((sparse_matrix_entry *)v2)->j;
}


/* Routine used for sorting nonzeros within a sparse matrix row;
 *  internally, a combination of qsort and manual sorting is utilized
 *  (parallel calls to qsort when multithreading, rows mapped to threads)
 *
 * A: sparse matrix for which to sort nonzeros within a row, stored in CSR format
 */
void Sort_Matrix_Rows( sparse_matrix * const A )
{
    unsigned int i, j, si, ei;
    sparse_matrix_entry *temp;

#ifdef _OPENMP
    //    #pragma omp parallel default(none) private(i, j, si, ei, temp) shared(stderr)
#endif
    {
        if ( ( temp = (sparse_matrix_entry *) malloc( A->n * sizeof(sparse_matrix_entry)) ) == NULL )
        {
            fprintf( stderr, "Not enough space for matrix row sort. Terminating...\n" );
            exit( INSUFFICIENT_MEMORY );
        }

        /* sort each row of A using column indices */
#ifdef _OPENMP
        //        #pragma omp for schedule(guided)
#endif
        for ( i = 0; i < A->n; ++i )
        {
            si = A->start[i];
            ei = A->start[i + 1];

            for ( j = 0; j < (ei - si); ++j )
            {
                (temp + j)->j = A->j[si + j];
                (temp + j)->val = A->val[si + j];
            }

            /* polymorphic sort in standard C library using column indices */
            qsort( temp, ei - si, sizeof(sparse_matrix_entry), compare_matrix_entry );

            for ( j = 0; j < (ei - si); ++j )
            {
                A->j[si + j] = (temp + j)->j;
                A->val[si + j] = (temp + j)->val;
            }
        }

        sfree( temp, "Sort_Matrix_Rows::temp" );
    }
}


/* Convert a symmetric, half-sored sparse matrix into
 * a full-stored sparse matrix
 *
 * A: symmetric sparse matrix, lower half stored in CSR
 * A_full: resultant full sparse matrix in CSR
 *   If A_full is NULL, allocate space, otherwise do not
 *
 * Assumptions:
 *   A has non-zero diagonals
 *   Each row of A has at least one non-zero (i.e., no rows with all zeros) */
static void compute_full_sparse_matrix( const sparse_matrix * const A,
                                        sparse_matrix ** A_full )
{
    int count, i, pj;
    sparse_matrix *A_t;

    if ( *A_full == NULL )
    {
        if ( Allocate_Matrix( A_full, A->n, 2 * A->m - A->n ) == FAILURE )
        {
            fprintf( stderr, "not enough memory for full A. terminating.\n" );
            exit( INSUFFICIENT_MEMORY );
        }
    }


    if ( Allocate_Matrix( &A_t, A->n, A->m ) == FAILURE )
    {
        fprintf( stderr, "not enough memory for full A. terminating.\n" );
        exit( INSUFFICIENT_MEMORY );
    }

    /* Set up the sparse matrix data structure for A. */
    Transpose( A, A_t );

    count = 0;
    for ( i = 0; i < A->n; ++i )
    {

        if ((*A_full)->start == NULL)
        {
        }
        (*A_full)->start[i] = count;

        /* A: symmetric, lower triangular portion only stored */
        for ( pj = A->start[i]; pj < A->start[i + 1]; ++pj )
        {
            (*A_full)->val[count] = A->val[pj];
            (*A_full)->j[count] = A->j[pj];
            ++count;
        }
        /* A^T: symmetric, upper triangular portion only stored;
         * skip diagonal from A^T, as included from A above */
        for ( pj = A_t->start[i] + 1; pj < A_t->start[i + 1]; ++pj )
        {
            (*A_full)->val[count] = A_t->val[pj];
            (*A_full)->j[count] = A_t->j[pj];
            ++count;
        }
    }
    (*A_full)->start[i] = count;

