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amd_2.c @ 1:c445c931472f

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1	/* ========================================================================= */
2	/* === AMD_2 =============================================================== */
3	/* ========================================================================= */
4
5	/* ------------------------------------------------------------------------- */
6	/* AMD, Copyright (c) Timothy A. Davis, */
7	/* Patrick R. Amestoy, and Iain S. Duff. See ../README.txt for License. */
8	/* email: davis at cise.ufl.edu CISE Department, Univ. of Florida. */
9	/* web: http://www.cise.ufl.edu/research/sparse/amd */
10	/* ------------------------------------------------------------------------- */
11
12	/* AMD_2: performs the AMD ordering on a symmetric sparse matrix A, followed
13	* by a postordering (via depth-first search) of the assembly tree using the
14	* AMD_postorder routine.
15	*/
16
17	#include "amd_internal.h"
18
19	/* ========================================================================= */
20	/* === clear_flag ========================================================== */
21	/* ========================================================================= */
22
23	static Int clear_flag (Int wflg, Int wbig, Int W [ ], Int n)
24	{
25	Int x ;
26	if (wflg < 2 \|\| wflg >= wbig)
27	{
28	for (x = 0 ; x < n ; x++)
29	{
30	if (W [x] != 0) W [x] = 1 ;
31	}
32	wflg = 2 ;
33	}
34	/* at this point, W [0..n-1] < wflg holds */
35	return (wflg) ;
36	}
37
38
39	/* ========================================================================= */
40	/* === AMD_2 =============================================================== */
41	/* ========================================================================= */
42
43	GLOBAL void AMD_2
44	(
45	Int n, /* A is n-by-n, where n > 0 */
46	Int Pe [ ], /* Pe [0..n-1]: index in Iw of row i on input */
47	Int Iw [ ], /* workspace of size iwlen. Iw [0..pfree-1]
48	* holds the matrix on input */
49	Int Len [ ], /* Len [0..n-1]: length for row/column i on input */
50	Int iwlen, /* length of Iw. iwlen >= pfree + n */
51	Int pfree, /* Iw [pfree ... iwlen-1] is empty on input */
52
53	/* 7 size-n workspaces, not defined on input: */
54	Int Nv [ ], /* the size of each supernode on output */
55	Int Next [ ], /* the output inverse permutation */
56	Int Last [ ], /* the output permutation */
57	Int Head [ ],
58	Int Elen [ ], /* the size columns of L for each supernode */
59	Int Degree [ ],
60	Int W [ ],
61
62	/* control parameters and output statistics */
63	double Control [ ], /* array of size AMD_CONTROL */
64	double Info [ ] /* array of size AMD_INFO */
65	)
66	{
67
68	/*
69	* Given a representation of the nonzero pattern of a symmetric matrix, A,
70	* (excluding the diagonal) perform an approximate minimum (UMFPACK/MA38-style)
71	* degree ordering to compute a pivot order such that the introduction of
72	* nonzeros (fill-in) in the Cholesky factors A = LL' is kept low. At each
73	* step, the pivot selected is the one with the minimum UMFAPACK/MA38-style
74	* upper-bound on the external degree. This routine can optionally perform
75	* aggresive absorption (as done by MC47B in the Harwell Subroutine
76	* Library).
77	*
78	* The approximate degree algorithm implemented here is the symmetric analog of
79	* the degree update algorithm in MA38 and UMFPACK (the Unsymmetric-pattern
80	* MultiFrontal PACKage, both by Davis and Duff). The routine is based on the
81	* MA27 minimum degree ordering algorithm by Iain Duff and John Reid.
82	*
83	* This routine is a translation of the original AMDBAR and MC47B routines,
84	* in Fortran, with the following modifications:
85	*
86	* (1) dense rows/columns are removed prior to ordering the matrix, and placed
87	* last in the output order. The presence of a dense row/column can
88	* increase the ordering time by up to O(n^2), unless they are removed
89	* prior to ordering.
90	*
91	* (2) the minimum degree ordering is followed by a postordering (depth-first
92	* search) of the assembly tree. Note that mass elimination (discussed
93	* below) combined with the approximate degree update can lead to the mass
94	* elimination of nodes with lower exact degree than the current pivot
95	* element. No additional fill-in is caused in the representation of the
96	* Schur complement. The mass-eliminated nodes merge with the current
97	* pivot element. They are ordered prior to the current pivot element.
98	* Because they can have lower exact degree than the current element, the
99	* merger of two or more of these nodes in the current pivot element can
100	* lead to a single element that is not a "fundamental supernode". The
101	* diagonal block can have zeros in it. Thus, the assembly tree used here
102	* is not guaranteed to be the precise supernodal elemination tree (with
103	* "funadmental" supernodes), and the postordering performed by this
104	* routine is not guaranteed to be a precise postordering of the
105	* elimination tree.
106	*
107	* (3) input parameters are added, to control aggressive absorption and the
108	* detection of "dense" rows/columns of A.
109	*
110	* (4) additional statistical information is returned, such as the number of
111	* nonzeros in L, and the flop counts for subsequent LDL' and LU
112	* factorizations. These are slight upper bounds, because of the mass
113	* elimination issue discussed above.
114	*
115	* (5) additional routines are added to interface this routine to MATLAB
116	* to provide a simple C-callable user-interface, to check inputs for
117	* errors, compute the symmetry of the pattern of A and the number of
118	* nonzeros in each row/column of A+A', to compute the pattern of A+A',
119	* to perform the assembly tree postordering, and to provide debugging
120	* ouput. Many of these functions are also provided by the Fortran
121	* Harwell Subroutine Library routine MC47A.
122	*
123	* (6) both int and UF_long versions are provided. In the descriptions below
124	* and integer is and int or UF_long depending on which version is
125	* being used.
126
127	**********************************************************************
128	*** CAUTION: ARGUMENTS ARE NOT CHECKED FOR ERRORS ON INPUT. ****
129	**********************************************************************
130	If you want error checking, a more versatile input format, and a
131	simpler user interface, use amd_order or amd_l_order instead.
132	This routine is not meant to be user-callable.
133	**********************************************************************
134
135	* ----------------------------------------------------------------------------
136	* References:
137	* ----------------------------------------------------------------------------
138	*
139	* [1] Timothy A. Davis and Iain Duff, "An unsymmetric-pattern multifrontal
140	* method for sparse LU factorization", SIAM J. Matrix Analysis and
141	* Applications, vol. 18, no. 1, pp. 140-158. Discusses UMFPACK / MA38,
142	* which first introduced the approximate minimum degree used by this
143	* routine.
144	*
145	* [2] Patrick Amestoy, Timothy A. Davis, and Iain S. Duff, "An approximate
146	* minimum degree ordering algorithm," SIAM J. Matrix Analysis and
147	* Applications, vol. 17, no. 4, pp. 886-905, 1996. Discusses AMDBAR and
148	* MC47B, which are the Fortran versions of this routine.
149	*
150	* [3] Alan George and Joseph Liu, "The evolution of the minimum degree
151	* ordering algorithm," SIAM Review, vol. 31, no. 1, pp. 1-19, 1989.
152	* We list below the features mentioned in that paper that this code
153	* includes:
154	*
155	* mass elimination:
156	* Yes. MA27 relied on supervariable detection for mass elimination.
157	*
158	* indistinguishable nodes:
159	* Yes (we call these "supervariables"). This was also in the MA27
160	* code - although we modified the method of detecting them (the
161	* previous hash was the true degree, which we no longer keep track
162	* of). A supervariable is a set of rows with identical nonzero
163	* pattern. All variables in a supervariable are eliminated together.
164	* Each supervariable has as its numerical name that of one of its
165	* variables (its principal variable).
166	*
167	* quotient graph representation:
168	* Yes. We use the term "element" for the cliques formed during
169	* elimination. This was also in the MA27 code. The algorithm can
170	* operate in place, but it will work more efficiently if given some
171	* "elbow room."
172	*
173	* element absorption:
174	* Yes. This was also in the MA27 code.
175	*
176	* external degree:
177	* Yes. The MA27 code was based on the true degree.
