internal Namespace Reference

Classes
struct	colamd_col
union	colamd_col.shared1
union	colamd_col.shared2
union	colamd_col.shared3
union	colamd_col.shared4
struct	Colamd_Row
union	Colamd_Row.shared1
union	Colamd_Row.shared2

Functions
template<typename IndexType >
IndexType	colamd_c (IndexType n_col)
template<typename IndexType >
IndexType	colamd_r (IndexType n_row)
template<typename IndexType >
static IndexType	init_rows_cols (IndexType n_row, IndexType n_col, Colamd_Row< IndexType > Row[], colamd_col< IndexType > col[], IndexType A[], IndexType p[], IndexType stats[COLAMD_STATS])
template<typename IndexType >
static void	init_scoring (IndexType n_row, IndexType n_col, Colamd_Row< IndexType > Row[], colamd_col< IndexType > Col[], IndexType A[], IndexType head[], double knobs[COLAMD_KNOBS], IndexType *p_n_row2, IndexType *p_n_col2, IndexType *p_max_deg)
template<typename IndexType >
static IndexType	find_ordering (IndexType n_row, IndexType n_col, IndexType Alen, Colamd_Row< IndexType > Row[], colamd_col< IndexType > Col[], IndexType A[], IndexType head[], IndexType n_col2, IndexType max_deg, IndexType pfree)
template<typename IndexType >
static void	order_children (IndexType n_col, colamd_col< IndexType > Col[], IndexType p[])
template<typename IndexType >
static void	detect_super_cols (colamd_col< IndexType > Col[], IndexType A[], IndexType head[], IndexType row_start, IndexType row_length)
template<typename IndexType >
static IndexType	garbage_collection (IndexType n_row, IndexType n_col, Colamd_Row< IndexType > Row[], colamd_col< IndexType > Col[], IndexType A[], IndexType *pfree)
template<typename IndexType >
static IndexType	clear_mark (IndexType n_row, Colamd_Row< IndexType > Row[])
template<typename IndexType >
IndexType	colamd_recommended (IndexType nnz, IndexType n_row, IndexType n_col)
	Returns the recommended value of Alen.
static void	colamd_set_defaults (double knobs[COLAMD_KNOBS])
	set default parameters The use of this routine is optional.
template<typename IndexType >
static bool	colamd (IndexType n_row, IndexType n_col, IndexType Alen, IndexType *A, IndexType *p, double knobs[COLAMD_KNOBS], IndexType stats[COLAMD_STATS])
	Computes a column ordering using the column approximate minimum degree ordering.

---

Class Documentation

internal::colamd_col

struct internal::colamd_col

template<typename IndexType>
struct internal::colamd_col< IndexType >

Class Members
IndexType	length
union colamd_col.shared1	shared1
union colamd_col.shared2	shared2
union colamd_col.shared3	shared3
union colamd_col.shared4	shared4
IndexType	start

internal::colamd_col.shared1

union internal::colamd_col.shared1

Class Members
IndexType	parent
IndexType	thickness

internal::colamd_col.shared2

union internal::colamd_col.shared2

Class Members
IndexType	order
IndexType	score

internal::colamd_col.shared3

union internal::colamd_col.shared3

Class Members
IndexType	hash
IndexType	headhash
IndexType	prev

internal::colamd_col.shared4

union internal::colamd_col.shared4

Class Members
IndexType	degree_next
IndexType	hash_next

internal::Colamd_Row

struct internal::Colamd_Row

template<typename IndexType>
struct internal::Colamd_Row< IndexType >

Class Members
IndexType	length
union Colamd_Row.shared1	shared1
union Colamd_Row.shared2	shared2
IndexType	start

internal::Colamd_Row.shared1

union internal::Colamd_Row.shared1

Class Members
IndexType	degree
IndexType	p

internal::Colamd_Row.shared2

union internal::Colamd_Row.shared2

Class Members
IndexType	first_column
IndexType	mark

Function Documentation

clear_mark()

template<typename IndexType >

static IndexType internal::clear_mark ( IndexType  n_row, Colamd_Row< IndexType >  Row[]  )

inlinestatic

{
  /* === Local variables ================================================== */
 
  IndexType r ;
 
  for (r = 0 ; r < n_row ; r++)
  {
    if (ROW_IS_ALIVE (r))
    {
      Row [r].shared2.mark = 0 ;
    }
  }
  return (1) ;
}

References ROW_IS_ALIVE.

colamd()

template<typename IndexType >

static bool internal::colamd ( IndexType  n_row, IndexType  n_col, IndexType  Alen, IndexType *  A, IndexType *  p, double  knobs[COLAMD_KNOBS], IndexType  stats[COLAMD_STATS]  )

static

Computes a column ordering using the column approximate minimum degree ordering.

Computes a column ordering (Q) of A such that P(AQ)=LU or (AQ)'AQ=LL' have less fill-in and require fewer floating point operations than factorizing the unpermuted matrix A or A'A, respectively.

Parameters

n_row	number of rows in A
n_col	number of columns in A
Alen,size	of the array A
A	row indices of the matrix, of size ALen
p	column pointers of A, of size n_col+1
knobs	parameter settings for colamd
stats	colamd output statistics and error codes

{
  /* === Local variables ================================================== */
  
  IndexType i ;     /* loop index */
  IndexType nnz ;     /* nonzeros in A */
  IndexType Row_size ;    /* size of Row [], in integers */
  IndexType Col_size ;    /* size of Col [], in integers */
  IndexType need ;      /* minimum required length of A */
  Colamd_Row<IndexType> *Row ;   /* pointer into A of Row [0..n_row] array */
  colamd_col<IndexType> *Col ;   /* pointer into A of Col [0..n_col] array */
  IndexType n_col2 ;    /* number of non-dense, non-empty columns */
  IndexType n_row2 ;    /* number of non-dense, non-empty rows */
  IndexType ngarbage ;    /* number of garbage collections performed */
  IndexType max_deg ;   /* maximum row degree */
  double default_knobs [COLAMD_KNOBS] ; /* default knobs array */
  
  
  /* === Check the input arguments ======================================== */
  
  if (!stats)
  {
    COLAMD_DEBUG0 (("colamd: stats not present\n")) ;
    return (false) ;
  }
  for (i = 0 ; i < COLAMD_STATS ; i++)
  {
    stats [i] = 0 ;
  }
  stats [COLAMD_STATUS] = COLAMD_OK ;
  stats [COLAMD_INFO1] = -1 ;
  stats [COLAMD_INFO2] = -1 ;
  
  if (!A)   /* A is not present */
  {
    stats [COLAMD_STATUS] = COLAMD_ERROR_A_not_present ;
    COLAMD_DEBUG0 (("colamd: A not present\n")) ;
    return (false) ;
  }
  
  if (!p)   /* p is not present */
  {
    stats [COLAMD_STATUS] = COLAMD_ERROR_p_not_present ;
    COLAMD_DEBUG0 (("colamd: p not present\n")) ;
    return (false) ;
  }
  
  if (n_row < 0)  /* n_row must be >= 0 */
  {
    stats [COLAMD_STATUS] = COLAMD_ERROR_nrow_negative ;
    stats [COLAMD_INFO1] = n_row ;
    COLAMD_DEBUG0 (("colamd: nrow negative %d\n", n_row)) ;
    return (false) ;
  }
  
  if (n_col < 0)  /* n_col must be >= 0 */
  {
    stats [COLAMD_STATUS] = COLAMD_ERROR_ncol_negative ;
    stats [COLAMD_INFO1] = n_col ;
    COLAMD_DEBUG0 (("colamd: ncol negative %d\n", n_col)) ;
    return (false) ;
  }
  
  nnz = p [n_col] ;
  if (nnz < 0)  /* nnz must be >= 0 */
  {
    stats [COLAMD_STATUS] = COLAMD_ERROR_nnz_negative ;
    stats [COLAMD_INFO1] = nnz ;
    COLAMD_DEBUG0 (("colamd: number of entries negative %d\n", nnz)) ;
    return (false) ;
  }
  
  if (p [0] != 0)
  {
    stats [COLAMD_STATUS] = COLAMD_ERROR_p0_nonzero ;
    stats [COLAMD_INFO1] = p [0] ;
    COLAMD_DEBUG0 (("colamd: p[0] not zero %d\n", p [0])) ;
    return (false) ;
  }
  