    Deallocate_Matrix( A_t );
}


/* Setup routines for sparse approximate inverse preconditioner
 *
 * A: symmetric sparse matrix, lower half stored in CSR
 * filter:
 * A_spar_patt:
 *
 * Assumptions:
 *   A has non-zero diagonals
 *   Each row of A has at least one non-zero (i.e., no rows with all zeros) */
void Setup_Sparsity_Pattern( const sparse_matrix * const A,
                             const real filter, sparse_matrix ** A_spar_patt )
{
    int i, pj, size;
    real min, max, threshold, val;

    min = 0.0;
    max = 0.0;

    if ( *A_spar_patt == NULL )
    {
        if ( Allocate_Matrix( A_spar_patt, A->n, A->m ) == FAILURE )
        {
            fprintf( stderr, "[SAI] Not enough memory for preconditioning matrices. terminating.\n" );
            exit( INSUFFICIENT_MEMORY );
        }
    }
    else if ( ((*A_spar_patt)->m) < (A->m) )
    {
        Deallocate_Matrix( *A_spar_patt );
        if ( Allocate_Matrix( A_spar_patt, A->n, A->m ) == FAILURE )
        {
            fprintf( stderr, "[SAI] Not enough memory for preconditioning matrices. terminating.\n" );
            exit( INSUFFICIENT_MEMORY );
        }
    }

    // find min and max element of the matrix
    for ( i = 0; i < A->n; ++i )
    {
        for ( pj = A->start[i]; pj < A->start[i + 1]; ++pj )
        {
            val = A->val[pj];
            if ( pj == 0 )
            {
                min = val;
                max = val;
            }
            else
            {
                if ( min > val )
                {
                    min = val;
                }
                if ( max < val )
                {
                    max = val;
                }
            }
        }
    }

    threshold = min + (max - min) * (1.0 - filter);
    // calculate the nnz of the sparsity pattern
    //    for ( size = 0, i = 0; i < A->n; ++i )
    //    {
    //        for ( pj = A->start[i]; pj < A->start[i + 1]; ++pj )
    //        {
    //            if ( threshold <= A->val[pj] )
    //                size++;
    //        }
    //    }
    //
    //    if ( Allocate_Matrix( A_spar_patt, A->n, size ) == NULL )
    //    {
    //        fprintf( stderr, "[SAI] Not enough memory for preconditioning matrices. terminating.\n" );
    //        exit( INSUFFICIENT_MEMORY );
    //    }

    //(*A_spar_patt)->start[(*A_spar_patt)->n] = size;

    // fill the sparsity pattern
    for ( size = 0, i = 0; i < A->n; ++i )
    {
        (*A_spar_patt)->start[i] = size;

        for ( pj = A->start[i]; pj < A->start[i + 1]; ++pj )
        {
            if ( ( A->val[pj] >= threshold )  || ( A->j[pj] == i ) )
            {
                (*A_spar_patt)->val[size] = A->val[pj];
                (*A_spar_patt)->j[size] = A->j[pj];
                size++;
            }
        }
    }
    (*A_spar_patt)->start[A->n] = size;
}

void Calculate_Droptol( const sparse_matrix * const A,
                        real * const droptol, const real dtol )
{
    int i, j, k;
    real val;
#ifdef _OPENMP
    static real *droptol_local;
    unsigned int tid;
#endif

#ifdef _OPENMP
    #pragma omp parallel default(none) private(i, j, k, val, tid), shared(droptol_local, stderr)
#endif
    {
#ifdef _OPENMP
        tid = omp_get_thread_num();

        #pragma omp master
        {
            /* keep b_local for program duration to avoid allocate/free
             * overhead per Sparse_MatVec call*/
            if ( droptol_local == NULL )
            {
                if ( (droptol_local = (real*) malloc( omp_get_num_threads() * A->n * sizeof(real))) == NULL )
                {
                    fprintf( stderr, "Not enough space for droptol. Terminating...\n" );
                    exit( INSUFFICIENT_MEMORY );
                }
            }
        }

