Squashed 'third_party/eigen/' content from commit 61d72f6
Change-Id: Iccc90fa0b55ab44037f018046d2fcffd90d9d025
git-subtree-dir: third_party/eigen
git-subtree-split: 61d72f6383cfa842868c53e30e087b0258177257
diff --git a/Eigen/src/OrderingMethods/Eigen_Colamd.h b/Eigen/src/OrderingMethods/Eigen_Colamd.h
new file mode 100644
index 0000000..44548f6
--- /dev/null
+++ b/Eigen/src/OrderingMethods/Eigen_Colamd.h
@@ -0,0 +1,1850 @@
+// // This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2012 Desire Nuentsa Wakam <desire.nuentsa_wakam@inria.fr>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+// This file is modified from the colamd/symamd library. The copyright is below
+
+// The authors of the code itself are Stefan I. Larimore and Timothy A.
+// Davis (davis@cise.ufl.edu), University of Florida. The algorithm was
+// developed in collaboration with John Gilbert, Xerox PARC, and Esmond
+// Ng, Oak Ridge National Laboratory.
+//
+// Date:
+//
+// September 8, 2003. Version 2.3.
+//
+// Acknowledgements:
+//
+// This work was supported by the National Science Foundation, under
+// grants DMS-9504974 and DMS-9803599.
+//
+// Notice:
+//
+// Copyright (c) 1998-2003 by the University of Florida.
+// 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, copy, modify, and/or distribute
+// this program, provided that the Copyright, this License, and the
+// Availability of the original version is retained on all copies and made
+// accessible to the end-user of any code or package that includes COLAMD
+// or any modified version of COLAMD.
+//
+// Availability:
+//
+// The colamd/symamd library is available at
+//
+// http://www.cise.ufl.edu/research/sparse/colamd/
+
+// This is the http://www.cise.ufl.edu/research/sparse/colamd/colamd.h
+// file. It is required by the colamd.c, colamdmex.c, and symamdmex.c
+// files, and by any C code that calls the routines whose prototypes are
+// listed below, or that uses the colamd/symamd definitions listed below.
+
+#ifndef EIGEN_COLAMD_H
+#define EIGEN_COLAMD_H
+
+namespace internal {
+/* Ensure that debugging is turned off: */
+#ifndef COLAMD_NDEBUG
+#define COLAMD_NDEBUG
+#endif /* NDEBUG */
+/* ========================================================================== */
+/* === Knob and statistics definitions ====================================== */
+/* ========================================================================== */
+
+/* size of the knobs [ ] array. Only knobs [0..1] are currently used. */
+#define COLAMD_KNOBS 20
+
+/* number of output statistics. Only stats [0..6] are currently used. */
+#define COLAMD_STATS 20
+
+/* knobs [0] and stats [0]: dense row knob and output statistic. */
+#define COLAMD_DENSE_ROW 0
+
+/* knobs [1] and stats [1]: dense column knob and output statistic. */
+#define COLAMD_DENSE_COL 1
+
+/* stats [2]: memory defragmentation count output statistic */
+#define COLAMD_DEFRAG_COUNT 2
+
+/* stats [3]: colamd status: zero OK, > 0 warning or notice, < 0 error */
+#define COLAMD_STATUS 3
+
+/* stats [4..6]: error info, or info on jumbled columns */
+#define COLAMD_INFO1 4
+#define COLAMD_INFO2 5
+#define COLAMD_INFO3 6
+
+/* error codes returned in stats [3]: */
+#define COLAMD_OK (0)
+#define COLAMD_OK_BUT_JUMBLED (1)
+#define COLAMD_ERROR_A_not_present (-1)
+#define COLAMD_ERROR_p_not_present (-2)
+#define COLAMD_ERROR_nrow_negative (-3)
+#define COLAMD_ERROR_ncol_negative (-4)
+#define COLAMD_ERROR_nnz_negative (-5)
+#define COLAMD_ERROR_p0_nonzero (-6)
+#define COLAMD_ERROR_A_too_small (-7)
+#define COLAMD_ERROR_col_length_negative (-8)
+#define COLAMD_ERROR_row_index_out_of_bounds (-9)
+#define COLAMD_ERROR_out_of_memory (-10)
+#define COLAMD_ERROR_internal_error (-999)
+
+/* ========================================================================== */
+/* === Definitions ========================================================== */