  /* === If no knobs, set default knobs =================================== */
  
  if (!knobs)
  {
    colamd_set_defaults (default_knobs) ;
    knobs = default_knobs ;
  }
  
  /* === Allocate the Row and Col arrays from array A ===================== */
  
  Col_size = colamd_c (n_col) ;
  Row_size = colamd_r (n_row) ;
  need = 2*nnz + n_col + Col_size + Row_size ;
  
  if (need > Alen)
  {
    /* not enough space in array A to perform the ordering */
    stats [COLAMD_STATUS] = COLAMD_ERROR_A_too_small ;
    stats [COLAMD_INFO1] = need ;
    stats [COLAMD_INFO2] = Alen ;
    COLAMD_DEBUG0 (("colamd: Need Alen >= %d, given only Alen = %d\n", need,Alen));
    return (false) ;
  }
  
  Alen -= Col_size + Row_size ;
  Col = (colamd_col<IndexType> *) &A [Alen] ;
  Row = (Colamd_Row<IndexType> *) &A [Alen + Col_size] ;
 
  /* === Construct the row and column data structures ===================== */
  
  if (!Eigen::internal::init_rows_cols (n_row, n_col, Row, Col, A, p, stats))
  {
    /* input matrix is invalid */
    COLAMD_DEBUG0 (("colamd: Matrix invalid\n")) ;
    return (false) ;
  }
  
  /* === Initialize scores, kill dense rows/columns ======================= */
 
  Eigen::internal::init_scoring (n_row, n_col, Row, Col, A, p, knobs,
    &n_row2, &n_col2, &max_deg) ;
  
  /* === Order the supercolumns =========================================== */
  
  ngarbage = Eigen::internal::find_ordering (n_row, n_col, Alen, Row, Col, A, p,
          n_col2, max_deg, 2*nnz) ;
  
  /* === Order the non-principal columns ================================== */
  
  Eigen::internal::order_children (n_col, Col, p) ;
  
  /* === Return statistics in stats ======================================= */
  
  stats [COLAMD_DENSE_ROW] = n_row - n_row2 ;
  stats [COLAMD_DENSE_COL] = n_col - n_col2 ;
  stats [COLAMD_DEFRAG_COUNT] = ngarbage ;
  COLAMD_DEBUG0 (("colamd: done.\n")) ; 
  return (true) ;
}

References colamd_c(), COLAMD_DEBUG0, COLAMD_DEFRAG_COUNT, COLAMD_DENSE_COL, COLAMD_DENSE_ROW, COLAMD_ERROR_A_not_present, COLAMD_ERROR_A_too_small, COLAMD_ERROR_ncol_negative, COLAMD_ERROR_nnz_negative, COLAMD_ERROR_nrow_negative, COLAMD_ERROR_p0_nonzero, COLAMD_ERROR_p_not_present, COLAMD_INFO1, COLAMD_INFO2, COLAMD_KNOBS, COLAMD_OK, colamd_r(), colamd_set_defaults(), COLAMD_STATS, and COLAMD_STATUS.

Referenced by Eigen::COLAMDOrdering< StorageIndex >::operator()().

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colamd_c()

template<typename IndexType >

IndexType internal::colamd_c ( IndexType  n_col )

inline

{ return IndexType( ((n_col) + 1) * sizeof (colamd_col<IndexType>) / sizeof (IndexType) ) ; }

Referenced by colamd(), and colamd_recommended().

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colamd_r()

template<typename IndexType >

IndexType internal::colamd_r ( IndexType  n_row )

inline

{ return IndexType(((n_row) + 1) * sizeof (Colamd_Row<IndexType>) / sizeof (IndexType)); }

Referenced by colamd(), and colamd_recommended().

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colamd_recommended()

template<typename IndexType >

IndexType internal::colamd_recommended ( IndexType  nnz, IndexType  n_row, IndexType  n_col  )

inline

Returns the recommended value of Alen.

Returns recommended value of Alen for use by colamd.
Returns -1 if any input argument is negative.
The use of this routine or macro is optional.
Note that the macro uses its arguments more than once, so be careful for side effects, if you pass expressions as arguments to COLAMD_RECOMMENDED.

Parameters

nnz	nonzeros in A
n_row	number of rows in A
n_col	number of columns in A

Returns

recommended value of Alen for use by colamd

{
  if ((nnz) < 0 || (n_row) < 0 || (n_col) < 0)
    return (-1);
  else
    return (2 * (nnz) + colamd_c (n_col) + colamd_r (n_row) + (n_col) + ((nnz) / 5)); 
}

References colamd_c(), and colamd_r().

Referenced by Eigen::COLAMDOrdering< StorageIndex >::operator()().

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colamd_set_defaults()

static void internal::colamd_set_defaults ( double  knobs[COLAMD_KNOBS] )

inlinestatic

set default parameters The use of this routine is optional.

Colamd: rows with more than (knobs [COLAMD_DENSE_ROW] * n_col) entries are removed prior to ordering. Columns with more than (knobs [COLAMD_DENSE_COL] * n_row) entries are removed prior to ordering, and placed last in the output column ordering.

COLAMD_DENSE_ROW and COLAMD_DENSE_COL are defined as 0 and 1, respectively, in colamd.h. Default values of these two knobs are both 0.5. Currently, only knobs [0] and knobs [1] are used, but future versions may use more knobs. If so, they will be properly set to their defaults by the future version of colamd_set_defaults, so that the code that calls colamd will not need to change, assuming that you either use colamd_set_defaults, or pass a (double *) NULL pointer as the knobs array to colamd or symamd.

Parameters

knobs

parameter settings for colamd

{
  /* === Local variables ================================================== */
  
  int i ;
 
  if (!knobs)
  {
    return ;      /* no knobs to initialize */
  }
  for (i = 0 ; i < COLAMD_KNOBS ; i++)
  {
    knobs [i] = 0 ;
  }
  knobs [COLAMD_DENSE_ROW] = 0.5 ;  /* ignore rows over 50% dense */
  knobs [COLAMD_DENSE_COL] = 0.5 ;  /* ignore columns over 50% dense */
}

References COLAMD_DENSE_COL, COLAMD_DENSE_ROW, and COLAMD_KNOBS.

Referenced by colamd(), and Eigen::COLAMDOrdering< StorageIndex >::operator()().