178	*
179	* incomplete degree update and multiple elimination:
180	* No. This was not in MA27, either. Our method of degree update
181	* within MC47B is element-based, not variable-based. It is thus
182	* not well-suited for use with incomplete degree update or multiple
183	* elimination.
184	*
185	* Authors, and Copyright (C) 2004 by:
186	* Timothy A. Davis, Patrick Amestoy, Iain S. Duff, John K. Reid.
187	*
188	* Acknowledgements: This work (and the UMFPACK package) was supported by the
189	* National Science Foundation (ASC-9111263, DMS-9223088, and CCR-0203270).
190	* The UMFPACK/MA38 approximate degree update algorithm, the unsymmetric analog
191	* which forms the basis of AMD, was developed while Tim Davis was supported by
192	* CERFACS (Toulouse, France) in a post-doctoral position. This C version, and
193	* the etree postorder, were written while Tim Davis was on sabbatical at
194	* Stanford University and Lawrence Berkeley National Laboratory.
195
196	* ----------------------------------------------------------------------------
197	* INPUT ARGUMENTS (unaltered):
198	* ----------------------------------------------------------------------------
199
200	* n: The matrix order. Restriction: n >= 1.
201	*
202	* iwlen: The size of the Iw array. On input, the matrix is stored in
203	* Iw [0..pfree-1]. However, Iw [0..iwlen-1] should be slightly larger
204	* than what is required to hold the matrix, at least iwlen >= pfree + n.
205	* Otherwise, excessive compressions will take place. The recommended
206	* value of iwlen is 1.2 * pfree + n, which is the value used in the
207	* user-callable interface to this routine (amd_order.c). The algorithm
208	* will not run at all if iwlen < pfree. Restriction: iwlen >= pfree + n.
209	* Note that this is slightly more restrictive than the actual minimum
210	* (iwlen >= pfree), but AMD_2 will be very slow with no elbow room.
211	* Thus, this routine enforces a bare minimum elbow room of size n.
212	*
213	* pfree: On input the tail end of the array, Iw [pfree..iwlen-1], is empty,
214	* and the matrix is stored in Iw [0..pfree-1]. During execution,
215	* additional data is placed in Iw, and pfree is modified so that
216	* Iw [pfree..iwlen-1] is always the unused part of Iw.
217	*
218	* Control: A double array of size AMD_CONTROL containing input parameters
219	* that affect how the ordering is computed. If NULL, then default
220	* settings are used.
221	*
222	* Control [AMD_DENSE] is used to determine whether or not a given input
223	* row is "dense". A row is "dense" if the number of entries in the row
224	* exceeds Control [AMD_DENSE] times sqrt (n), except that rows with 16 or
225	* fewer entries are never considered "dense". To turn off the detection
226	* of dense rows, set Control [AMD_DENSE] to a negative number, or to a
227	* number larger than sqrt (n). The default value of Control [AMD_DENSE]
228	* is AMD_DEFAULT_DENSE, which is defined in amd.h as 10.
229	*
230	* Control [AMD_AGGRESSIVE] is used to determine whether or not aggressive
231	* absorption is to be performed. If nonzero, then aggressive absorption
232	* is performed (this is the default).
233
234	* ----------------------------------------------------------------------------
235	* INPUT/OUPUT ARGUMENTS:
236	* ----------------------------------------------------------------------------
237	*
238	* Pe: An integer array of size n. On input, Pe [i] is the index in Iw of
239	* the start of row i. Pe [i] is ignored if row i has no off-diagonal
240	* entries. Thus Pe [i] must be in the range 0 to pfree-1 for non-empty
241	* rows.
242	*
243	* During execution, it is used for both supervariables and elements:
244	*
245	* Principal supervariable i: index into Iw of the description of
246	* supervariable i. A supervariable represents one or more rows of
247	* the matrix with identical nonzero pattern. In this case,
248	* Pe [i] >= 0.
249	*
250	* Non-principal supervariable i: if i has been absorbed into another
251	* supervariable j, then Pe [i] = FLIP (j), where FLIP (j) is defined
252	* as (-(j)-2). Row j has the same pattern as row i. Note that j
253	* might later be absorbed into another supervariable j2, in which
254	* case Pe [i] is still FLIP (j), and Pe [j] = FLIP (j2) which is
255	* < EMPTY, where EMPTY is defined as (-1) in amd_internal.h.
256	*
257	* Unabsorbed element e: the index into Iw of the description of element
258	* e, if e has not yet been absorbed by a subsequent element. Element
259	* e is created when the supervariable of the same name is selected as
260	* the pivot. In this case, Pe [i] >= 0.
261	*
262	* Absorbed element e: if element e is absorbed into element e2, then
263	* Pe [e] = FLIP (e2). This occurs when the pattern of e (which we
264	* refer to as Le) is found to be a subset of the pattern of e2 (that
265	* is, Le2). In this case, Pe [i] < EMPTY. If element e is "null"
266	* (it has no nonzeros outside its pivot block), then Pe [e] = EMPTY,
267	* and e is the root of an assembly subtree (or the whole tree if
268	* there is just one such root).
269	*
270	* Dense variable i: if i is "dense", then Pe [i] = EMPTY.
271	*
272	* On output, Pe holds the assembly tree/forest, which implicitly
273	* represents a pivot order with identical fill-in as the actual order
274	* (via a depth-first search of the tree), as follows. If Nv [i] > 0,
275	* then i represents a node in the assembly tree, and the parent of i is
276	* Pe [i], or EMPTY if i is a root. If Nv [i] = 0, then (i, Pe [i])
277	* represents an edge in a subtree, the root of which is a node in the
278	* assembly tree. Note that i refers to a row/column in the original
279	* matrix, not the permuted matrix.
280	*
281	* Info: A double array of size AMD_INFO. If present, (that is, not NULL),
282	* then statistics about the ordering are returned in the Info array.
283	* See amd.h for a description.
284
285	* ----------------------------------------------------------------------------
286	* INPUT/MODIFIED (undefined on output):
287	* ----------------------------------------------------------------------------
288	*
289	* Len: An integer array of size n. On input, Len [i] holds the number of
290	* entries in row i of the matrix, excluding the diagonal. The contents
291	* of Len are undefined on output.
292	*
293	* Iw: An integer array of size iwlen. On input, Iw [0..pfree-1] holds the
294	* description of each row i in the matrix. The matrix must be symmetric,
295	* and both upper and lower triangular parts must be present. The
296	* diagonal must not be present. Row i is held as follows:
297	*
298	* Len [i]: the length of the row i data structure in the Iw array.
299	* Iw [Pe [i] ... Pe [i] + Len [i] - 1]:
300	* the list of column indices for nonzeros in row i (simple
301	* supervariables), excluding the diagonal. All supervariables
302	* start with one row/column each (supervariable i is just row i).
303	* If Len [i] is zero on input, then Pe [i] is ignored on input.
304	*
305	* Note that the rows need not be in any particular order, and there
306	* may be empty space between the rows.
307	*
308	* During execution, the supervariable i experiences fill-in. This is
309	* represented by placing in i a list of the elements that cause fill-in
310	* in supervariable i:
311	*
312	* Len [i]: the length of supervariable i in the Iw array.
313	* Iw [Pe [i] ... Pe [i] + Elen [i] - 1]:
314	* the list of elements that contain i. This list is kept short
315	* by removing absorbed elements.
316	* Iw [Pe [i] + Elen [i] ... Pe [i] + Len [i] - 1]:
317	* the list of supervariables in i. This list is kept short by
318	* removing nonprincipal variables, and any entry j that is also
319	* contained in at least one of the elements (j in Le) in the list
320	* for i (e in row i).
321	*
322	* When supervariable i is selected as pivot, we create an element e of
323	* the same name (e=i):
324	*
325	* Len [e]: the length of element e in the Iw array.
326	* Iw [Pe [e] ... Pe [e] + Len [e] - 1]:
327	* the list of supervariables in element e.