        #pragma omp barrier
#endif

        /* init droptol to 0 */
        for ( i = 0; i < A->n; ++i )
        {
#ifdef _OPENMP
            droptol_local[tid * A->n + i] = 0.0;
#else
            droptol[i] = 0.0;
#endif
        }

#ifdef _OPENMP
        #pragma omp barrier
#endif

        /* calculate sqaure of the norm of each row */
#ifdef _OPENMP
        #pragma omp for schedule(static)
#endif
        for ( i = 0; i < A->n; ++i )
        {
            for ( k = A->start[i]; k < A->start[i + 1] - 1; ++k )
            {
                j = A->j[k];
                val = A->val[k];

#ifdef _OPENMP
                droptol_local[tid * A->n + i] += val * val;
                droptol_local[tid * A->n + j] += val * val;
#else
                droptol[i] += val * val;
                droptol[j] += val * val;
#endif
            }

            // diagonal entry
            val = A->val[k];
#ifdef _OPENMP
            droptol_local[tid * A->n + i] += val * val;
#else
            droptol[i] += val * val;
#endif
        }

#ifdef _OPENMP
        #pragma omp barrier

        #pragma omp for schedule(static)
        for ( i = 0; i < A->n; ++i )
        {
            droptol[i] = 0.0;
            for ( k = 0; k < omp_get_num_threads(); ++k )
            {
                droptol[i] += droptol_local[k * A->n + i];
            }
        }

        #pragma omp barrier
#endif

        /* calculate local droptol for each row */
        //fprintf( stderr, "droptol: " );
#ifdef _OPENMP
        #pragma omp for schedule(static)
#endif
        for ( i = 0; i < A->n; ++i )
        {
            //fprintf( stderr, "%f-->", droptol[i] );
            droptol[i] = SQRT( droptol[i] ) * dtol;
            //fprintf( stderr, "%f  ", droptol[i] );
        }
        //fprintf( stderr, "\n" );
    }
}


int Estimate_LU_Fill( const sparse_matrix * const A, const real * const droptol )
{
    int i, pj;
    int fillin;
    real val;

    fillin = 0;

#ifdef _OPENMP
    #pragma omp parallel for schedule(static) \
    default(none) private(i, pj, val) reduction(+: fillin)
#endif
    for ( i = 0; i < A->n; ++i )
    {
        for ( pj = A->start[i]; pj < A->start[i + 1] - 1; ++pj )
        {
            val = A->val[pj];

            if ( FABS(val) > droptol[i] )
            {
                ++fillin;
            }
        }
    }

    return fillin + A->n;
}


#if defined(HAVE_SUPERLU_MT)
real SuperLU_Factorize( const sparse_matrix * const A,
                        sparse_matrix * const L, sparse_matrix * const U )
{
    unsigned int i, pj, count, *Ltop, *Utop, r;
    sparse_matrix *A_t;
    SuperMatrix A_S, AC_S, L_S, U_S;
    NCformat *A_S_store;
    SCPformat *L_S_store;
    NCPformat *U_S_store;
    superlumt_options_t superlumt_options;
    pxgstrf_shared_t pxgstrf_shared;
    pdgstrf_threadarg_t *pdgstrf_threadarg;
    int_t nprocs;
    fact_t fact;
    trans_t trans;
    yes_no_t refact, usepr;
    real u, drop_tol;
    real *a, *at;
    int_t *asub, *atsub, *xa, *xat;
    int_t *perm_c; /* column permutation vector */
    int_t *perm_r; /* row permutations from partial pivoting */
    void *work;
    int_t info, lwork;
    int_t permc_spec, panel_size, relax;
    Gstat_t Gstat;
    flops_t flopcnt;

    /* Default parameters to control factorization. */
#ifdef _OPENMP
    //TODO: set as global parameter and use
    #pragma omp parallel \
    default(none) shared(nprocs)
    {
        #pragma omp master
        {
            /* SuperLU_MT spawns threads internally, so set and pass parameter */
            nprocs = omp_get_num_threads();
        }
    }
#else
    nprocs = 1;
#endif