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detect_super_cols()

template<typename IndexType >

static void internal::detect_super_cols ( colamd_col< IndexType >  Col[], IndexType  A[], IndexType  head[], IndexType  row_start, IndexType  row_length  )

static

{
  /* === Local variables ================================================== */
 
  IndexType hash ;      /* hash value for a column */
  IndexType *rp ;     /* pointer to a row */
  IndexType c ;     /* a column index */
  IndexType super_c ;   /* column index of the column to absorb into */
  IndexType *cp1 ;      /* column pointer for column super_c */
  IndexType *cp2 ;      /* column pointer for column c */
  IndexType length ;    /* length of column super_c */
  IndexType prev_c ;    /* column preceding c in hash bucket */
  IndexType i ;     /* loop counter */
  IndexType *rp_end ;   /* pointer to the end of the row */
  IndexType col ;     /* a column index in the row to check */
  IndexType head_column ;   /* first column in hash bucket or degree list */
  IndexType first_col ;   /* first column in hash bucket */
 
  /* === Consider each column in the row ================================== */
 
  rp = &A [row_start] ;
  rp_end = rp + row_length ;
  while (rp < rp_end)
  {
    col = *rp++ ;
    if (COL_IS_DEAD (col))
    {
      continue ;
    }
 
    /* get hash number for this column */
    hash = Col [col].shared3.hash ;
    COLAMD_ASSERT (hash <= n_col) ;
 
    /* === Get the first column in this hash bucket ===================== */
 
    head_column = head [hash] ;
    if (head_column > COLAMD_EMPTY)
    {
      first_col = Col [head_column].shared3.headhash ;
    }
    else
    {
      first_col = - (head_column + 2) ;
    }
 
    /* === Consider each column in the hash bucket ====================== */
 
    for (super_c = first_col ; super_c != COLAMD_EMPTY ;
   super_c = Col [super_c].shared4.hash_next)
    {
      COLAMD_ASSERT (COL_IS_ALIVE (super_c)) ;
      COLAMD_ASSERT (Col [super_c].shared3.hash == hash) ;
      length = Col [super_c].length ;
 
      /* prev_c is the column preceding column c in the hash bucket */
      prev_c = super_c ;
 
      /* === Compare super_c with all columns after it ================ */
 
      for (c = Col [super_c].shared4.hash_next ;
     c != COLAMD_EMPTY ; c = Col [c].shared4.hash_next)
      {
  COLAMD_ASSERT (c != super_c) ;
  COLAMD_ASSERT (COL_IS_ALIVE (c)) ;
  COLAMD_ASSERT (Col [c].shared3.hash == hash) ;
 
  /* not identical if lengths or scores are different */
  if (Col [c].length != length ||
      Col [c].shared2.score != Col [super_c].shared2.score)
  {
    prev_c = c ;
    continue ;
  }
 
  /* compare the two columns */
  cp1 = &A [Col [super_c].start] ;
  cp2 = &A [Col [c].start] ;
 
  for (i = 0 ; i < length ; i++)
  {
    /* the columns are "clean" (no dead rows) */
    COLAMD_ASSERT (ROW_IS_ALIVE (*cp1))  ;
    COLAMD_ASSERT (ROW_IS_ALIVE (*cp2))  ;
    /* row indices will same order for both supercols, */
    /* no gather scatter nessasary */
    if (*cp1++ != *cp2++)
    {
      break ;
    }
  }
 
  /* the two columns are different if the for-loop "broke" */
  if (i != length)
  {
    prev_c = c ;
    continue ;
  }
 
  /* === Got it!  two columns are identical =================== */
 
  COLAMD_ASSERT (Col [c].shared2.score == Col [super_c].shared2.score) ;
 
  Col [super_c].shared1.thickness += Col [c].shared1.thickness ;
  Col [c].shared1.parent = super_c ;
  KILL_NON_PRINCIPAL_COL (c) ;
  /* order c later, in order_children() */
  Col [c].shared2.order = COLAMD_EMPTY ;
  /* remove c from hash bucket */
  Col [prev_c].shared4.hash_next = Col [c].shared4.hash_next ;
      }
    }
 
    /* === Empty this hash bucket ======================================= */
 
    if (head_column > COLAMD_EMPTY)
    {
      /* corresponding degree list "hash" is not empty */
      Col [head_column].shared3.headhash = COLAMD_EMPTY ;
    }
    else
    {
      /* corresponding degree list "hash" is empty */
      head [hash] = COLAMD_EMPTY ;
    }
  }
}

References col(), COL_IS_ALIVE, COL_IS_DEAD, COLAMD_ASSERT, COLAMD_EMPTY, head(), KILL_NON_PRINCIPAL_COL, internal::colamd_col< IndexType >::length, ROW_IS_ALIVE, internal::colamd_col< IndexType >::shared2, internal::colamd_col< IndexType >::shared3, internal::colamd_col< IndexType >::shared4, and internal::colamd_col< IndexType >::start.

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find_ordering()

template<typename IndexType >

static IndexType internal::find_ordering ( IndexType  n_row, IndexType  n_col, IndexType  Alen, Colamd_Row< IndexType >  Row[], colamd_col< IndexType >  Col[], IndexType  A[], IndexType  head[], IndexType  n_col2, IndexType  max_deg, IndexType  pfree  )

static

{
  /* === Local variables ================================================== */
 
  IndexType k ;     /* current pivot ordering step */
  IndexType pivot_col ;   /* current pivot column */
  IndexType *cp ;     /* a column pointer */
  IndexType *rp ;     /* a row pointer */
  IndexType pivot_row ;   /* current pivot row */
  IndexType *new_cp ;   /* modified column pointer */
  IndexType *new_rp ;   /* modified row pointer */
  IndexType pivot_row_start ; /* pointer to start of pivot row */
  IndexType pivot_row_degree ;  /* number of columns in pivot row */
  IndexType pivot_row_length ;  /* number of supercolumns in pivot row */
  IndexType pivot_col_score ; /* score of pivot column */
  IndexType needed_memory ;   /* free space needed for pivot row */
  IndexType *cp_end ;   /* pointer to the end of a column */
  IndexType *rp_end ;   /* pointer to the end of a row */
  IndexType row ;     /* a row index */
  IndexType col ;     /* a column index */
  IndexType max_score ;   /* maximum possible score */
  IndexType cur_score ;   /* score of current column */
  unsigned int hash ;   /* hash value for supernode detection */
  IndexType head_column ;   /* head of hash bucket */
  IndexType first_col ;   /* first column in hash bucket */
  IndexType tag_mark ;    /* marker value for mark array */
  IndexType row_mark ;    /* Row [row].shared2.mark */
  IndexType set_difference ;  /* set difference size of row with pivot row */
  IndexType min_score ;   /* smallest column score */
  IndexType col_thickness ;   /* "thickness" (no. of columns in a supercol) */
  IndexType max_mark ;    /* maximum value of tag_mark */
  IndexType pivot_col_thickness ; /* number of columns represented by pivot col */
  IndexType prev_col ;    /* Used by Dlist operations. */
  IndexType next_col ;    /* Used by Dlist operations. */
  IndexType ngarbage ;    /* number of garbage collections performed */
 
 
  /* === Initialization and clear mark ==================================== */
 
  max_mark = INT_MAX - n_col ;  /* INT_MAX defined in <limits.h> */
  tag_mark = Eigen::internal::clear_mark (n_row, Row) ;
  min_score = 0 ;
  ngarbage = 0 ;
  COLAMD_DEBUG1 (("colamd: Ordering, n_col2=%d\n", n_col2)) ;
 