+/* ========================================================================== */
+
+#define COLAMD_MAX(a,b) (((a) > (b)) ? (a) : (b))
+#define COLAMD_MIN(a,b) (((a) < (b)) ? (a) : (b))
+
+#define ONES_COMPLEMENT(r) (-(r)-1)
+
+/* -------------------------------------------------------------------------- */
+
+#define COLAMD_EMPTY (-1)
+
+/* Row and column status */
+#define ALIVE (0)
+#define DEAD (-1)
+
+/* Column status */
+#define DEAD_PRINCIPAL (-1)
+#define DEAD_NON_PRINCIPAL (-2)
+
+/* Macros for row and column status update and checking. */
+#define ROW_IS_DEAD(r) ROW_IS_MARKED_DEAD (Row[r].shared2.mark)
+#define ROW_IS_MARKED_DEAD(row_mark) (row_mark < ALIVE)
+#define ROW_IS_ALIVE(r) (Row [r].shared2.mark >= ALIVE)
+#define COL_IS_DEAD(c) (Col [c].start < ALIVE)
+#define COL_IS_ALIVE(c) (Col [c].start >= ALIVE)
+#define COL_IS_DEAD_PRINCIPAL(c) (Col [c].start == DEAD_PRINCIPAL)
+#define KILL_ROW(r) { Row [r].shared2.mark = DEAD ; }
+#define KILL_PRINCIPAL_COL(c) { Col [c].start = DEAD_PRINCIPAL ; }
+#define KILL_NON_PRINCIPAL_COL(c) { Col [c].start = DEAD_NON_PRINCIPAL ; }
+
+/* ========================================================================== */
+/* === Colamd reporting mechanism =========================================== */
+/* ========================================================================== */
+
+// == Row and Column structures ==
+template <typename Index>
+struct colamd_col
+{
+ Index start ; /* index for A of first row in this column, or DEAD */
+ /* if column is dead */
+ Index length ; /* number of rows in this column */
+ union
+ {
+ Index thickness ; /* number of original columns represented by this */
+ /* col, if the column is alive */
+ Index parent ; /* parent in parent tree super-column structure, if */
+ /* the column is dead */
+ } shared1 ;
+ union
+ {
+ Index score ; /* the score used to maintain heap, if col is alive */
+ Index order ; /* pivot ordering of this column, if col is dead */
+ } shared2 ;
+ union
+ {
+ Index headhash ; /* head of a hash bucket, if col is at the head of */
+ /* a degree list */
+ Index hash ; /* hash value, if col is not in a degree list */
+ Index prev ; /* previous column in degree list, if col is in a */
+ /* degree list (but not at the head of a degree list) */
+ } shared3 ;
+ union
+ {
+ Index degree_next ; /* next column, if col is in a degree list */
+ Index hash_next ; /* next column, if col is in a hash list */
+ } shared4 ;
+
+};
+
+template <typename Index>
+struct Colamd_Row
+{
+ Index start ; /* index for A of first col in this row */
+ Index length ; /* number of principal columns in this row */
+ union
+ {
+ Index degree ; /* number of principal & non-principal columns in row */
+ Index p ; /* used as a row pointer in init_rows_cols () */
+ } shared1 ;
+ union
+ {
+ Index mark ; /* for computing set differences and marking dead rows*/
+ Index first_column ;/* first column in row (used in garbage collection) */
+ } shared2 ;
+
+};
+
+/* ========================================================================== */
+/* === Colamd recommended memory size ======================================= */
+/* ========================================================================== */
+
+/*
+ The recommended length Alen of the array A passed to colamd is given by
+ the COLAMD_RECOMMENDED (nnz, n_row, n_col) macro. It returns -1 if any
+ argument is negative. 2*nnz space is required for the row and column
+ indices of the matrix. colamd_c (n_col) + colamd_r (n_row) space is
+ required for the Col and Row arrays, respectively, which are internal to
+ colamd. An additional n_col space is the minimal amount of "elbow room",
+ and nnz/5 more space is recommended for run time efficiency.
+
+ This macro is not needed when using symamd.
+
+ Explicit typecast to Index added Sept. 23, 2002, COLAMD version 2.2, to avoid
+ gcc -pedantic warning messages.