    //    fact = EQUILIBRATE; /* equilibrate A (i.e., scale rows & cols to have unit norm), then factorize */
    fact = DOFACT; /* factor from scratch */
    trans = NOTRANS;
    refact = NO; /* first time factorization */
    //TODO: add to control file and use the value there to set these
    panel_size = sp_ienv(1); /* # consec. cols treated as unit task */
    relax = sp_ienv(2); /* # cols grouped as relaxed supernode */
    u = 1.0; /* diagonal pivoting threshold */
    usepr = NO;
    drop_tol = 0.0;
    work = NULL;
    lwork = 0;

#if defined(DEBUG)
    fprintf( stderr, "nprocs = %d\n", nprocs );
    fprintf( stderr, "Panel size = %d\n", panel_size );
    fprintf( stderr, "Relax = %d\n", relax );
#endif

    if ( !(perm_r = intMalloc(A->n)) )
    {
        SUPERLU_ABORT("Malloc fails for perm_r[].");
    }
    if ( !(perm_c = intMalloc(A->n)) )
    {
        SUPERLU_ABORT("Malloc fails for perm_c[].");
    }
    if ( !(superlumt_options.etree = intMalloc(A->n)) )
    {
        SUPERLU_ABORT("Malloc fails for etree[].");
    }
    if ( !(superlumt_options.colcnt_h = intMalloc(A->n)) )
    {
        SUPERLU_ABORT("Malloc fails for colcnt_h[].");
    }
    if ( !(superlumt_options.part_super_h = intMalloc(A->n)) )
    {
        SUPERLU_ABORT("Malloc fails for part_super__h[].");
    }
    if ( ( (a = (real*) malloc( (2 * A->start[A->n] - A->n) * sizeof(real))) == NULL )
            || ( (asub = (int_t*) malloc( (2 * A->start[A->n] - A->n) * sizeof(int_t))) == NULL )
            || ( (xa = (int_t*) malloc( (A->n + 1) * sizeof(int_t))) == NULL )
            || ( (Ltop = (unsigned int*) malloc( (A->n + 1) * sizeof(unsigned int))) == NULL )
            || ( (Utop = (unsigned int*) malloc( (A->n + 1) * sizeof(unsigned int))) == NULL ) )
    {
        fprintf( stderr, "Not enough space for SuperLU factorization. Terminating...\n" );
        exit( INSUFFICIENT_MEMORY );
    }
    if ( Allocate_Matrix( &A_t, A->n, A->m ) == FAILURE )
    {
        fprintf( stderr, "not enough memory for preconditioning matrices. terminating.\n" );
        exit( INSUFFICIENT_MEMORY );
    }

    /* Set up the sparse matrix data structure for A. */
    Transpose( A, A_t );

    count = 0;
    for ( i = 0; i < A->n; ++i )
    {
        xa[i] = count;
        for ( pj = A->start[i]; pj < A->start[i + 1]; ++pj )
        {
            a[count] = A->entries[pj].val;
            asub[count] = A->entries[pj].j;
            ++count;
        }
        for ( pj = A_t->start[i] + 1; pj < A_t->start[i + 1]; ++pj )
        {
            a[count] = A_t->entries[pj].val;
            asub[count] = A_t->entries[pj].j;
            ++count;
        }
    }
    xa[i] = count;

    dCompRow_to_CompCol( A->n, A->n, 2 * A->start[A->n] - A->n, a, asub, xa,
                         &at, &atsub, &xat );

    for ( i = 0; i < (2 * A->start[A->n] - A->n); ++i )
        fprintf( stderr, "%6d", asub[i] );
    fprintf( stderr, "\n" );
    for ( i = 0; i < (2 * A->start[A->n] - A->n); ++i )
        fprintf( stderr, "%6.1f", a[i] );
    fprintf( stderr, "\n" );
    for ( i = 0; i <= A->n; ++i )
        fprintf( stderr, "%6d", xa[i] );
    fprintf( stderr, "\n" );
    for ( i = 0; i < (2 * A->start[A->n] - A->n); ++i )
        fprintf( stderr, "%6d", atsub[i] );
    fprintf( stderr, "\n" );
    for ( i = 0; i < (2 * A->start[A->n] - A->n); ++i )
        fprintf( stderr, "%6.1f", at[i] );
    fprintf( stderr, "\n" );
    for ( i = 0; i <= A->n; ++i )
        fprintf( stderr, "%6d", xat[i] );
    fprintf( stderr, "\n" );