  /* === Order the columns ================================================ */
 
  for (k = 0 ; k < n_col2 ; /* 'k' is incremented below */)
  {
 
    /* === Select pivot column, and order it ============================ */
 
    /* make sure degree list isn't empty */
    COLAMD_ASSERT (min_score >= 0) ;
    COLAMD_ASSERT (min_score <= n_col) ;
    COLAMD_ASSERT (head [min_score] >= COLAMD_EMPTY) ;
 
    /* get pivot column from head of minimum degree list */
    while (min_score < n_col && head [min_score] == COLAMD_EMPTY)
    {
      min_score++ ;
    }
    pivot_col = head [min_score] ;
    COLAMD_ASSERT (pivot_col >= 0 && pivot_col <= n_col) ;
    next_col = Col [pivot_col].shared4.degree_next ;
    head [min_score] = next_col ;
    if (next_col != COLAMD_EMPTY)
    {
      Col [next_col].shared3.prev = COLAMD_EMPTY ;
    }
 
    COLAMD_ASSERT (COL_IS_ALIVE (pivot_col)) ;
    COLAMD_DEBUG3 (("Pivot col: %d\n", pivot_col)) ;
 
    /* remember score for defrag check */
    pivot_col_score = Col [pivot_col].shared2.score ;
 
    /* the pivot column is the kth column in the pivot order */
    Col [pivot_col].shared2.order = k ;
 
    /* increment order count by column thickness */
    pivot_col_thickness = Col [pivot_col].shared1.thickness ;
    k += pivot_col_thickness ;
    COLAMD_ASSERT (pivot_col_thickness > 0) ;
 
    /* === Garbage_collection, if necessary ============================= */
 
    needed_memory = numext::mini(pivot_col_score, n_col - k) ;
    if (pfree + needed_memory >= Alen)
    {
      pfree = Eigen::internal::garbage_collection (n_row, n_col, Row, Col, A, &A [pfree]) ;
      ngarbage++ ;
      /* after garbage collection we will have enough */
      COLAMD_ASSERT (pfree + needed_memory < Alen) ;
      /* garbage collection has wiped out the Row[].shared2.mark array */
      tag_mark = Eigen::internal::clear_mark (n_row, Row) ;
 
    }
 
    /* === Compute pivot row pattern ==================================== */
 
    /* get starting location for this new merged row */
    pivot_row_start = pfree ;
 
    /* initialize new row counts to zero */
    pivot_row_degree = 0 ;
 
    /* tag pivot column as having been visited so it isn't included */
    /* in merged pivot row */
    Col [pivot_col].shared1.thickness = -pivot_col_thickness ;
 
    /* pivot row is the union of all rows in the pivot column pattern */
    cp = &A [Col [pivot_col].start] ;
    cp_end = cp + Col [pivot_col].length ;
    while (cp < cp_end)
    {
      /* get a row */
      row = *cp++ ;
      COLAMD_DEBUG4 (("Pivot col pattern %d %d\n", ROW_IS_ALIVE (row), row)) ;
      /* skip if row is dead */
      if (ROW_IS_DEAD (row))
      {
  continue ;
      }
      rp = &A [Row [row].start] ;
      rp_end = rp + Row [row].length ;
      while (rp < rp_end)
      {
  /* get a column */
  col = *rp++ ;
  /* add the column, if alive and untagged */
  col_thickness = Col [col].shared1.thickness ;
  if (col_thickness > 0 && COL_IS_ALIVE (col))
  {
    /* tag column in pivot row */
    Col [col].shared1.thickness = -col_thickness ;
    COLAMD_ASSERT (pfree < Alen) ;
    /* place column in pivot row */
    A [pfree++] = col ;
    pivot_row_degree += col_thickness ;
  }
      }
    }
 
    /* clear tag on pivot column */
    Col [pivot_col].shared1.thickness = pivot_col_thickness ;
    max_deg = numext::maxi(max_deg, pivot_row_degree) ;
 
 
    /* === Kill all rows used to construct pivot row ==================== */
 
    /* also kill pivot row, temporarily */
    cp = &A [Col [pivot_col].start] ;
    cp_end = cp + Col [pivot_col].length ;
    while (cp < cp_end)
    {
      /* may be killing an already dead row */
      row = *cp++ ;
      COLAMD_DEBUG3 (("Kill row in pivot col: %d\n", row)) ;
      KILL_ROW (row) ;
    }
 
    /* === Select a row index to use as the new pivot row =============== */
 
    pivot_row_length = pfree - pivot_row_start ;
    if (pivot_row_length > 0)
    {
      /* pick the "pivot" row arbitrarily (first row in col) */
      pivot_row = A [Col [pivot_col].start] ;
      COLAMD_DEBUG3 (("Pivotal row is %d\n", pivot_row)) ;
    }
    else
    {
      /* there is no pivot row, since it is of zero length */
      pivot_row = COLAMD_EMPTY ;
      COLAMD_ASSERT (pivot_row_length == 0) ;
    }
    COLAMD_ASSERT (Col [pivot_col].length > 0 || pivot_row_length == 0) ;
 
    /* === Approximate degree computation =============================== */
 
    /* Here begins the computation of the approximate degree.  The column */
    /* score is the sum of the pivot row "length", plus the size of the */
    /* set differences of each row in the column minus the pattern of the */
    /* pivot row itself.  The column ("thickness") itself is also */
    /* excluded from the column score (we thus use an approximate */
    /* external degree). */
 
    /* The time taken by the following code (compute set differences, and */
    /* add them up) is proportional to the size of the data structure */
    /* being scanned - that is, the sum of the sizes of each column in */
    /* the pivot row.  Thus, the amortized time to compute a column score */
    /* is proportional to the size of that column (where size, in this */
    /* context, is the column "length", or the number of row indices */
    /* in that column).  The number of row indices in a column is */
    /* monotonically non-decreasing, from the length of the original */
    /* column on input to colamd. */
 
    /* === Compute set differences ====================================== */
 
    COLAMD_DEBUG3 (("** Computing set differences phase. **\n")) ;
 
    /* pivot row is currently dead - it will be revived later. */
 
    COLAMD_DEBUG3 (("Pivot row: ")) ;
    /* for each column in pivot row */
    rp = &A [pivot_row_start] ;
    rp_end = rp + pivot_row_length ;
    while (rp < rp_end)
    {
      col = *rp++ ;
      COLAMD_ASSERT (COL_IS_ALIVE (col) && col != pivot_col) ;
      COLAMD_DEBUG3 (("Col: %d\n", col)) ;
 
      /* clear tags used to construct pivot row pattern */
      col_thickness = -Col [col].shared1.thickness ;
      COLAMD_ASSERT (col_thickness > 0) ;
      Col [col].shared1.thickness = col_thickness ;
 
      /* === Remove column from degree list =========================== */
 
      cur_score = Col [col].shared2.score ;
      prev_col = Col [col].shared3.prev ;
      next_col = Col [col].shared4.degree_next ;
      COLAMD_ASSERT (cur_score >= 0) ;
      COLAMD_ASSERT (cur_score <= n_col) ;
      COLAMD_ASSERT (cur_score >= COLAMD_EMPTY) ;
      if (prev_col == COLAMD_EMPTY)
      {
  head [cur_score] = next_col ;
      }
      else
      {
  Col [prev_col].shared4.degree_next = next_col ;
      }
      if (next_col != COLAMD_EMPTY)
      {
  Col [next_col].shared3.prev = prev_col ;
      }
 