328	*
329	* An element represents the fill-in that occurs when supervariable i is
330	* selected as pivot (which represents the selection of row i and all
331	* non-principal variables whose principal variable is i). We use the
332	* term Le to denote the set of all supervariables in element e. Absorbed
333	* supervariables and elements are pruned from these lists when
334	* computationally convenient.
335	*
336	* CAUTION: THE INPUT MATRIX IS OVERWRITTEN DURING COMPUTATION.
337	* The contents of Iw are undefined on output.
338
339	* ----------------------------------------------------------------------------
340	* OUTPUT (need not be set on input):
341	* ----------------------------------------------------------------------------
342	*
343	* Nv: An integer array of size n. During execution, ABS (Nv [i]) is equal to
344	* the number of rows that are represented by the principal supervariable
345	* i. If i is a nonprincipal or dense variable, then Nv [i] = 0.
346	* Initially, Nv [i] = 1 for all i. Nv [i] < 0 signifies that i is a
347	* principal variable in the pattern Lme of the current pivot element me.
348	* After element me is constructed, Nv [i] is set back to a positive
349	* value.
350	*
351	* On output, Nv [i] holds the number of pivots represented by super
352	* row/column i of the original matrix, or Nv [i] = 0 for non-principal
353	* rows/columns. Note that i refers to a row/column in the original
354	* matrix, not the permuted matrix.
355	*
356	* Elen: An integer array of size n. See the description of Iw above. At the
357	* start of execution, Elen [i] is set to zero for all rows i. During
358	* execution, Elen [i] is the number of elements in the list for
359	* supervariable i. When e becomes an element, Elen [e] = FLIP (esize) is
360	* set, where esize is the size of the element (the number of pivots, plus
361	* the number of nonpivotal entries). Thus Elen [e] < EMPTY.
362	* Elen (i) = EMPTY set when variable i becomes nonprincipal.
363	*
364	* For variables, Elen (i) >= EMPTY holds until just before the
365	* postordering and permutation vectors are computed. For elements,
366	* Elen [e] < EMPTY holds.
367	*
368	* On output, Elen [i] is the degree of the row/column in the Cholesky
369	* factorization of the permuted matrix, corresponding to the original row
370	* i, if i is a super row/column. It is equal to EMPTY if i is
371	* non-principal. Note that i refers to a row/column in the original
372	* matrix, not the permuted matrix.
373	*
374	* Note that the contents of Elen on output differ from the Fortran
375	* version (Elen holds the inverse permutation in the Fortran version,
376	* which is instead returned in the Next array in this C version,
377	* described below).
378	*
379	* Last: In a degree list, Last [i] is the supervariable preceding i, or EMPTY
380	* if i is the head of the list. In a hash bucket, Last [i] is the hash
381	* key for i.
382	*
383	* Last [Head [hash]] is also used as the head of a hash bucket if
384	* Head [hash] contains a degree list (see the description of Head,
385	* below).
386	*
387	* On output, Last [0..n-1] holds the permutation. That is, if
388	* i = Last [k], then row i is the kth pivot row (where k ranges from 0 to
389	* n-1). Row Last [k] of A is the kth row in the permuted matrix, PAP'.
390	*
391	* Next: Next [i] is the supervariable following i in a link list, or EMPTY if
392	* i is the last in the list. Used for two kinds of lists: degree lists
393	* and hash buckets (a supervariable can be in only one kind of list at a
394	* time).
395	*
396	* On output Next [0..n-1] holds the inverse permutation. That is, if
397	* k = Next [i], then row i is the kth pivot row. Row i of A appears as
398	* the (Next[i])-th row in the permuted matrix, PAP'.
399	*
400	* Note that the contents of Next on output differ from the Fortran
401	* version (Next is undefined on output in the Fortran version).
402
403	* ----------------------------------------------------------------------------
404	* LOCAL WORKSPACE (not input or output - used only during execution):
405	* ----------------------------------------------------------------------------
406	*
407	* Degree: An integer array of size n. If i is a supervariable, then
408	* Degree [i] holds the current approximation of the external degree of
409	* row i (an upper bound). The external degree is the number of nonzeros
410	* in row i, minus ABS (Nv [i]), the diagonal part. The bound is equal to
411	* the exact external degree if Elen [i] is less than or equal to two.
412	*
413	* We also use the term "external degree" for elements e to refer to
414	* \|Le \ Lme\|. If e is an element, then Degree [e] is \|Le\|, which is the
415	* degree of the off-diagonal part of the element e (not including the
416	* diagonal part).
417	*
418	* Head: An integer array of size n. Head is used for degree lists.
419	* Head [deg] is the first supervariable in a degree list. All
420	* supervariables i in a degree list Head [deg] have the same approximate
421	* degree, namely, deg = Degree [i]. If the list Head [deg] is empty then
422	* Head [deg] = EMPTY.
423	*
424	* During supervariable detection Head [hash] also serves as a pointer to
425	* a hash bucket. If Head [hash] >= 0, there is a degree list of degree
426	* hash. The hash bucket head pointer is Last [Head [hash]]. If
427	* Head [hash] = EMPTY, then the degree list and hash bucket are both
428	* empty. If Head [hash] < EMPTY, then the degree list is empty, and
429	* FLIP (Head [hash]) is the head of the hash bucket. After supervariable
430	* detection is complete, all hash buckets are empty, and the
431	* (Last [Head [hash]] = EMPTY) condition is restored for the non-empty
432	* degree lists.
433	*
434	* W: An integer array of size n. The flag array W determines the status of
435	* elements and variables, and the external degree of elements.
436	*
437	* for elements:
438	* if W [e] = 0, then the element e is absorbed.
439	* if W [e] >= wflg, then W [e] - wflg is the size of the set
440	* \|Le \ Lme\|, in terms of nonzeros (the sum of ABS (Nv [i]) for
441	* each principal variable i that is both in the pattern of
442	* element e and NOT in the pattern of the current pivot element,
443	* me).
444	* if wflg > W [e] > 0, then e is not absorbed and has not yet been
445	* seen in the scan of the element lists in the computation of
446	* \|Le\Lme\| in Scan 1 below.
447	*
448	* for variables:
449	* during supervariable detection, if W [j] != wflg then j is
450	* not in the pattern of variable i.
451	*
452	* The W array is initialized by setting W [i] = 1 for all i, and by
453	* setting wflg = 2. It is reinitialized if wflg becomes too large (to
454	* ensure that wflg+n does not cause integer overflow).
455
456	* ----------------------------------------------------------------------------
457	* LOCAL INTEGERS:
458	* ----------------------------------------------------------------------------
459	*/
460
461	Int deg, degme, dext, lemax, e, elenme, eln, i, ilast, inext, j,
462	jlast, jnext, k, knt1, knt2, knt3, lenj, ln, me, mindeg, nel, nleft,
463	nvi, nvj, nvpiv, slenme, wbig, we, wflg, wnvi, ok, ndense, ncmpa,
464	dense, aggressive ;
465
466	unsigned Int hash ; /* unsigned, so that hash % n is well defined.*/
467
468	/*
469	* deg: the degree of a variable or element
470	* degme: size, \|Lme\|, of the current element, me (= Degree [me])
471	* dext: external degree, \|Le \ Lme\|, of some element e
472	* lemax: largest \|Le\| seen so far (called dmax in Fortran version)
473	* e: an element
474	* elenme: the length, Elen [me], of element list of pivotal variable
475	* eln: the length, Elen [...], of an element list
476	* hash: the computed value of the hash function
477	* i: a supervariable
478	* ilast: the entry in a link list preceding i
479	* inext: the entry in a link list following i
480	* j: a supervariable
481	* jlast: the entry in a link list preceding j
482	* jnext: the entry in a link list, or path, following j
483	* k: the pivot order of an element or variable
484	* knt1: loop counter used during element construction
485	* knt2: loop counter used during element construction
486	* knt3: loop counter used during compression
487	* lenj: Len [j]
488	* ln: length of a supervariable list
489	* me: current supervariable being eliminated, and the current
490	* element created by eliminating that supervariable
491	* mindeg: current minimum degree
492	* nel: number of pivots selected so far
493	* nleft: n - nel, the number of nonpivotal rows/columns remaining
494	* nvi: the number of variables in a supervariable i (= Nv [i])
495	* nvj: the number of variables in a supervariable j (= Nv [j])
496	* nvpiv: number of pivots in current element
497	* slenme: number of variables in variable list of pivotal variable
498	* wbig: = INT_MAX - n for the int version, UF_long_max - n for the
499	* UF_long version. wflg is not allowed to be >= wbig.