    A_S.Stype = SLU_NC; /* column-wise, no supernode */
    A_S.Dtype = SLU_D; /* double-precision */
    A_S.Mtype = SLU_GE; /* full (general) matrix -- required for parallel factorization */
    A_S.nrow = A->n;
    A_S.ncol = A->n;
    A_S.Store = (void *) SUPERLU_MALLOC( sizeof(NCformat) );
    A_S_store = (NCformat *) A_S.Store;
    A_S_store->nnz = 2 * A->start[A->n] - A->n;
    A_S_store->nzval = at;
    A_S_store->rowind = atsub;
    A_S_store->colptr = xat;

    /* ------------------------------------------------------------
       Allocate storage and initialize statistics variables.
       ------------------------------------------------------------*/
    StatAlloc( A->n, nprocs, panel_size, relax, &Gstat );
    StatInit( A->n, nprocs, &Gstat );

    /* ------------------------------------------------------------
       Get column permutation vector perm_c[], according to permc_spec:
       permc_spec = 0: natural ordering
       permc_spec = 1: minimum degree ordering on structure of A'*A
       permc_spec = 2: minimum degree ordering on structure of A'+A
       permc_spec = 3: approximate minimum degree for unsymmetric matrices
       ------------------------------------------------------------*/
    permc_spec = 0;
    get_perm_c( permc_spec, &A_S, perm_c );

    /* ------------------------------------------------------------
       Initialize the option structure superlumt_options using the
       user-input parameters;
       Apply perm_c to the columns of original A to form AC.
       ------------------------------------------------------------*/
    pdgstrf_init( nprocs, fact, trans, refact, panel_size, relax,
                  u, usepr, drop_tol, perm_c, perm_r,
                  work, lwork, &A_S, &AC_S, &superlumt_options, &Gstat );

    for ( i = 0; i < ((NCPformat*)AC_S.Store)->nnz; ++i )
        fprintf( stderr, "%6.1f", ((real*)(((NCPformat*)AC_S.Store)->nzval))[i] );
    fprintf( stderr, "\n" );

    /* ------------------------------------------------------------
       Compute the LU factorization of A.
       The following routine will create nprocs threads.
       ------------------------------------------------------------*/
    pdgstrf( &superlumt_options, &AC_S, perm_r, &L_S, &U_S, &Gstat, &info );

    fprintf( stderr, "INFO: %d\n", info );

    flopcnt = 0;
    for (i = 0; i < nprocs; ++i)
    {
        flopcnt += Gstat.procstat[i].fcops;
    }
    Gstat.ops[FACT] = flopcnt;

#if defined(DEBUG)
    printf("\n** Result of sparse LU **\n");
    L_S_store = (SCPformat *) L_S.Store;
    U_S_store = (NCPformat *) U_S.Store;
    printf( "No of nonzeros in factor L = " IFMT "\n", L_S_store->nnz );
    printf( "No of nonzeros in factor U = " IFMT "\n", U_S_store->nnz );
    fflush( stdout );
#endif