      /* === Scan the column ========================================== */
 
      cp = &A [Col [col].start] ;
      cp_end = cp + Col [col].length ;
      while (cp < cp_end)
      {
  /* get a row */
  row = *cp++ ;
  row_mark = Row [row].shared2.mark ;
  /* skip if dead */
  if (ROW_IS_MARKED_DEAD (row_mark))
  {
    continue ;
  }
  COLAMD_ASSERT (row != pivot_row) ;
  set_difference = row_mark - tag_mark ;
  /* check if the row has been seen yet */
  if (set_difference < 0)
  {
    COLAMD_ASSERT (Row [row].shared1.degree <= max_deg) ;
    set_difference = Row [row].shared1.degree ;
  }
  /* subtract column thickness from this row's set difference */
  set_difference -= col_thickness ;
  COLAMD_ASSERT (set_difference >= 0) ;
  /* absorb this row if the set difference becomes zero */
  if (set_difference == 0)
  {
    COLAMD_DEBUG3 (("aggressive absorption. Row: %d\n", row)) ;
    KILL_ROW (row) ;
  }
  else
  {
    /* save the new mark */
    Row [row].shared2.mark = set_difference + tag_mark ;
  }
      }
    }
 
 
    /* === Add up set differences for each column ======================= */
 
    COLAMD_DEBUG3 (("** Adding set differences phase. **\n")) ;
 
    /* for each column in pivot row */
    rp = &A [pivot_row_start] ;
    rp_end = rp + pivot_row_length ;
    while (rp < rp_end)
    {
      /* get a column */
      col = *rp++ ;
      COLAMD_ASSERT (COL_IS_ALIVE (col) && col != pivot_col) ;
      hash = 0 ;
      cur_score = 0 ;
      cp = &A [Col [col].start] ;
      /* compact the column */
      new_cp = cp ;
      cp_end = cp + Col [col].length ;
 
      COLAMD_DEBUG4 (("Adding set diffs for Col: %d.\n", col)) ;
 
      while (cp < cp_end)
      {
  /* get a row */
  row = *cp++ ;
  COLAMD_ASSERT(row >= 0 && row < n_row) ;
  row_mark = Row [row].shared2.mark ;
  /* skip if dead */
  if (ROW_IS_MARKED_DEAD (row_mark))
  {
    continue ;
  }
  COLAMD_ASSERT (row_mark > tag_mark) ;
  /* compact the column */
  *new_cp++ = row ;
  /* compute hash function */
  hash += row ;
  /* add set difference */
  cur_score += row_mark - tag_mark ;
  /* integer overflow... */
  cur_score = numext::mini(cur_score, n_col) ;
      }
 
      /* recompute the column's length */
      Col [col].length = (IndexType) (new_cp - &A [Col [col].start]) ;
 
      /* === Further mass elimination ================================= */
 
      if (Col [col].length == 0)
      {
  COLAMD_DEBUG4 (("further mass elimination. Col: %d\n", col)) ;
  /* nothing left but the pivot row in this column */
  KILL_PRINCIPAL_COL (col) ;
  pivot_row_degree -= Col [col].shared1.thickness ;
  COLAMD_ASSERT (pivot_row_degree >= 0) ;
  /* order it */
  Col [col].shared2.order = k ;
  /* increment order count by column thickness */
  k += Col [col].shared1.thickness ;
      }
      else
      {
  /* === Prepare for supercolumn detection ==================== */
 
  COLAMD_DEBUG4 (("Preparing supercol detection for Col: %d.\n", col)) ;
 
  /* save score so far */
  Col [col].shared2.score = cur_score ;
 
  /* add column to hash table, for supercolumn detection */
  hash %= n_col + 1 ;
 
  COLAMD_DEBUG4 ((" Hash = %d, n_col = %d.\n", hash, n_col)) ;
  COLAMD_ASSERT (hash <= n_col) ;
 
  head_column = head [hash] ;
  if (head_column > COLAMD_EMPTY)
  {
    /* degree list "hash" is non-empty, use prev (shared3) of */
    /* first column in degree list as head of hash bucket */
    first_col = Col [head_column].shared3.headhash ;
    Col [head_column].shared3.headhash = col ;
  }
  else
  {
    /* degree list "hash" is empty, use head as hash bucket */
    first_col = - (head_column + 2) ;
    head [hash] = - (col + 2) ;
  }
  Col [col].shared4.hash_next = first_col ;
 
  /* save hash function in Col [col].shared3.hash */
  Col [col].shared3.hash = (IndexType) hash ;
  COLAMD_ASSERT (COL_IS_ALIVE (col)) ;
      }
    }
 
    /* The approximate external column degree is now computed.  */
 
    /* === Supercolumn detection ======================================== */
 
    COLAMD_DEBUG3 (("** Supercolumn detection phase. **\n")) ;
 
    Eigen::internal::detect_super_cols (Col, A, head, pivot_row_start, pivot_row_length) ;
 
    /* === Kill the pivotal column ====================================== */
 
    KILL_PRINCIPAL_COL (pivot_col) ;
 
    /* === Clear mark =================================================== */
 
    tag_mark += (max_deg + 1) ;
    if (tag_mark >= max_mark)
    {
      COLAMD_DEBUG2 (("clearing tag_mark\n")) ;
      tag_mark = Eigen::internal::clear_mark (n_row, Row) ;
    }
 
    /* === Finalize the new pivot row, and column scores ================ */
 
    COLAMD_DEBUG3 (("** Finalize scores phase. **\n")) ;
 
    /* for each column in pivot row */
    rp = &A [pivot_row_start] ;
    /* compact the pivot row */
    new_rp = rp ;
    rp_end = rp + pivot_row_length ;
    while (rp < rp_end)
    {
      col = *rp++ ;
      /* skip dead columns */
      if (COL_IS_DEAD (col))
      {
  continue ;
      }
      *new_rp++ = col ;
      /* add new pivot row to column */
      A [Col [col].start + (Col [col].length++)] = pivot_row ;
 
      /* retrieve score so far and add on pivot row's degree. */
      /* (we wait until here for this in case the pivot */
      /* row's degree was reduced due to mass elimination). */
      cur_score = Col [col].shared2.score + pivot_row_degree ;
 
      /* calculate the max possible score as the number of */
      /* external columns minus the 'k' value minus the */
      /* columns thickness */
      max_score = n_col - k - Col [col].shared1.thickness ;
 
      /* make the score the external degree of the union-of-rows */
      cur_score -= Col [col].shared1.thickness ;
 
      /* make sure score is less or equal than the max score */
      cur_score = numext::mini(cur_score, max_score) ;
      COLAMD_ASSERT (cur_score >= 0) ;
 
      /* store updated score */
      Col [col].shared2.score = cur_score ;
 
      /* === Place column back in degree list ========================= */
 
      COLAMD_ASSERT (min_score >= 0) ;
      COLAMD_ASSERT (min_score <= n_col) ;
      COLAMD_ASSERT (cur_score >= 0) ;
      COLAMD_ASSERT (cur_score <= n_col) ;
      COLAMD_ASSERT (head [cur_score] >= COLAMD_EMPTY) ;
      next_col = head [cur_score] ;
      Col [col].shared4.degree_next = next_col ;
      Col [col].shared3.prev = COLAMD_EMPTY ;
      if (next_col != COLAMD_EMPTY)
      {
  Col [next_col].shared3.prev = col ;
      }
      head [cur_score] = col ;
 
      /* see if this score is less than current min */
      min_score = numext::mini(min_score, cur_score) ;
 
    }
 
    /* === Resurrect the new pivot row ================================== */
 
    if (pivot_row_degree > 0)
    {
      /* update pivot row length to reflect any cols that were killed */
      /* during super-col detection and mass elimination */
      Row [pivot_row].start  = pivot_row_start ;
      Row [pivot_row].length = (IndexType) (new_rp - &A[pivot_row_start]) ;
      Row [pivot_row].shared1.degree = pivot_row_degree ;
      Row [pivot_row].shared2.mark = 0 ;
      /* pivot row is no longer dead */
    }
  }
 
  /* === All principal columns have now been ordered ====================== */
 
  return (ngarbage) ;
}

References col(), COL_IS_ALIVE, COL_IS_DEAD, COLAMD_ASSERT, COLAMD_DEBUG1, COLAMD_DEBUG2, COLAMD_DEBUG3, COLAMD_DEBUG4, COLAMD_EMPTY, head(), KILL_PRINCIPAL_COL, KILL_ROW, row(), ROW_IS_ALIVE, ROW_IS_DEAD, and ROW_IS_MARKED_DEAD.