500	* we: W [e]
501	* wflg: used for flagging the W array. See description of Iw.
502	* wnvi: wflg - Nv [i]
503	* x: either a supervariable or an element
504	*
505	* ok: true if supervariable j can be absorbed into i
506	* ndense: number of "dense" rows/columns
507	* dense: rows/columns with initial degree > dense are considered "dense"
508	* aggressive: true if aggressive absorption is being performed
509	* ncmpa: number of garbage collections
510
511	* ----------------------------------------------------------------------------
512	* LOCAL DOUBLES, used for statistical output only (except for alpha):
513	* ----------------------------------------------------------------------------
514	*/
515
516	double f, r, ndiv, s, nms_lu, nms_ldl, dmax, alpha, lnz, lnzme ;
517
518	/*
519	* f: nvpiv
520	* r: degme + nvpiv
521	* ndiv: number of divisions for LU or LDL' factorizations
522	* s: number of multiply-subtract pairs for LU factorization, for the
523	* current element me
524	* nms_lu number of multiply-subtract pairs for LU factorization
525	* nms_ldl number of multiply-subtract pairs for LDL' factorization
526	* dmax: the largest number of entries in any column of L, including the
527	* diagonal
528	* alpha: "dense" degree ratio
529	* lnz: the number of nonzeros in L (excluding the diagonal)
530	* lnzme: the number of nonzeros in L (excl. the diagonal) for the
531	* current element me
532
533	* ----------------------------------------------------------------------------
534	* LOCAL "POINTERS" (indices into the Iw array)
535	* ----------------------------------------------------------------------------
536	*/
537
538	Int p, p1, p2, p3, p4, pdst, pend, pj, pme, pme1, pme2, pn, psrc ;
539
540	/*
541	* Any parameter (Pe [...] or pfree) or local variable starting with "p" (for
542	* Pointer) is an index into Iw, and all indices into Iw use variables starting
543	* with "p." The only exception to this rule is the iwlen input argument.
544	*
545	* p: pointer into lots of things
546	* p1: Pe [i] for some variable i (start of element list)
547	* p2: Pe [i] + Elen [i] - 1 for some variable i
548	* p3: index of first supervariable in clean list
549	* p4:
550	* pdst: destination pointer, for compression
551	* pend: end of memory to compress
552	* pj: pointer into an element or variable
553	* pme: pointer into the current element (pme1...pme2)
554	* pme1: the current element, me, is stored in Iw [pme1...pme2]
555	* pme2: the end of the current element
556	* pn: pointer into a "clean" variable, also used to compress
557	* psrc: source pointer, for compression
558	*/
559
560	/* ========================================================================= */
561	/* INITIALIZATIONS */
562	/* ========================================================================= */
563
564	/* Note that this restriction on iwlen is slightly more restrictive than
565	* what is actually required in AMD_2. AMD_2 can operate with no elbow
566	* room at all, but it will be slow. For better performance, at least
567	* size-n elbow room is enforced. */
568	ASSERT (iwlen >= pfree + n) ;
569	ASSERT (n > 0) ;
570
571	/* initialize output statistics */
572	lnz = 0 ;
573	ndiv = 0 ;
574	nms_lu = 0 ;
575	nms_ldl = 0 ;
576	dmax = 1 ;
577	me = EMPTY ;
578
579	mindeg = 0 ;
580	ncmpa = 0 ;
581	nel = 0 ;
582	lemax = 0 ;
583
584	/* get control parameters */
585	if (Control != (double *) NULL)
586	{
587	alpha = Control [AMD_DENSE] ;
588	aggressive = (Control [AMD_AGGRESSIVE] != 0) ;
589	}
590	else
591	{
592	alpha = AMD_DEFAULT_DENSE ;
593	aggressive = AMD_DEFAULT_AGGRESSIVE ;
594	}
595	/* Note: if alpha is NaN, this is undefined: */
596	if (alpha < 0)
597	{
598	/* only remove completely dense rows/columns */
599	dense = n-2 ;
600	}
601	else
602	{
603	dense = alpha * sqrt ((double) n) ;
604	}
605	dense = MAX (16, dense) ;
606	dense = MIN (n, dense) ;
607	AMD_DEBUG1 (("\n\nAMD (debug), alpha %g, aggr. "ID"\n",
608	alpha, aggressive)) ;
609
610	for (i = 0 ; i < n ; i++)
611	{
612	Last [i] = EMPTY ;
613	Head [i] = EMPTY ;
614	Next [i] = EMPTY ;
615	/* if separate Hhead array is used for hash buckets: *
616	Hhead [i] = EMPTY ;
617	*/
618	Nv [i] = 1 ;
619	W [i] = 1 ;
620	Elen [i] = 0 ;
621	Degree [i] = Len [i] ;
622	}
623
624	#ifndef NDEBUG
625	AMD_DEBUG1 (("\n======Nel "ID" initial\n", nel)) ;
626	AMD_dump (n, Pe, Iw, Len, iwlen, pfree, Nv, Next, Last,
627	Head, Elen, Degree, W, -1) ;
628	#endif
629
630	/* initialize wflg */
631	wbig = Int_MAX - n ;
632	wflg = clear_flag (0, wbig, W, n) ;
633
634	/* --------------------------------------------------------------------- */
635	/* initialize degree lists and eliminate dense and empty rows */
636	/* --------------------------------------------------------------------- */
637
638	ndense = 0 ;
639
640	for (i = 0 ; i < n ; i++)
641	{
642	deg = Degree [i] ;
643	ASSERT (deg >= 0 && deg < n) ;
644	if (deg == 0)
645	{
646
647	/* -------------------------------------------------------------
648	* we have a variable that can be eliminated at once because
649	* there is no off-diagonal non-zero in its row. Note that
650	* Nv [i] = 1 for an empty variable i. It is treated just
651	* the same as an eliminated element i.
652	* ------------------------------------------------------------- */
653
654	Elen [i] = FLIP (1) ;
655	nel++ ;
656	Pe [i] = EMPTY ;
657	W [i] = 0 ;
658
659	}
660	else if (deg > dense)
661	{
662
663	/* -------------------------------------------------------------
664	* Dense variables are not treated as elements, but as unordered,
665	* non-principal variables that have no parent. They do not take
666	* part in the postorder, since Nv [i] = 0. Note that the Fortran
667	* version does not have this option.