+*/
+template <typename Index>
+inline Index colamd_c(Index n_col)
+{ return Index( ((n_col) + 1) * sizeof (colamd_col<Index>) / sizeof (Index) ) ; }
+
+template <typename Index>
+inline Index colamd_r(Index n_row)
+{ return Index(((n_row) + 1) * sizeof (Colamd_Row<Index>) / sizeof (Index)); }
+
+// Prototypes of non-user callable routines
+template <typename Index>
+static Index init_rows_cols (Index n_row, Index n_col, Colamd_Row<Index> Row [], colamd_col<Index> col [], Index A [], Index p [], Index stats[COLAMD_STATS] );
+
+template <typename Index>
+static void init_scoring (Index n_row, Index n_col, Colamd_Row<Index> Row [], colamd_col<Index> Col [], Index A [], Index head [], double knobs[COLAMD_KNOBS], Index *p_n_row2, Index *p_n_col2, Index *p_max_deg);
+
+template <typename Index>
+static Index find_ordering (Index n_row, Index n_col, Index Alen, Colamd_Row<Index> Row [], colamd_col<Index> Col [], Index A [], Index head [], Index n_col2, Index max_deg, Index pfree);
+
+template <typename Index>
+static void order_children (Index n_col, colamd_col<Index> Col [], Index p []);
+
+template <typename Index>
+static void detect_super_cols (colamd_col<Index> Col [], Index A [], Index head [], Index row_start, Index row_length ) ;
+
+template <typename Index>
+static Index garbage_collection (Index n_row, Index n_col, Colamd_Row<Index> Row [], colamd_col<Index> Col [], Index A [], Index *pfree) ;
+
+template <typename Index>
+static inline Index clear_mark (Index n_row, Colamd_Row<Index> Row [] ) ;
+
+/* === No debugging ========================================================= */
+
+#define COLAMD_DEBUG0(params) ;
+#define COLAMD_DEBUG1(params) ;
+#define COLAMD_DEBUG2(params) ;
+#define COLAMD_DEBUG3(params) ;
+#define COLAMD_DEBUG4(params) ;
+
+#define COLAMD_ASSERT(expression) ((void) 0)
+
+
+/**
+ * \brief 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.
+ *
+ * \param nnz nonzeros in A
+ * \param n_row number of rows in A
+ * \param n_col number of columns in A
+ * \return recommended value of Alen for use by colamd
+ */
+template <typename Index>
+inline Index colamd_recommended ( Index nnz, Index n_row, Index n_col)
+{
+ 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));
+}
+
+/**
+ * \brief 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.
+ *
+ * \param knobs parameter settings for colamd
+ */
+
+static inline void colamd_set_defaults(double knobs[COLAMD_KNOBS])
+{
+ /* === 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 */
+}
+
+/**
+ * \brief 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.
+ *
+ *
+ * \param n_row number of rows in A
+ * \param n_col number of columns in A
+ * \param Alen, size of the array A
+ * \param A row indices of the matrix, of size ALen
+ * \param p column pointers of A, of size n_col+1
+ * \param knobs parameter settings for colamd
+ * \param stats colamd output statistics and error codes
+ */
+template <typename Index>
+static bool colamd(Index n_row, Index n_col, Index Alen, Index *A, Index *p, double knobs[COLAMD_KNOBS], Index stats[COLAMD_STATS])
+{
+ /* === Local variables ================================================== */
+
+ Index i ; /* loop index */
+ Index nnz ; /* nonzeros in A */
+ Index Row_size ; /* size of Row [], in integers */
+ Index Col_size ; /* size of Col [], in integers */
+ Index need ; /* minimum required length of A */
+ Colamd_Row<Index> *Row ; /* pointer into A of Row [0..n_row] array */
+ colamd_col<Index> *Col ; /* pointer into A of Col [0..n_col] array */
+ Index n_col2 ; /* number of non-dense, non-empty columns */
+ Index n_row2 ; /* number of non-dense, non-empty rows */
+ Index ngarbage ; /* number of garbage collections performed */
+ Index 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<Index> *) &A [Alen] ;
+ Row = (Colamd_Row<Index> *) &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) ;
+}
+
+/* ========================================================================== */
+/* === NON-USER-CALLABLE ROUTINES: ========================================== */
+/* ========================================================================== */
+
+/* There are no user-callable routines beyond this point in the file */
+
+
+/* ========================================================================== */
+/* === init_rows_cols ======================================================= */
+/* ========================================================================== */
+
+/*
+ Takes the column form of the matrix in A and creates the row form of the
+ matrix. Also, row and column attributes are stored in the Col and Row
+ structs. If the columns are un-sorted or contain duplicate row indices,
+ this routine will also sort and remove duplicate row indices from the
+ column form of the matrix. Returns false if the matrix is invalid,
+ true otherwise. Not user-callable.