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garbage_collection()

template<typename IndexType >

static IndexType internal::garbage_collection ( IndexType  n_row, IndexType  n_col, Colamd_Row< IndexType >  Row[], colamd_col< IndexType >  Col[], IndexType  A[], IndexType *  pfree  )

static

{
  /* === Local variables ================================================== */
 
  IndexType *psrc ;     /* source pointer */
  IndexType *pdest ;    /* destination pointer */
  IndexType j ;     /* counter */
  IndexType r ;     /* a row index */
  IndexType c ;     /* a column index */
  IndexType length ;    /* length of a row or column */
 
  /* === Defragment the columns =========================================== */
 
  pdest = &A[0] ;
  for (c = 0 ; c < n_col ; c++)
  {
    if (COL_IS_ALIVE (c))
    {
      psrc = &A [Col [c].start] ;
 
      /* move and compact the column */
      COLAMD_ASSERT (pdest <= psrc) ;
      Col [c].start = (IndexType) (pdest - &A [0]) ;
      length = Col [c].length ;
      for (j = 0 ; j < length ; j++)
      {
  r = *psrc++ ;
  if (ROW_IS_ALIVE (r))
  {
    *pdest++ = r ;
  }
      }
      Col [c].length = (IndexType) (pdest - &A [Col [c].start]) ;
    }
  }
 
  /* === Prepare to defragment the rows =================================== */
 
  for (r = 0 ; r < n_row ; r++)
  {
    if (ROW_IS_ALIVE (r))
    {
      if (Row [r].length == 0)
      {
  /* this row is of zero length.  cannot compact it, so kill it */
  COLAMD_DEBUG3 (("Defrag row kill\n")) ;
  KILL_ROW (r) ;
      }
      else
      {
  /* save first column index in Row [r].shared2.first_column */
  psrc = &A [Row [r].start] ;
  Row [r].shared2.first_column = *psrc ;
  COLAMD_ASSERT (ROW_IS_ALIVE (r)) ;
  /* flag the start of the row with the one's complement of row */
  *psrc = ONES_COMPLEMENT (r) ;
 
      }
    }
  }
 
  /* === Defragment the rows ============================================== */
 
  psrc = pdest ;
  while (psrc < pfree)
  {
    /* find a negative number ... the start of a row */
    if (*psrc++ < 0)
    {
      psrc-- ;
      /* get the row index */
      r = ONES_COMPLEMENT (*psrc) ;
      COLAMD_ASSERT (r >= 0 && r < n_row) ;
      /* restore first column index */
      *psrc = Row [r].shared2.first_column ;
      COLAMD_ASSERT (ROW_IS_ALIVE (r)) ;
 
      /* move and compact the row */
      COLAMD_ASSERT (pdest <= psrc) ;
      Row [r].start = (IndexType) (pdest - &A [0]) ;
      length = Row [r].length ;
      for (j = 0 ; j < length ; j++)
      {
  c = *psrc++ ;
  if (COL_IS_ALIVE (c))
  {
    *pdest++ = c ;
  }
      }
      Row [r].length = (IndexType) (pdest - &A [Row [r].start]) ;
 
    }
  }
  /* ensure we found all the rows */
  COLAMD_ASSERT (debug_rows == 0) ;
 
  /* === Return the new value of pfree ==================================== */
 
  return ((IndexType) (pdest - &A [0])) ;
}

References COL_IS_ALIVE, COLAMD_ASSERT, COLAMD_DEBUG3, KILL_ROW, ONES_COMPLEMENT, and ROW_IS_ALIVE.

init_rows_cols()

template<typename IndexType >

static IndexType internal::init_rows_cols ( IndexType  n_row, IndexType  n_col, Colamd_Row< IndexType >  Row[], colamd_col< IndexType >  col[], IndexType  A[], IndexType  p[], IndexType  stats[COLAMD_STATS]  )

static

{
  /* === Local variables ================================================== */
 
  IndexType col ;     /* a column index */
  IndexType row ;     /* a row index */
  IndexType *cp ;     /* a column pointer */
  IndexType *cp_end ;   /* a pointer to the end of a column */
  IndexType *rp ;     /* a row pointer */
  IndexType *rp_end ;   /* a pointer to the end of a row */
  IndexType last_row ;    /* previous row */
 
  /* === Initialize columns, and check column pointers ==================== */
 
  for (col = 0 ; col < n_col ; col++)
  {
    Col [col].start = p [col] ;
    Col [col].length = p [col+1] - p [col] ;
 
    if ((Col [col].length) < 0) // extra parentheses to work-around gcc bug 10200
    {
      /* column pointers must be non-decreasing */
      stats [COLAMD_STATUS] = COLAMD_ERROR_col_length_negative ;
      stats [COLAMD_INFO1] = col ;
      stats [COLAMD_INFO2] = Col [col].length ;
      COLAMD_DEBUG0 (("colamd: col %d length %d < 0\n", col, Col [col].length)) ;
      return (false) ;
    }
 
    Col [col].shared1.thickness = 1 ;
    Col [col].shared2.score = 0 ;
    Col [col].shared3.prev = COLAMD_EMPTY ;
    Col [col].shared4.degree_next = COLAMD_EMPTY ;
  }
 
  /* p [0..n_col] no longer needed, used as "head" in subsequent routines */
 
  /* === Scan columns, compute row degrees, and check row indices ========= */
 
  stats [COLAMD_INFO3] = 0 ;  /* number of duplicate or unsorted row indices*/
 
  for (row = 0 ; row < n_row ; row++)
  {
    Row [row].length = 0 ;
    Row [row].shared2.mark = -1 ;
  }
 
  for (col = 0 ; col < n_col ; col++)
  {
    last_row = -1 ;
 
    cp = &A [p [col]] ;
    cp_end = &A [p [col+1]] ;
 
    while (cp < cp_end)
    {
      row = *cp++ ;
 