668	* ------------------------------------------------------------- */
669
670	AMD_DEBUG1 (("Dense node "ID" degree "ID"\n", i, deg)) ;
671	ndense++ ;
672	Nv [i] = 0 ; /* do not postorder this node */
673	Elen [i] = EMPTY ;
674	nel++ ;
675	Pe [i] = EMPTY ;
676
677	}
678	else
679	{
680
681	/* -------------------------------------------------------------
682	* place i in the degree list corresponding to its degree
683	* ------------------------------------------------------------- */
684
685	inext = Head [deg] ;
686	ASSERT (inext >= EMPTY && inext < n) ;
687	if (inext != EMPTY) Last [inext] = i ;
688	Next [i] = inext ;
689	Head [deg] = i ;
690
691	}
692	}
693
694	/* ========================================================================= */
695	/* WHILE (selecting pivots) DO */
696	/* ========================================================================= */
697
698	while (nel < n)
699	{
700
701	#ifndef NDEBUG
702	AMD_DEBUG1 (("\n======Nel "ID"\n", nel)) ;
703	if (AMD_debug >= 2)
704	{
705	AMD_dump (n, Pe, Iw, Len, iwlen, pfree, Nv, Next,
706	Last, Head, Elen, Degree, W, nel) ;
707	}
708	#endif
709
710	/* ========================================================================= */
711	/* GET PIVOT OF MINIMUM DEGREE */
712	/* ========================================================================= */
713
714	/* ----------------------------------------------------------------- */
715	/* find next supervariable for elimination */
716	/* ----------------------------------------------------------------- */
717
718	ASSERT (mindeg >= 0 && mindeg < n) ;
719	for (deg = mindeg ; deg < n ; deg++)
720	{
721	me = Head [deg] ;
722	if (me != EMPTY) break ;
723	}
724	mindeg = deg ;
725	ASSERT (me >= 0 && me < n) ;
726	AMD_DEBUG1 (("=================me: "ID"\n", me)) ;
727
728	/* ----------------------------------------------------------------- */
729	/* remove chosen variable from link list */
730	/* ----------------------------------------------------------------- */
731
732	inext = Next [me] ;
733	ASSERT (inext >= EMPTY && inext < n) ;
734	if (inext != EMPTY) Last [inext] = EMPTY ;
735	Head [deg] = inext ;
736
737	/* ----------------------------------------------------------------- */
738	/* me represents the elimination of pivots nel to nel+Nv[me]-1. */
739	/* place me itself as the first in this set. */
740	/* ----------------------------------------------------------------- */
741
742	elenme = Elen [me] ;
743	nvpiv = Nv [me] ;
744	ASSERT (nvpiv > 0) ;
745	nel += nvpiv ;
746
747	/* ========================================================================= */
748	/* CONSTRUCT NEW ELEMENT */
749	/* ========================================================================= */
750
751	/* -----------------------------------------------------------------
752	* At this point, me is the pivotal supervariable. It will be
753	* converted into the current element. Scan list of the pivotal
754	* supervariable, me, setting tree pointers and constructing new list
755	* of supervariables for the new element, me. p is a pointer to the
756	* current position in the old list.
757	* ----------------------------------------------------------------- */
758
759	/* flag the variable "me" as being in Lme by negating Nv [me] */
760	Nv [me] = -nvpiv ;
761	degme = 0 ;
762	ASSERT (Pe [me] >= 0 && Pe [me] < iwlen) ;
763
764	if (elenme == 0)
765	{
766
767	/* ------------------------------------------------------------- */
768	/* construct the new element in place */
769	/* ------------------------------------------------------------- */
770
771	pme1 = Pe [me] ;
772	pme2 = pme1 - 1 ;
773
774	for (p = pme1 ; p <= pme1 + Len [me] - 1 ; p++)
775	{
776	i = Iw [p] ;
777	ASSERT (i >= 0 && i < n && Nv [i] >= 0) ;
778	nvi = Nv [i] ;
779	if (nvi > 0)
780	{
781
782	/* ----------------------------------------------------- */
783	/* i is a principal variable not yet placed in Lme. */
784	/* store i in new list */
785	/* ----------------------------------------------------- */
786
787	/* flag i as being in Lme by negating Nv [i] */
788	degme += nvi ;
789	Nv [i] = -nvi ;
790	Iw [++pme2] = i ;
791
792	/* ----------------------------------------------------- */
793	/* remove variable i from degree list. */
794	/* ----------------------------------------------------- */
795
796	ilast = Last [i] ;
797	inext = Next [i] ;
798	ASSERT (ilast >= EMPTY && ilast < n) ;
799	ASSERT (inext >= EMPTY && inext < n) ;
800	if (inext != EMPTY) Last [inext] = ilast ;
801	if (ilast != EMPTY)
802	{
803	Next [ilast] = inext ;
804	}
805	else
806	{
807	/* i is at the head of the degree list */
808	ASSERT (Degree [i] >= 0 && Degree [i] < n) ;
809	Head [Degree [i]] = inext ;
810	}
811	}
812	}
813	}
814	else
815	{
816
817	/* ------------------------------------------------------------- */
818	/* construct the new element in empty space, Iw [pfree ...] */
819	/* ------------------------------------------------------------- */
820
821	p = Pe [me] ;
822	pme1 = pfree ;
823	slenme = Len [me] - elenme ;
824
825	for (knt1 = 1 ; knt1 <= elenme + 1 ; knt1++)
826	{
827
828	if (knt1 > elenme)
829	{
830	/* search the supervariables in me. */
831	e = me ;
832	pj = p ;
833	ln = slenme ;
834	AMD_DEBUG2 (("Search sv: "ID" "ID" "ID"\n", me,pj,ln)) ;
835	}
836	else
837	{
838	/* search the elements in me. */
839	e = Iw [p++] ;
840	ASSERT (e >= 0 && e < n) ;
841	pj = Pe [e] ;
842	ln = Len [e] ;
843	AMD_DEBUG2 (("Search element e "ID" in me "ID"\n", e,me)) ;
844	ASSERT (Elen [e] < EMPTY && W [e] > 0 && pj >= 0) ;
845	}
846	ASSERT (ln >= 0 && (ln == 0 \|\| (pj >= 0 && pj < iwlen))) ;
847
848	/* ---------------------------------------------------------
849	* search for different supervariables and add them to the
850	* new list, compressing when necessary. this loop is
851	* executed once for each element in the list and once for
852	* all the supervariables in the list.
853	* --------------------------------------------------------- */
854
855	for (knt2 = 1 ; knt2 <= ln ; knt2++)
856	{
857	i = Iw [pj++] ;
858	ASSERT (i >= 0 && i < n && (i == me \|\| Elen [i] >= EMPTY));
859	nvi = Nv [i] ;
860	AMD_DEBUG2 ((": "ID" "ID" "ID" "ID"\n",
861	i, Elen [i], Nv [i], wflg)) ;
862
863	if (nvi > 0)
864	{
865
866	/* ------------------------------------------------- */
867	/* compress Iw, if necessary */
868	/* ------------------------------------------------- */
869
870	if (pfree >= iwlen)
871	{
872
873	AMD_DEBUG1 (("GARBAGE COLLECTION\n")) ;
874