    /* convert L and R from SuperLU formats to CSR */
    memset( Ltop, 0, (A->n + 1) * sizeof(int) );
    memset( Utop, 0, (A->n + 1) * sizeof(int) );
    memset( L->start, 0, (A->n + 1) * sizeof(int) );
    memset( U->start, 0, (A->n + 1) * sizeof(int) );

    for ( i = 0; i < 2 * L_S_store->nnz; ++i )
        fprintf( stderr, "%6.1f", ((real*)(L_S_store->nzval))[i] );
    fprintf( stderr, "\n" );
    for ( i = 0; i < 2 * U_S_store->nnz; ++i )
        fprintf( stderr, "%6.1f", ((real*)(U_S_store->nzval))[i] );
    fprintf( stderr, "\n" );

    printf( "No of supernodes in factor L = " IFMT "\n", L_S_store->nsuper );
    for ( i = 0; i < A->n; ++i )
    {
        fprintf( stderr, "nzval_col_beg[%5d] = %d\n", i, L_S_store->nzval_colbeg[i] );
        fprintf( stderr, "nzval_col_end[%5d] = %d\n", i, L_S_store->nzval_colend[i] );
        //TODO: correct for SCPformat for L?
        //for( pj = L_S_store->rowind_colbeg[i]; pj < L_S_store->rowind_colend[i]; ++pj )
        //        for( pj = 0; pj < L_S_store->rowind_colend[i] - L_S_store->rowind_colbeg[i]; ++pj )
        //        {
        //            ++Ltop[L_S_store->rowind[L_S_store->rowind_colbeg[i] + pj] + 1];
        //        }
        fprintf( stderr, "col_beg[%5d] = %d\n", i, U_S_store->colbeg[i] );
        fprintf( stderr, "col_end[%5d] = %d\n", i, U_S_store->colend[i] );
        for ( pj = U_S_store->colbeg[i]; pj < U_S_store->colend[i]; ++pj )
        {
            ++Utop[U_S_store->rowind[pj] + 1];
            fprintf( stderr, "Utop[%5d]     = %d\n", U_S_store->rowind[pj] + 1, Utop[U_S_store->rowind[pj] + 1] );
        }
    }
    for ( i = 1; i <= A->n; ++i )
    {
        //        Ltop[i] = L->start[i] = Ltop[i] + Ltop[i - 1];
        Utop[i] = U->start[i] = Utop[i] + Utop[i - 1];
        //        fprintf( stderr, "Utop[%5d]     = %d\n", i, Utop[i] );
        //        fprintf( stderr, "U->start[%5d] = %d\n", i, U->start[i] );
    }
    for ( i = 0; i < A->n; ++i )
    {
        //        for( pj = 0; pj < L_S_store->nzval_colend[i] - L_S_store->nzval_colbeg[i]; ++pj )
        //        {
        //            r = L_S_store->rowind[L_S_store->rowind_colbeg[i] + pj];
        //            L->entries[Ltop[r]].j = r;
        //            L->entries[Ltop[r]].val = ((real*)L_S_store->nzval)[L_S_store->nzval_colbeg[i] + pj];
        //            ++Ltop[r];
        //        }
        for ( pj = U_S_store->colbeg[i]; pj < U_S_store->colend[i]; ++pj )
        {
            r = U_S_store->rowind[pj];
            U->entries[Utop[r]].j = i;
            U->entries[Utop[r]].val = ((real*)U_S_store->nzval)[pj];
            ++Utop[r];
        }
    }

    /* ------------------------------------------------------------
       Deallocate storage after factorization.
       ------------------------------------------------------------*/
    pxgstrf_finalize( &superlumt_options, &AC_S );
    Deallocate_Matrix( A_t );
    sfree( xa, "SuperLU_Factorize::xa" );
    sfree( asub, "SuperLU_Factorize::asub" );
    sfree( a, "SuperLU_Factorize::a" );
    SUPERLU_FREE( perm_r );
    SUPERLU_FREE( perm_c );
    SUPERLU_FREE( ((NCformat *)A_S.Store)->rowind );
    SUPERLU_FREE( ((NCformat *)A_S.Store)->colptr );
    SUPERLU_FREE( ((NCformat *)A_S.Store)->nzval );
    SUPERLU_FREE( A_S.Store );
    if ( lwork == 0 )
    {
        Destroy_SuperNode_SCP(&L_S);
        Destroy_CompCol_NCP(&U_S);
    }
    else if ( lwork > 0 )
    {
        SUPERLU_FREE(work);
    }
    StatFree(&Gstat);

    sfree( Utop, "SuperLU_Factorize::Utop" );
    sfree( Ltop, "SuperLU_Factorize::Ltop" );