      /* make sure row indices within range */
      if (row < 0 || row >= n_row)
      {
  stats [COLAMD_STATUS] = COLAMD_ERROR_row_index_out_of_bounds ;
  stats [COLAMD_INFO1] = col ;
  stats [COLAMD_INFO2] = row ;
  stats [COLAMD_INFO3] = n_row ;
  COLAMD_DEBUG0 (("colamd: row %d col %d out of bounds\n", row, col)) ;
  return (false) ;
      }
 
      if (row <= last_row || Row [row].shared2.mark == col)
      {
  /* row index are unsorted or repeated (or both), thus col */
  /* is jumbled.  This is a notice, not an error condition. */
  stats [COLAMD_STATUS] = COLAMD_OK_BUT_JUMBLED ;
  stats [COLAMD_INFO1] = col ;
  stats [COLAMD_INFO2] = row ;
  (stats [COLAMD_INFO3]) ++ ;
  COLAMD_DEBUG1 (("colamd: row %d col %d unsorted/duplicate\n",row,col));
      }
 
      if (Row [row].shared2.mark != col)
      {
  Row [row].length++ ;
      }
      else
      {
  /* this is a repeated entry in the column, */
  /* it will be removed */
  Col [col].length-- ;
      }
 
      /* mark the row as having been seen in this column */
      Row [row].shared2.mark = col ;
 
      last_row = row ;
    }
  }
 
  /* === Compute row pointers ============================================= */
 
  /* row form of the matrix starts directly after the column */
  /* form of matrix in A */
  Row [0].start = p [n_col] ;
  Row [0].shared1.p = Row [0].start ;
  Row [0].shared2.mark = -1 ;
  for (row = 1 ; row < n_row ; row++)
  {
    Row [row].start = Row [row-1].start + Row [row-1].length ;
    Row [row].shared1.p = Row [row].start ;
    Row [row].shared2.mark = -1 ;
  }
 
  /* === Create row form ================================================== */
 
  if (stats [COLAMD_STATUS] == COLAMD_OK_BUT_JUMBLED)
  {
    /* if cols jumbled, watch for repeated row indices */
    for (col = 0 ; col < n_col ; col++)
    {
      cp = &A [p [col]] ;
      cp_end = &A [p [col+1]] ;
      while (cp < cp_end)
      {
  row = *cp++ ;
  if (Row [row].shared2.mark != col)
  {
    A [(Row [row].shared1.p)++] = col ;
    Row [row].shared2.mark = col ;
  }
      }
    }
  }
  else
  {
    /* if cols not jumbled, we don't need the mark (this is faster) */
    for (col = 0 ; col < n_col ; col++)
    {
      cp = &A [p [col]] ;
      cp_end = &A [p [col+1]] ;
      while (cp < cp_end)
      {
  A [(Row [*cp++].shared1.p)++] = col ;
      }
    }
  }
 
  /* === Clear the row marks and set row degrees ========================== */
 
  for (row = 0 ; row < n_row ; row++)
  {
    Row [row].shared2.mark = 0 ;
    Row [row].shared1.degree = Row [row].length ;
  }
 
  /* === See if we need to re-create columns ============================== */
 
  if (stats [COLAMD_STATUS] == COLAMD_OK_BUT_JUMBLED)
  {
    COLAMD_DEBUG0 (("colamd: reconstructing column form, matrix jumbled\n")) ;
 
 
    /* === Compute col pointers ========================================= */
 
    /* col form of the matrix starts at A [0]. */
    /* Note, we may have a gap between the col form and the row */
    /* form if there were duplicate entries, if so, it will be */
    /* removed upon the first garbage collection */
    Col [0].start = 0 ;
    p [0] = Col [0].start ;
    for (col = 1 ; col < n_col ; col++)
    {
      /* note that the lengths here are for pruned columns, i.e. */
      /* no duplicate row indices will exist for these columns */
      Col [col].start = Col [col-1].start + Col [col-1].length ;
      p [col] = Col [col].start ;
    }
 
    /* === Re-create col form =========================================== */
 
    for (row = 0 ; row < n_row ; row++)
    {
      rp = &A [Row [row].start] ;
      rp_end = rp + Row [row].length ;
      while (rp < rp_end)
      {
  A [(p [*rp++])++] = row ;
      }
    }
  }
 
  /* === Done.  Matrix is not (or no longer) jumbled ====================== */
 
  return (true) ;
}

References col(), COLAMD_DEBUG0, COLAMD_DEBUG1, COLAMD_EMPTY, COLAMD_ERROR_col_length_negative, COLAMD_ERROR_row_index_out_of_bounds, COLAMD_INFO1, COLAMD_INFO2, COLAMD_INFO3, COLAMD_OK_BUT_JUMBLED, COLAMD_STATUS, and row().

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init_scoring()

template<typename IndexType >

static void internal::init_scoring ( IndexType  n_row, IndexType  n_col, Colamd_Row< IndexType >  Row[], colamd_col< IndexType >  Col[], IndexType  A[], IndexType  head[], double  knobs[COLAMD_KNOBS], IndexType *  p_n_row2, IndexType *  p_n_col2, IndexType *  p_max_deg  )

static

{
  /* === Local variables ================================================== */
 
  IndexType c ;     /* a column index */
  IndexType r, row ;    /* a row index */
  IndexType *cp ;     /* a column pointer */
  IndexType deg ;     /* degree of a row or column */
  IndexType *cp_end ;   /* a pointer to the end of a column */
  IndexType *new_cp ;   /* new column pointer */
  IndexType col_length ;    /* length of pruned column */
  IndexType score ;     /* current column score */
  IndexType n_col2 ;    /* number of non-dense, non-empty columns */
  IndexType n_row2 ;    /* number of non-dense, non-empty rows */
  IndexType dense_row_count ; /* remove rows with more entries than this */
  IndexType dense_col_count ; /* remove cols with more entries than this */
  IndexType min_score ;   /* smallest column score */
  IndexType max_deg ;   /* maximum row degree */
  IndexType next_col ;    /* Used to add to degree list.*/
 
 
  /* === Extract knobs ==================================================== */
 
  dense_row_count = numext::maxi(IndexType(0), numext::mini(IndexType(knobs [COLAMD_DENSE_ROW] * n_col), n_col)) ;
  dense_col_count = numext::maxi(IndexType(0), numext::mini(IndexType(knobs [COLAMD_DENSE_COL] * n_row), n_row)) ;
  COLAMD_DEBUG1 (("colamd: densecount: %d %d\n", dense_row_count, dense_col_count)) ;
  max_deg = 0 ;
  n_col2 = n_col ;
  n_row2 = n_row ;
 
  /* === Kill empty columns =============================================== */
 
  /* Put the empty columns at the end in their natural order, so that LU */
  /* factorization can proceed as far as possible. */
  for (c = n_col-1 ; c >= 0 ; c--)
  {
    deg = Col [c].length ;
    if (deg == 0)
    {
      /* this is a empty column, kill and order it last */
      Col [c].shared2.order = --n_col2 ;
      KILL_PRINCIPAL_COL (c) ;
    }
  }
  COLAMD_DEBUG1 (("colamd: null columns killed: %d\n", n_col - n_col2)) ;
 
  /* === Kill dense columns =============================================== */
 
  /* Put the dense columns at the end, in their natural order */
  for (c = n_col-1 ; c >= 0 ; c--)
  {
    /* skip any dead columns */
    if (COL_IS_DEAD (c))
    {
      continue ;
    }
    deg = Col [c].length ;
    if (deg > dense_col_count)
    {
      /* this is a dense column, kill and order it last */
      Col [c].shared2.order = --n_col2 ;
      /* decrement the row degrees */
      cp = &A [Col [c].start] ;
      cp_end = cp + Col [c].length ;
      while (cp < cp_end)
      {
  Row [*cp++].shared1.degree-- ;
      }
      KILL_PRINCIPAL_COL (c) ;
    }
  }
  COLAMD_DEBUG1 (("colamd: Dense and null columns killed: %d\n", n_col - n_col2)) ;
 