875	/* prepare for compressing Iw by adjusting pointers
876	* and lengths so that the lists being searched in
877	* the inner and outer loops contain only the
878	* remaining entries. */
879
880	Pe [me] = p ;
881	Len [me] -= knt1 ;
882	/* check if nothing left of supervariable me */
883	if (Len [me] == 0) Pe [me] = EMPTY ;
884	Pe [e] = pj ;
885	Len [e] = ln - knt2 ;
886	/* nothing left of element e */
887	if (Len [e] == 0) Pe [e] = EMPTY ;
888
889	ncmpa++ ; /* one more garbage collection */
890
891	/* store first entry of each object in Pe */
892	/* FLIP the first entry in each object */
893	for (j = 0 ; j < n ; j++)
894	{
895	pn = Pe [j] ;
896	if (pn >= 0)
897	{
898	ASSERT (pn >= 0 && pn < iwlen) ;
899	Pe [j] = Iw [pn] ;
900	Iw [pn] = FLIP (j) ;
901	}
902	}
903
904	/* psrc/pdst point to source/destination */
905	psrc = 0 ;
906	pdst = 0 ;
907	pend = pme1 - 1 ;
908
909	while (psrc <= pend)
910	{
911	/* search for next FLIP'd entry */
912	j = FLIP (Iw [psrc++]) ;
913	if (j >= 0)
914	{
915	AMD_DEBUG2 (("Got object j: "ID"\n", j)) ;
916	Iw [pdst] = Pe [j] ;
917	Pe [j] = pdst++ ;
918	lenj = Len [j] ;
919	/* copy from source to destination */
920	for (knt3 = 0 ; knt3 <= lenj - 2 ; knt3++)
921	{
922	Iw [pdst++] = Iw [psrc++] ;
923	}
924	}
925	}
926
927	/* move the new partially-constructed element */
928	p1 = pdst ;
929	for (psrc = pme1 ; psrc <= pfree-1 ; psrc++)
930	{
931	Iw [pdst++] = Iw [psrc] ;
932	}
933	pme1 = p1 ;
934	pfree = pdst ;
935	pj = Pe [e] ;
936	p = Pe [me] ;
937
938	}
939
940	/* ------------------------------------------------- */
941	/* i is a principal variable not yet placed in Lme */
942	/* store i in new list */
943	/* ------------------------------------------------- */
944
945	/* flag i as being in Lme by negating Nv [i] */
946	degme += nvi ;
947	Nv [i] = -nvi ;
948	Iw [pfree++] = i ;
949	AMD_DEBUG2 ((" s: "ID" nv "ID"\n", i, Nv [i]));
950
951	/* ------------------------------------------------- */
952	/* remove variable i from degree link list */
953	/* ------------------------------------------------- */
954
955	ilast = Last [i] ;
956	inext = Next [i] ;
957	ASSERT (ilast >= EMPTY && ilast < n) ;
958	ASSERT (inext >= EMPTY && inext < n) ;
959	if (inext != EMPTY) Last [inext] = ilast ;
960	if (ilast != EMPTY)
961	{
962	Next [ilast] = inext ;
963	}
964	else
965	{
966	/* i is at the head of the degree list */
967	ASSERT (Degree [i] >= 0 && Degree [i] < n) ;
968	Head [Degree [i]] = inext ;
969	}
970	}
971	}
972
973	if (e != me)
974	{
975	/* set tree pointer and flag to indicate element e is
976	* absorbed into new element me (the parent of e is me) */
977	AMD_DEBUG1 ((" Element "ID" => "ID"\n", e, me)) ;
978	Pe [e] = FLIP (me) ;
979	W [e] = 0 ;
980	}
981	}
982
983	pme2 = pfree - 1 ;
984	}
985
986	/* ----------------------------------------------------------------- */
987	/* me has now been converted into an element in Iw [pme1..pme2] */
988	/* ----------------------------------------------------------------- */
989
990	/* degme holds the external degree of new element */
991	Degree [me] = degme ;
992	Pe [me] = pme1 ;
993	Len [me] = pme2 - pme1 + 1 ;
994	ASSERT (Pe [me] >= 0 && Pe [me] < iwlen) ;
995
996	Elen [me] = FLIP (nvpiv + degme) ;
997	/* FLIP (Elen (me)) is now the degree of pivot (including
998	* diagonal part). */
999
1000	#ifndef NDEBUG
1001	AMD_DEBUG2 (("New element structure: length= "ID"\n", pme2-pme1+1)) ;
1002	for (pme = pme1 ; pme <= pme2 ; pme++) AMD_DEBUG3 ((" "ID"", Iw[pme]));
1003	AMD_DEBUG3 (("\n")) ;
1004	#endif
1005
1006	/* ----------------------------------------------------------------- */
1007	/* make sure that wflg is not too large. */
1008	/* ----------------------------------------------------------------- */
1009
1010	/* With the current value of wflg, wflg+n must not cause integer
1011	* overflow */
1012
1013	wflg = clear_flag (wflg, wbig, W, n) ;
1014
1015	/* ========================================================================= */
1016	/* COMPUTE (W [e] - wflg) = \|Le\Lme\| FOR ALL ELEMENTS */
1017	/* ========================================================================= */
1018
1019	/* -----------------------------------------------------------------
1020	* Scan 1: compute the external degrees of previous elements with
1021	* respect to the current element. That is:
1022	* (W [e] - wflg) = \|Le \ Lme\|
1023	* for each element e that appears in any supervariable in Lme. The
1024	* notation Le refers to the pattern (list of supervariables) of a
1025	* previous element e, where e is not yet absorbed, stored in
1026	* Iw [Pe [e] + 1 ... Pe [e] + Len [e]]. The notation Lme
1027	* refers to the pattern of the current element (stored in
1028	* Iw [pme1..pme2]). If aggressive absorption is enabled, and
1029	* (W [e] - wflg) becomes zero, then the element e will be absorbed
1030	* in Scan 2.
1031	* ----------------------------------------------------------------- */
1032
1033	AMD_DEBUG2 (("me: ")) ;
1034	for (pme = pme1 ; pme <= pme2 ; pme++)
1035	{
1036	i = Iw [pme] ;
1037	ASSERT (i >= 0 && i < n) ;
1038	eln = Elen [i] ;
1039	AMD_DEBUG3 ((""ID" Elen "ID": \n", i, eln)) ;
1040	if (eln > 0)
1041	{
1042	/* note that Nv [i] has been negated to denote i in Lme: */
1043	nvi = -Nv [i] ;
1044	ASSERT (nvi > 0 && Pe [i] >= 0 && Pe [i] < iwlen) ;
1045	wnvi = wflg - nvi ;
1046	for (p = Pe [i] ; p <= Pe [i] + eln - 1 ; p++)
1047	{
1048	e = Iw [p] ;
1049	ASSERT (e >= 0 && e < n) ;
1050	we = W [e] ;
1051	AMD_DEBUG4 ((" e "ID" we "ID" ", e, we)) ;
1052	if (we >= wflg)
1053	{
1054	/* unabsorbed element e has been seen in this loop */
1055	AMD_DEBUG4 ((" unabsorbed, first time seen")) ;
1056	we -= nvi ;
1057	}
1058	else if (we != 0)
1059	{
1060	/* e is an unabsorbed element */
1061	/* this is the first we have seen e in all of Scan 1 */
1062	AMD_DEBUG4 ((" unabsorbed")) ;
1063	we = Degree [e] + wnvi ;
1064	}
1065	AMD_DEBUG4 (("\n")) ;
1066	W [e] = we ;
1067	}
1068	}
1069	}
1070	AMD_DEBUG2 (("\n")) ;
1071
1072	/* ========================================================================= */
1073	/* DEGREE UPDATE AND ELEMENT ABSORPTION */
1074	/* ========================================================================= */
1075
1076	/* -----------------------------------------------------------------
1077	* Scan 2: for each i in Lme, sum up the degree of Lme (which is
1078	* degme), plus the sum of the external degrees of each Le for the
1079	* elements e appearing within i, plus the supervariables in i.
1080	* Place i in hash list.