+*/
+template <typename Index>
+static Index init_rows_cols /* returns true if OK, or false otherwise */
+ (
+ /* === Parameters ======================================================= */
+
+ Index n_row, /* number of rows of A */
+ Index n_col, /* number of columns of A */
+ Colamd_Row<Index> Row [], /* of size n_row+1 */
+ colamd_col<Index> Col [], /* of size n_col+1 */
+ Index A [], /* row indices of A, of size Alen */
+ Index p [], /* pointers to columns in A, of size n_col+1 */
+ Index stats [COLAMD_STATS] /* colamd statistics */
+ )
+{
+ /* === Local variables ================================================== */
+
+ Index col ; /* a column index */
+ Index row ; /* a row index */
+ Index *cp ; /* a column pointer */
+ Index *cp_end ; /* a pointer to the end of a column */
+ Index *rp ; /* a row pointer */
+ Index *rp_end ; /* a pointer to the end of a row */
+ Index 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)
+ {
+ /* 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) ;
+}
+
+
+/* ========================================================================== */
+/* === init_scoring ========================================================= */
+/* ========================================================================== */
+
+/*
+ Kills dense or empty columns and rows, calculates an initial score for
+ each column, and places all columns in the degree lists. Not user-callable.
+*/
+template <typename Index>
+static void init_scoring
+ (
+ /* === Parameters ======================================================= */
+
+ Index n_row, /* number of rows of A */
+ Index n_col, /* number of columns of A */
+ Colamd_Row<Index> Row [], /* of size n_row+1 */
+ colamd_col<Index> Col [], /* of size n_col+1 */
+ Index A [], /* column form and row form of A */
+ Index head [], /* of size n_col+1 */
+ double knobs [COLAMD_KNOBS],/* parameters */
+ Index *p_n_row2, /* number of non-dense, non-empty rows */
+ Index *p_n_col2, /* number of non-dense, non-empty columns */
+ Index *p_max_deg /* maximum row degree */
+ )
+{
+ /* === Local variables ================================================== */
+
+ Index c ; /* a column index */
+ Index r, row ; /* a row index */
+ Index *cp ; /* a column pointer */
+ Index deg ; /* degree of a row or column */
+ Index *cp_end ; /* a pointer to the end of a column */
+ Index *new_cp ; /* new column pointer */
+ Index col_length ; /* length of pruned column */
+ Index score ; /* current column score */
+ Index n_col2 ; /* number of non-dense, non-empty columns */
+ Index n_row2 ; /* number of non-dense, non-empty rows */
+ Index dense_row_count ; /* remove rows with more entries than this */
+ Index dense_col_count ; /* remove cols with more entries than this */
+ Index min_score ; /* smallest column score */
+ Index max_deg ; /* maximum row degree */
+ Index next_col ; /* Used to add to degree list.*/
+
+
+ /* === Extract knobs ==================================================== */
+
+ dense_row_count = COLAMD_MAX (0, COLAMD_MIN (knobs [COLAMD_DENSE_ROW] * n_col, n_col)) ;
+ dense_col_count = COLAMD_MAX (0, COLAMD_MIN (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 = COLAMD_MAX (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 = COLAMD_MIN (score, n_col) ;
+ }
+ /* determine pruned column length */
+ col_length = (Index) (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 = COLAMD_MIN (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 ;
+}
+
+
+/* ========================================================================== */
+/* === find_ordering ======================================================== */
+/* ========================================================================== */
+
+/*
+ Order the principal columns of the supercolumn form of the matrix
+ (no supercolumns on input). Uses a minimum approximate column minimum
+ degree ordering method. Not user-callable.