    //TODO: return iters
    return 0.;
}
#endif


/* Diagonal (Jacobi) preconditioner computation */
real diag_pre_comp( const sparse_matrix * const H, real * const Hdia_inv )
{
    unsigned int i;
    real start;

    start = Get_Time( );

#ifdef _OPENMP
    #pragma omp parallel for schedule(static) \
    default(none) private(i)
#endif
    for ( i = 0; i < H->n; ++i )
    {
        if ( H->val[H->start[i + 1] - 1] != 0.0 )
        {
            Hdia_inv[i] = 1.0 / H->val[H->start[i + 1] - 1];
        }
        else
        {
            Hdia_inv[i] = 1.0;
        }
    }

    return Get_Timing_Info( start );
}


/* Incomplete Cholesky factorization with dual thresholding */
real ICHOLT( const sparse_matrix * const A, const real * const droptol,
             sparse_matrix * const L, sparse_matrix * const U )
{
    int *tmp_j;
    real *tmp_val;
    int i, j, pj, k1, k2, tmptop, Ltop;
    real val, start;
    unsigned int *Utop;

    start = Get_Time( );

    if ( ( Utop = (unsigned int*) malloc((A->n + 1) * sizeof(unsigned int)) ) == NULL ||
            ( tmp_j = (int*) malloc(A->n * sizeof(int)) ) == NULL ||
            ( tmp_val = (real*) malloc(A->n * sizeof(real)) ) == NULL )
    {
        fprintf( stderr, "[ICHOLT] Not enough memory for preconditioning matrices. terminating.\n" );
        exit( INSUFFICIENT_MEMORY );
    }

    // clear variables
    Ltop = 0;
    tmptop = 0;
    memset( L->start, 0, (A->n + 1) * sizeof(unsigned int) );
    memset( U->start, 0, (A->n + 1) * sizeof(unsigned int) );
    memset( Utop, 0, A->n * sizeof(unsigned int) );

    for ( i = 0; i < A->n; ++i )
    {
        L->start[i] = Ltop;
        tmptop = 0;

        for ( pj = A->start[i]; pj < A->start[i + 1] - 1; ++pj )
        {
            j = A->j[pj];
            val = A->val[pj];

            if ( FABS(val) > droptol[i] )
            {
                k1 = 0;
                k2 = L->start[j];
                while ( k1 < tmptop && k2 < L->start[j + 1] )
                {
                    if ( tmp_j[k1] < L->j[k2] )
                    {
                        ++k1;
                    }
                    else if ( tmp_j[k1] > L->j[k2] )
                    {
                        ++k2;
                    }
                    else
                    {
                        val -= (tmp_val[k1++] * L->val[k2++]);
                    }
                }

                // L matrix is lower triangular,
                // so right before the start of next row comes jth diagonal
                val /= L->val[L->start[j + 1] - 1];

                tmp_j[tmptop] = j;
                tmp_val[tmptop] = val;
                ++tmptop;
            }
        }

        // sanity check
        if ( A->j[pj] != i )
        {
            fprintf( stderr, "[ICHOLT] badly built A matrix!\n (i = %d) ", i );
            exit( NUMERIC_BREAKDOWN );
        }

        // compute the ith diagonal in L
        val = A->val[pj];
        for ( k1 = 0; k1 < tmptop; ++k1 )
        {
            val -= (tmp_val[k1] * tmp_val[k1]);
        }

#if defined(DEBUG)
        if ( val < 0.0 )
        {
            fprintf( stderr, "[ICHOLT] Numeric breakdown (SQRT of negative on diagonal i = %d). Terminating.\n", i );
            exit( NUMERIC_BREAKDOWN );

        }
#endif

        tmp_j[tmptop] = i;
        tmp_val[tmptop] = SQRT( val );