  /* === Kill dense and empty rows ======================================== */
 
  for (r = 0 ; r < n_row ; r++)
  {
    deg = Row [r].shared1.degree ;
    COLAMD_ASSERT (deg >= 0 && deg <= n_col) ;
    if (deg > dense_row_count || deg == 0)
    {
      /* kill a dense or empty row */
      KILL_ROW (r) ;
      --n_row2 ;
    }
    else
    {
      /* keep track of max degree of remaining rows */
      max_deg = numext::maxi(max_deg, deg) ;
    }
  }
  COLAMD_DEBUG1 (("colamd: Dense and null rows killed: %d\n", n_row - n_row2)) ;
 
  /* === Compute initial column scores ==================================== */
 
  /* At this point the row degrees are accurate.  They reflect the number */
  /* of "live" (non-dense) columns in each row.  No empty rows exist. */
  /* Some "live" columns may contain only dead rows, however.  These are */
  /* pruned in the code below. */
 
  /* now find the initial matlab score for each column */
  for (c = n_col-1 ; c >= 0 ; c--)
  {
    /* skip dead column */
    if (COL_IS_DEAD (c))
    {
      continue ;
    }
    score = 0 ;
    cp = &A [Col [c].start] ;
    new_cp = cp ;
    cp_end = cp + Col [c].length ;
    while (cp < cp_end)
    {
      /* get a row */
      row = *cp++ ;
      /* skip if dead */
      if (ROW_IS_DEAD (row))
      {
  continue ;
      }
      /* compact the column */
      *new_cp++ = row ;
      /* add row's external degree */
      score += Row [row].shared1.degree - 1 ;
      /* guard against integer overflow */
      score = numext::mini(score, n_col) ;
    }
    /* determine pruned column length */
    col_length = (IndexType) (new_cp - &A [Col [c].start]) ;
    if (col_length == 0)
    {
      /* a newly-made null column (all rows in this col are "dense" */
      /* and have already been killed) */
      COLAMD_DEBUG2 (("Newly null killed: %d\n", c)) ;
      Col [c].shared2.order = --n_col2 ;
      KILL_PRINCIPAL_COL (c) ;
    }
    else
    {
      /* set column length and set score */
      COLAMD_ASSERT (score >= 0) ;
      COLAMD_ASSERT (score <= n_col) ;
      Col [c].length = col_length ;
      Col [c].shared2.score = score ;
    }
  }
  COLAMD_DEBUG1 (("colamd: Dense, null, and newly-null columns killed: %d\n",
      n_col-n_col2)) ;
 
  /* At this point, all empty rows and columns are dead.  All live columns */
  /* are "clean" (containing no dead rows) and simplicial (no supercolumns */
  /* yet).  Rows may contain dead columns, but all live rows contain at */
  /* least one live column. */
 
  /* === Initialize degree lists ========================================== */
 
 
  /* clear the hash buckets */
  for (c = 0 ; c <= n_col ; c++)
  {
    head [c] = COLAMD_EMPTY ;
  }
  min_score = n_col ;
  /* place in reverse order, so low column indices are at the front */
  /* of the lists.  This is to encourage natural tie-breaking */
  for (c = n_col-1 ; c >= 0 ; c--)
  {
    /* only add principal columns to degree lists */
    if (COL_IS_ALIVE (c))
    {
      COLAMD_DEBUG4 (("place %d score %d minscore %d ncol %d\n",
          c, Col [c].shared2.score, min_score, n_col)) ;
 
      /* === Add columns score to DList =============================== */
 
      score = Col [c].shared2.score ;
 
      COLAMD_ASSERT (min_score >= 0) ;
      COLAMD_ASSERT (min_score <= n_col) ;
      COLAMD_ASSERT (score >= 0) ;
      COLAMD_ASSERT (score <= n_col) ;
      COLAMD_ASSERT (head [score] >= COLAMD_EMPTY) ;
 
      /* now add this column to dList at proper score location */
      next_col = head [score] ;
      Col [c].shared3.prev = COLAMD_EMPTY ;
      Col [c].shared4.degree_next = next_col ;
 
      /* if there already was a column with the same score, set its */
      /* previous pointer to this new column */
      if (next_col != COLAMD_EMPTY)
      {
  Col [next_col].shared3.prev = c ;
      }
      head [score] = c ;
 
      /* see if this score is less than current min */
      min_score = numext::mini(min_score, score) ;
 
 
    }
  }
 
 
  /* === Return number of remaining columns, and max row degree =========== */
 
  *p_n_col2 = n_col2 ;
  *p_n_row2 = n_row2 ;
  *p_max_deg = max_deg ;
}

References COL_IS_ALIVE, COL_IS_DEAD, COLAMD_ASSERT, COLAMD_DEBUG1, COLAMD_DEBUG2, COLAMD_DEBUG4, COLAMD_DENSE_COL, COLAMD_DENSE_ROW, COLAMD_EMPTY, head(), KILL_PRINCIPAL_COL, KILL_ROW, internal::colamd_col< IndexType >::length, row(), ROW_IS_DEAD, internal::Colamd_Row< IndexType >::shared1, internal::colamd_col< IndexType >::shared2, internal::colamd_col< IndexType >::shared3, internal::colamd_col< IndexType >::shared4, and internal::colamd_col< IndexType >::start.

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order_children()

template<typename IndexType >

static void internal::order_children ( IndexType  n_col, colamd_col< IndexType >  Col[], IndexType  p[]  )

inlinestatic

{
  /* === Local variables ================================================== */
 
  IndexType i ;     /* loop counter for all columns */
  IndexType c ;     /* column index */
  IndexType parent ;    /* index of column's parent */
  IndexType order ;     /* column's order */
 
  /* === Order each non-principal column ================================== */
 
  for (i = 0 ; i < n_col ; i++)
  {
    /* find an un-ordered non-principal column */
    COLAMD_ASSERT (COL_IS_DEAD (i)) ;
    if (!COL_IS_DEAD_PRINCIPAL (i) && Col [i].shared2.order == COLAMD_EMPTY)
    {
      parent = i ;
      /* once found, find its principal parent */
      do
      {
  parent = Col [parent].shared1.parent ;
      } while (!COL_IS_DEAD_PRINCIPAL (parent)) ;
 
      /* now, order all un-ordered non-principal columns along path */
      /* to this parent.  collapse tree at the same time */
      c = i ;
      /* get order of parent */
      order = Col [parent].shared2.order ;
 
      do
      {
  COLAMD_ASSERT (Col [c].shared2.order == COLAMD_EMPTY) ;
 
  /* order this column */
  Col [c].shared2.order = order++ ;
  /* collaps tree */
  Col [c].shared1.parent = parent ;
 
  /* get immediate parent of this column */
  c = Col [c].shared1.parent ;
 
  /* continue until we hit an ordered column.  There are */
  /* guarranteed not to be anymore unordered columns */
  /* above an ordered column */
      } while (Col [c].shared2.order == COLAMD_EMPTY) ;
 
      /* re-order the super_col parent to largest order for this group */
      Col [parent].shared2.order = order ;
    }
  }
 
  /* === Generate the permutation ========================================= */
 
  for (c = 0 ; c < n_col ; c++)
  {
    p [Col [c].shared2.order] = c ;
  }
}

References COL_IS_DEAD, COL_IS_DEAD_PRINCIPAL, COLAMD_ASSERT, COLAMD_EMPTY, internal::colamd_col< IndexType >::shared1, and internal::colamd_col< IndexType >::shared2.