+*/
+template <typename Index>
+static Index find_ordering /* return the number of garbage collections */
+ (
+ /* === Parameters ======================================================= */
+
+ Index n_row, /* number of rows of A */
+ Index n_col, /* number of columns of A */
+ Index Alen, /* size of A, 2*nnz + n_col or larger */
+ Colamd_Row<Index> Row [], /* of size n_row+1 */
+ colamd_col<Index> Col [], /* of size n_col+1 */
+ Index A [], /* column form and row form of A */
+ Index head [], /* of size n_col+1 */
+ Index n_col2, /* Remaining columns to order */
+ Index max_deg, /* Maximum row degree */
+ Index pfree /* index of first free slot (2*nnz on entry) */
+ )
+{
+ /* === Local variables ================================================== */
+
+ Index k ; /* current pivot ordering step */
+ Index pivot_col ; /* current pivot column */
+ Index *cp ; /* a column pointer */
+ Index *rp ; /* a row pointer */
+ Index pivot_row ; /* current pivot row */
+ Index *new_cp ; /* modified column pointer */
+ Index *new_rp ; /* modified row pointer */
+ Index pivot_row_start ; /* pointer to start of pivot row */
+ Index pivot_row_degree ; /* number of columns in pivot row */
+ Index pivot_row_length ; /* number of supercolumns in pivot row */
+ Index pivot_col_score ; /* score of pivot column */
+ Index needed_memory ; /* free space needed for pivot row */
+ Index *cp_end ; /* pointer to the end of a column */
+ Index *rp_end ; /* pointer to the end of a row */
+ Index row ; /* a row index */
+ Index col ; /* a column index */
+ Index max_score ; /* maximum possible score */
+ Index cur_score ; /* score of current column */
+ unsigned int hash ; /* hash value for supernode detection */
+ Index head_column ; /* head of hash bucket */
+ Index first_col ; /* first column in hash bucket */
+ Index tag_mark ; /* marker value for mark array */
+ Index row_mark ; /* Row [row].shared2.mark */
+ Index set_difference ; /* set difference size of row with pivot row */
+ Index min_score ; /* smallest column score */
+ Index col_thickness ; /* "thickness" (no. of columns in a supercol) */
+ Index max_mark ; /* maximum value of tag_mark */
+ Index pivot_col_thickness ; /* number of columns represented by pivot col */
+ Index prev_col ; /* Used by Dlist operations. */
+ Index next_col ; /* Used by Dlist operations. */
+ Index 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 (head [min_score] == COLAMD_EMPTY && min_score < n_col)
+ {
+ 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 = COLAMD_MIN (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 = COLAMD_MAX (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 = COLAMD_MIN (cur_score, n_col) ;
+ }
+
+ /* recompute the column's length */
+ Col [col].length = (Index) (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 = (Index) 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 = COLAMD_MIN (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 = COLAMD_MIN (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 = (Index) (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) ;
+}
+
+
+/* ========================================================================== */
+/* === order_children ======================================================= */
+/* ========================================================================== */
+
+/*
+ The find_ordering routine has ordered all of the principal columns (the
+ representatives of the supercolumns). The non-principal columns have not
+ yet been ordered. This routine orders those columns by walking up the
+ parent tree (a column is a child of the column which absorbed it). The
+ final permutation vector is then placed in p [0 ... n_col-1], with p [0]
+ being the first column, and p [n_col-1] being the last. It doesn't look
+ like it at first glance, but be assured that this routine takes time linear
+ in the number of columns. Although not immediately obvious, the time
+ taken by this routine is O (n_col), that is, linear in the number of
+ columns. Not user-callable.
+*/
+template <typename Index>
+static inline void order_children
+(
+ /* === Parameters ======================================================= */
+
+ Index n_col, /* number of columns of A */
+ colamd_col<Index> Col [], /* of size n_col+1 */
+ Index p [] /* p [0 ... n_col-1] is the column permutation*/
+ )
+{
+ /* === Local variables ================================================== */
+
+ Index i ; /* loop counter for all columns */
+ Index c ; /* column index */
+ Index parent ; /* index of column's parent */
+ Index 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 ;
+ }
+}
+
+
+/* ========================================================================== */
+/* === detect_super_cols ==================================================== */
+/* ========================================================================== */
+
+/*
+ Detects supercolumns by finding matches between columns in the hash buckets.
+ Check amongst columns in the set A [row_start ... row_start + row_length-1].
+ The columns under consideration are currently *not* in the degree lists,
+ and have already been placed in the hash buckets.
+
+ The hash bucket for columns whose hash function is equal to h is stored
+ as follows:
+
+ if head [h] is >= 0, then head [h] contains a degree list, so:
+
+ head [h] is the first column in degree bucket h.