        // apply the dropping rule once again
        //fprintf( stderr, "row%d: tmptop: %d\n", i, tmptop );
        //for( k1 = 0; k1<= tmptop; ++k1 )
        //  fprintf( stderr, "%d(%f)  ", tmp[k1].j, tmp[k1].val );
        //fprintf( stderr, "\n" );
        //fprintf( stderr, "row(%d): droptol=%.4f\n", i+1, droptol[i] );
        for ( k1 = 0; k1 < tmptop; ++k1 )
        {
            if ( FABS(tmp_val[k1]) > droptol[i] / tmp_val[tmptop] )
            {
                L->j[Ltop] = tmp_j[k1];
                L->val[Ltop] = tmp_val[k1];
                U->start[tmp_j[k1] + 1]++;
                ++Ltop;
                //fprintf( stderr, "%d(%.4f)  ", tmp[k1].j+1, tmp[k1].val );
            }
        }
        // keep the diagonal in any case
        L->j[Ltop] = tmp_j[k1];
        L->val[Ltop] = tmp_val[k1];
        ++Ltop;
        //fprintf( stderr, "%d(%.4f)\n", tmp[k1].j+1,  tmp[k1].val );
    }

    L->start[i] = Ltop;
    //    fprintf( stderr, "nnz(L): %d, max: %d\n", Ltop, L->n * 50 );

    /* U = L^T (Cholesky factorization) */
    Transpose( L, U );
    //    for ( i = 1; i <= U->n; ++i )
    //    {
    //        Utop[i] = U->start[i] = U->start[i] + U->start[i - 1] + 1;
    //    }
    //    for ( i = 0; i < L->n; ++i )
    //    {
    //        for ( pj = L->start[i]; pj < L->start[i + 1]; ++pj )
    //        {
    //            j = L->j[pj];
    //            U->j[Utop[j]] = i;
    //            U->val[Utop[j]] = L->val[pj];
    //            Utop[j]++;
    //        }
    //    }

    //    fprintf( stderr, "nnz(U): %d, max: %d\n", Utop[U->n], U->n * 50 );

    sfree( tmp_val, "ICHOLT::tmp_val" );
    sfree( tmp_j, "ICHOLT::tmp_j" );
    sfree( Utop, "ICHOLT::Utop" );

    return Get_Timing_Info( start );
}


/* Fine-grained (parallel) incomplete Cholesky factorization
 *
 * Reference:
 * Edmond Chow and Aftab Patel
 * Fine-Grained Parallel Incomplete LU Factorization
 * SIAM J. Sci. Comp. */
#if defined(TESTING)
real ICHOL_PAR( const sparse_matrix * const A, const unsigned int sweeps,
                sparse_matrix * const U_t, sparse_matrix * const U )
{
    unsigned int i, j, k, pj, x = 0, y = 0, ei_x, ei_y;
    real *D, *D_inv, sum, start;
    sparse_matrix *DAD;
    int *Utop;

    start = Get_Time( );

    if ( Allocate_Matrix( &DAD, A->n, A->m ) == FAILURE ||
            ( D = (real*) malloc(A->n * sizeof(real)) ) == NULL ||
            ( D_inv = (real*) malloc(A->n * sizeof(real)) ) == NULL ||
            ( Utop = (int*) malloc((A->n + 1) * sizeof(int)) ) == NULL )
    {
        fprintf( stderr, "not enough memory for ICHOL_PAR preconditioning matrices. terminating.\n" );
        exit( INSUFFICIENT_MEMORY );
    }

#ifdef _OPENMP
    #pragma omp parallel for schedule(static) \
    default(none) shared(D_inv, D) private(i)
#endif
    for ( i = 0; i < A->n; ++i )
    {
        D_inv[i] = SQRT( A->val[A->start[i + 1] - 1] );
        D[i] = 1. / D_inv[i];
    }

    memset( U->start, 0, sizeof(unsigned int) * (A->n + 1) );
    memset( Utop, 0, sizeof(unsigned int) * (A->n + 1) );