1081	* ----------------------------------------------------------------- */
1082
1083	for (pme = pme1 ; pme <= pme2 ; pme++)
1084	{
1085	i = Iw [pme] ;
1086	ASSERT (i >= 0 && i < n && Nv [i] < 0 && Elen [i] >= 0) ;
1087	AMD_DEBUG2 (("Updating: i "ID" "ID" "ID"\n", i, Elen[i], Len [i]));
1088	p1 = Pe [i] ;
1089	p2 = p1 + Elen [i] - 1 ;
1090	pn = p1 ;
1091	hash = 0 ;
1092	deg = 0 ;
1093	ASSERT (p1 >= 0 && p1 < iwlen && p2 >= -1 && p2 < iwlen) ;
1094
1095	/* ------------------------------------------------------------- */
1096	/* scan the element list associated with supervariable i */
1097	/* ------------------------------------------------------------- */
1098
1099	/* UMFPACK/MA38-style approximate degree: */
1100	if (aggressive)
1101	{
1102	for (p = p1 ; p <= p2 ; p++)
1103	{
1104	e = Iw [p] ;
1105	ASSERT (e >= 0 && e < n) ;
1106	we = W [e] ;
1107	if (we != 0)
1108	{
1109	/* e is an unabsorbed element */
1110	/* dext = \| Le \ Lme \| */
1111	dext = we - wflg ;
1112	if (dext > 0)
1113	{
1114	deg += dext ;
1115	Iw [pn++] = e ;
1116	hash += e ;
1117	AMD_DEBUG4 ((" e: "ID" hash = "ID"\n",e,hash)) ;
1118	}
1119	else
1120	{
1121	/* external degree of e is zero, absorb e into me*/
1122	AMD_DEBUG1 ((" Element "ID" =>"ID" (aggressive)\n",
1123	e, me)) ;
1124	ASSERT (dext == 0) ;
1125	Pe [e] = FLIP (me) ;
1126	W [e] = 0 ;
1127	}
1128	}
1129	}
1130	}
1131	else
1132	{
1133	for (p = p1 ; p <= p2 ; p++)
1134	{
1135	e = Iw [p] ;
1136	ASSERT (e >= 0 && e < n) ;
1137	we = W [e] ;
1138	if (we != 0)
1139	{
1140	/* e is an unabsorbed element */
1141	dext = we - wflg ;
1142	ASSERT (dext >= 0) ;
1143	deg += dext ;
1144	Iw [pn++] = e ;
1145	hash += e ;
1146	AMD_DEBUG4 ((" e: "ID" hash = "ID"\n",e,hash)) ;
1147	}
1148	}
1149	}
1150
1151	/* count the number of elements in i (including me): */
1152	Elen [i] = pn - p1 + 1 ;
1153
1154	/* ------------------------------------------------------------- */
1155	/* scan the supervariables in the list associated with i */
1156	/* ------------------------------------------------------------- */
1157
1158	/* The bulk of the AMD run time is typically spent in this loop,
1159	* particularly if the matrix has many dense rows that are not
1160	* removed prior to ordering. */
1161	p3 = pn ;
1162	p4 = p1 + Len [i] ;
1163	for (p = p2 + 1 ; p < p4 ; p++)
1164	{
1165	j = Iw [p] ;
1166	ASSERT (j >= 0 && j < n) ;
1167	nvj = Nv [j] ;
1168	if (nvj > 0)
1169	{
1170	/* j is unabsorbed, and not in Lme. */
1171	/* add to degree and add to new list */
1172	deg += nvj ;
1173	Iw [pn++] = j ;
1174	hash += j ;
1175	AMD_DEBUG4 ((" s: "ID" hash "ID" Nv[j]= "ID"\n",
1176	j, hash, nvj)) ;
1177	}
1178	}
1179
1180	/* ------------------------------------------------------------- */
1181	/* update the degree and check for mass elimination */
1182	/* ------------------------------------------------------------- */
1183
1184	/* with aggressive absorption, deg==0 is identical to the
1185	* Elen [i] == 1 && p3 == pn test, below. */
1186	ASSERT (IMPLIES (aggressive, (deg==0) == (Elen[i]==1 && p3==pn))) ;
1187
1188	if (Elen [i] == 1 && p3 == pn)
1189	{
1190
1191	/* --------------------------------------------------------- */
1192	/* mass elimination */
1193	/* --------------------------------------------------------- */
1194
1195	/* There is nothing left of this node except for an edge to
1196	* the current pivot element. Elen [i] is 1, and there are
1197	* no variables adjacent to node i. Absorb i into the
1198	* current pivot element, me. Note that if there are two or
1199	* more mass eliminations, fillin due to mass elimination is
1200	* possible within the nvpiv-by-nvpiv pivot block. It is this
1201	* step that causes AMD's analysis to be an upper bound.
1202	*
1203	* The reason is that the selected pivot has a lower
1204	* approximate degree than the true degree of the two mass
1205	* eliminated nodes. There is no edge between the two mass
1206	* eliminated nodes. They are merged with the current pivot
1207	* anyway.
1208	*
1209	* No fillin occurs in the Schur complement, in any case,
1210	* and this effect does not decrease the quality of the
1211	* ordering itself, just the quality of the nonzero and
1212	* flop count analysis. It also means that the post-ordering
1213	* is not an exact elimination tree post-ordering. */
1214
1215	AMD_DEBUG1 ((" MASS i "ID" => parent e "ID"\n", i, me)) ;
1216	Pe [i] = FLIP (me) ;
1217	nvi = -Nv [i] ;
1218	degme -= nvi ;
1219	nvpiv += nvi ;
1220	nel += nvi ;
1221	Nv [i] = 0 ;
1222	Elen [i] = EMPTY ;
1223
1224	}
1225	else
1226	{
1227
1228	/* --------------------------------------------------------- */
1229	/* update the upper-bound degree of i */
1230	/* --------------------------------------------------------- */
1231
1232	/* the following degree does not yet include the size
1233	* of the current element, which is added later: */
1234
1235	Degree [i] = MIN (Degree [i], deg) ;
1236
1237	/* --------------------------------------------------------- */
1238	/* add me to the list for i */
1239	/* --------------------------------------------------------- */
1240
1241	/* move first supervariable to end of list */
1242	Iw [pn] = Iw [p3] ;
1243	/* move first element to end of element part of list */
1244	Iw [p3] = Iw [p1] ;
1245	/* add new element, me, to front of list. */
1246	Iw [p1] = me ;
1247	/* store the new length of the list in Len [i] */
1248	Len [i] = pn - p1 + 1 ;
1249
1250	/* --------------------------------------------------------- */
1251	/* place in hash bucket. Save hash key of i in Last [i]. */
1252	/* --------------------------------------------------------- */
1253
1254	/* NOTE: this can fail if hash is negative, because the ANSI C
1255	* standard does not define a % b when a and/or b are negative.
1256	* That's why hash is defined as an unsigned Int, to avoid this
1257	* problem. */
1258	hash = hash % n ;
1259	ASSERT (((Int) hash) >= 0 && ((Int) hash) < n) ;
1260
1261	/* if the Hhead array is not used: */
1262	j = Head [hash] ;
1263	if (j <= EMPTY)
1264	{
1265	/* degree list is empty, hash head is FLIP (j) */
1266	Next [i] = FLIP (j) ;
1267	Head [hash] = FLIP (i) ;
1268	}
1269	else
1270	{
1271	/* degree list is not empty, use Last [Head [hash]] as
1272	* hash head. */
1273	Next [i] = Last [j] ;
1274	Last [j] = i ;
1275	}
1276
1277	/* if a separate Hhead array is used: *
1278	Next [i] = Hhead [hash] ;
1279	Hhead [hash] = i ;
1280	*/
1281
1282	Last [i] = hash ;
1283	}
1284	}
1285
1286	Degree [me] = degme ;
1287
1288	/* ----------------------------------------------------------------- */
1289	/* Clear the counter array, W [...], by incrementing wflg. */
1290	/* ----------------------------------------------------------------- */
1291
1292	/* make sure that wflg+n does not cause integer overflow */
1293	lemax = MAX (lemax, degme) ;
1294	wflg += lemax ;
1295	wflg = clear_flag (wflg, wbig, W, n) ;
1296	/* at this point, W [0..n-1] < wflg holds */
1297
1298	/* ========================================================================= */
1299	/* SUPERVARIABLE DETECTION */
1300	/* ========================================================================= */
1301
1302	AMD_DEBUG1 (("Detecting supervariables:\n")) ;
1303	for (pme = pme1 ; pme <= pme2 ; pme++)
1304	{
1305	i = Iw [pme] ;
1306	ASSERT (i >= 0 && i < n) ;
1307	AMD_DEBUG2 (("Consider i "ID" nv "ID"\n", i, Nv [i])) ;
1308	if (Nv [i] < 0)
1309	{
1310	/* i is a principal variable in Lme */
1311
1312	/* ---------------------------------------------------------
1313	* examine all hash buckets with 2 or more variables. We do
1314	* this by examing all unique hash keys for supervariables in
1315	* the pattern Lme of the current element, me
1316	* --------------------------------------------------------- */
1317
1318	/* let i = head of hash bucket, and empty the hash bucket */
1319	ASSERT (Last [i] >= 0 && Last [i] < n) ;
1320	hash = Last [i] ;
1321
1322	/* if Hhead array is not used: */
1323	j = Head [hash] ;
1324	if (j == EMPTY)
1325	{
1326	/* hash bucket and degree list are both empty */
1327	i = EMPTY ;
1328	}
1329	else if (j < EMPTY)
1330	{
1331	/* degree list is empty */
1332	i = FLIP (j) ;
1333	Head [hash] = EMPTY ;
1334	}
1335	else
1336	{
1337	/* degree list is not empty, restore Last [j] of head j */
1338	i = Last [j] ;
1339	Last [j] = EMPTY ;
1340	}
1341
1342	/* if separate Hhead array is used: *
1343	i = Hhead [hash] ;
1344	Hhead [hash] = EMPTY ;
1345	*/
1346
1347	ASSERT (i >= EMPTY && i < n) ;
1348	AMD_DEBUG2 (("----i "ID" hash "ID"\n", i, hash)) ;
1349
1350	while (i != EMPTY && Next [i] != EMPTY)
1351	{
1352
1353	/* -----------------------------------------------------
1354	* this bucket has one or more variables following i.