+ Col [head [h]].headhash gives the first column in hash bucket h.
+
+ otherwise, the degree list is empty, and:
+
+ -(head [h] + 2) is the first column in hash bucket h.
+
+ For a column c in a hash bucket, Col [c].shared3.prev is NOT a "previous
+ column" pointer. Col [c].shared3.hash is used instead as the hash number
+ for that column. The value of Col [c].shared4.hash_next is the next column
+ in the same hash bucket.
+
+ Assuming no, or "few" hash collisions, the time taken by this routine is
+ linear in the sum of the sizes (lengths) of each column whose score has
+ just been computed in the approximate degree computation.
+ Not user-callable.
+*/
+template <typename Index>
+static void detect_super_cols
+(
+ /* === Parameters ======================================================= */
+
+ colamd_col<Index> Col [], /* of size n_col+1 */
+ Index A [], /* row indices of A */
+ Index head [], /* head of degree lists and hash buckets */
+ Index row_start, /* pointer to set of columns to check */
+ Index row_length /* number of columns to check */
+)
+{
+ /* === Local variables ================================================== */
+
+ Index hash ; /* hash value for a column */
+ Index *rp ; /* pointer to a row */
+ Index c ; /* a column index */
+ Index super_c ; /* column index of the column to absorb into */
+ Index *cp1 ; /* column pointer for column super_c */
+ Index *cp2 ; /* column pointer for column c */
+ Index length ; /* length of column super_c */
+ Index prev_c ; /* column preceding c in hash bucket */
+ Index i ; /* loop counter */
+ Index *rp_end ; /* pointer to the end of the row */
+ Index col ; /* a column index in the row to check */
+ Index head_column ; /* first column in hash bucket or degree list */
+ Index 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 ;
+ }
+ }
+}
+
+
+/* ========================================================================== */
+/* === garbage_collection =================================================== */
+/* ========================================================================== */
+
+/*
+ Defragments and compacts columns and rows in the workspace A. Used when
+ all avaliable memory has been used while performing row merging. Returns
+ the index of the first free position in A, after garbage collection. The
+ time taken by this routine is linear is the size of the array A, which is
+ itself linear in the number of nonzeros in the input matrix.
+ Not user-callable.
+*/
+template <typename Index>
+static Index garbage_collection /* returns the new value of pfree */
+ (
+ /* === Parameters ======================================================= */
+
+ Index n_row, /* number of rows */
+ Index n_col, /* number of columns */
+ Colamd_Row<Index> Row [], /* row info */
+ colamd_col<Index> Col [], /* column info */
+ Index A [], /* A [0 ... Alen-1] holds the matrix */
+ Index *pfree /* &A [0] ... pfree is in use */
+ )
+{
+ /* === Local variables ================================================== */
+
+ Index *psrc ; /* source pointer */
+ Index *pdest ; /* destination pointer */
+ Index j ; /* counter */
+ Index r ; /* a row index */
+ Index c ; /* a column index */
+ Index 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 = (Index) (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 = (Index) (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 = (Index) (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 = (Index) (pdest - &A [Row [r].start]) ;
+
+ }
+ }
+ /* ensure we found all the rows */
+ COLAMD_ASSERT (debug_rows == 0) ;
+
+ /* === Return the new value of pfree ==================================== */
+
+ return ((Index) (pdest - &A [0])) ;
+}
+
+
+/* ========================================================================== */
+/* === clear_mark =========================================================== */
+/* ========================================================================== */
+
+/*
+ Clears the Row [].shared2.mark array, and returns the new tag_mark.
+ Return value is the new tag_mark. Not user-callable.
+*/
+template <typename Index>
+static inline Index clear_mark /* return the new value for tag_mark */
+ (
+ /* === Parameters ======================================================= */
+
+ Index n_row, /* number of rows in A */
+ Colamd_Row<Index> Row [] /* Row [0 ... n_row-1].shared2.mark is set to zero */
+ )
+{
+ /* === Local variables ================================================== */
+
+ Index r ;
+
+ for (r = 0 ; r < n_row ; r++)
+ {
+ if (ROW_IS_ALIVE (r))
+ {
+ Row [r].shared2.mark = 0 ;
+ }
+ }
+ return (1) ;
+}
+
+
+} // namespace internal
+#endif