1355	* scan all of them to see if i can absorb any entries
1356	* that follow i in hash bucket. Scatter i into w.
1357	* ----------------------------------------------------- */
1358
1359	ln = Len [i] ;
1360	eln = Elen [i] ;
1361	ASSERT (ln >= 0 && eln >= 0) ;
1362	ASSERT (Pe [i] >= 0 && Pe [i] < iwlen) ;
1363	/* do not flag the first element in the list (me) */
1364	for (p = Pe [i] + 1 ; p <= Pe [i] + ln - 1 ; p++)
1365	{
1366	ASSERT (Iw [p] >= 0 && Iw [p] < n) ;
1367	W [Iw [p]] = wflg ;
1368	}
1369
1370	/* ----------------------------------------------------- */
1371	/* scan every other entry j following i in bucket */
1372	/* ----------------------------------------------------- */
1373
1374	jlast = i ;
1375	j = Next [i] ;
1376	ASSERT (j >= EMPTY && j < n) ;
1377
1378	while (j != EMPTY)
1379	{
1380	/* ------------------------------------------------- */
1381	/* check if j and i have identical nonzero pattern */
1382	/* ------------------------------------------------- */
1383
1384	AMD_DEBUG3 (("compare i "ID" and j "ID"\n", i,j)) ;
1385
1386	/* check if i and j have the same Len and Elen */
1387	ASSERT (Len [j] >= 0 && Elen [j] >= 0) ;
1388	ASSERT (Pe [j] >= 0 && Pe [j] < iwlen) ;
1389	ok = (Len [j] == ln) && (Elen [j] == eln) ;
1390	/* skip the first element in the list (me) */
1391	for (p = Pe [j] + 1 ; ok && p <= Pe [j] + ln - 1 ; p++)
1392	{
1393	ASSERT (Iw [p] >= 0 && Iw [p] < n) ;
1394	if (W [Iw [p]] != wflg) ok = 0 ;
1395	}
1396	if (ok)
1397	{
1398	/* --------------------------------------------- */
1399	/* found it! j can be absorbed into i */
1400	/* --------------------------------------------- */
1401
1402	AMD_DEBUG1 (("found it! j "ID" => i "ID"\n", j,i));
1403	Pe [j] = FLIP (i) ;
1404	/* both Nv [i] and Nv [j] are negated since they */
1405	/* are in Lme, and the absolute values of each */
1406	/* are the number of variables in i and j: */
1407	Nv [i] += Nv [j] ;
1408	Nv [j] = 0 ;
1409	Elen [j] = EMPTY ;
1410	/* delete j from hash bucket */
1411	ASSERT (j != Next [j]) ;
1412	j = Next [j] ;
1413	Next [jlast] = j ;
1414
1415	}
1416	else
1417	{
1418	/* j cannot be absorbed into i */
1419	jlast = j ;
1420	ASSERT (j != Next [j]) ;
1421	j = Next [j] ;
1422	}
1423	ASSERT (j >= EMPTY && j < n) ;
1424	}
1425
1426	/* -----------------------------------------------------
1427	* no more variables can be absorbed into i
1428	* go to next i in bucket and clear flag array
1429	* ----------------------------------------------------- */
1430
1431	wflg++ ;
1432	i = Next [i] ;
1433	ASSERT (i >= EMPTY && i < n) ;
1434
1435	}
1436	}
1437	}
1438	AMD_DEBUG2 (("detect done\n")) ;
1439
1440	/* ========================================================================= */
1441	/* RESTORE DEGREE LISTS AND REMOVE NONPRINCIPAL SUPERVARIABLES FROM ELEMENT */
1442	/* ========================================================================= */
1443
1444	p = pme1 ;
1445	nleft = n - nel ;
1446	for (pme = pme1 ; pme <= pme2 ; pme++)
1447	{
1448	i = Iw [pme] ;
1449	ASSERT (i >= 0 && i < n) ;
1450	nvi = -Nv [i] ;
1451	AMD_DEBUG3 (("Restore i "ID" "ID"\n", i, nvi)) ;
1452	if (nvi > 0)
1453	{
1454	/* i is a principal variable in Lme */
1455	/* restore Nv [i] to signify that i is principal */
1456	Nv [i] = nvi ;
1457
1458	/* --------------------------------------------------------- */
1459	/* compute the external degree (add size of current element) */
1460	/* --------------------------------------------------------- */
1461
1462	deg = Degree [i] + degme - nvi ;
1463	deg = MIN (deg, nleft - nvi) ;
1464	ASSERT (IMPLIES (aggressive, deg > 0) && deg >= 0 && deg < n) ;
1465
1466	/* --------------------------------------------------------- */
1467	/* place the supervariable at the head of the degree list */
1468	/* --------------------------------------------------------- */
1469
1470	inext = Head [deg] ;
1471	ASSERT (inext >= EMPTY && inext < n) ;
1472	if (inext != EMPTY) Last [inext] = i ;
1473	Next [i] = inext ;
1474	Last [i] = EMPTY ;
1475	Head [deg] = i ;
1476
1477	/* --------------------------------------------------------- */
1478	/* save the new degree, and find the minimum degree */
1479	/* --------------------------------------------------------- */
1480
1481	mindeg = MIN (mindeg, deg) ;
1482	Degree [i] = deg ;
1483
1484	/* --------------------------------------------------------- */
1485	/* place the supervariable in the element pattern */
1486	/* --------------------------------------------------------- */
1487
1488	Iw [p++] = i ;
1489
1490	}
1491	}
1492	AMD_DEBUG2 (("restore done\n")) ;
1493
1494	/* ========================================================================= */
1495	/* FINALIZE THE NEW ELEMENT */
1496	/* ========================================================================= */
1497
1498	AMD_DEBUG2 (("ME = "ID" DONE\n", me)) ;
1499	Nv [me] = nvpiv ;
1500	/* save the length of the list for the new element me */
1501	Len [me] = p - pme1 ;
1502	if (Len [me] == 0)
1503	{
1504	/* there is nothing left of the current pivot element */
1505	/* it is a root of the assembly tree */
1506	Pe [me] = EMPTY ;
1507	W [me] = 0 ;
1508	}
1509	if (elenme != 0)
1510	{
1511	/* element was not constructed in place: deallocate part of */
1512	/* it since newly nonprincipal variables may have been removed */
1513	pfree = p ;
1514	}
1515
1516	/* The new element has nvpiv pivots and the size of the contribution
1517	* block for a multifrontal method is degme-by-degme, not including
1518	* the "dense" rows/columns. If the "dense" rows/columns are included,
1519	* the frontal matrix is no larger than
1520	* (degme+ndense)-by-(degme+ndense).
1521	*/
1522
1523	if (Info != (double *) NULL)
1524	{
1525	f = nvpiv ;
1526	r = degme + ndense ;
1527	dmax = MAX (dmax, f + r) ;
1528
1529	/* number of nonzeros in L (excluding the diagonal) */
1530	lnzme = fr + (f-1)f/2 ;
1531	lnz += lnzme ;
1532
1533	/* number of divide operations for LDL' and for LU */
1534	ndiv += lnzme ;
1535
1536	/* number of multiply-subtract pairs for LU */
1537	s = frr + r(f-1)f + (f-1)f(2*f-1)/6 ;
1538	nms_lu += s ;
1539
1540	/* number of multiply-subtract pairs for LDL' */
1541	nms_ldl += (s + lnzme)/2 ;
1542	}
1543
1544	#ifndef NDEBUG
1545	AMD_DEBUG2 (("finalize done nel "ID" n "ID"\n ::::\n", nel, n)) ;
1546	for (pme = Pe [me] ; pme <= Pe [me] + Len [me] - 1 ; pme++)
1547	{
1548	AMD_DEBUG3 ((" "ID"", Iw [pme])) ;
1549	}
1550	AMD_DEBUG3 (("\n")) ;
1551	#endif
1552
1553	}
1554
1555	/* ========================================================================= */
1556	/* DONE SELECTING PIVOTS */
1557	/* ========================================================================= */
1558
1559	if (Info != (double *) NULL)
1560	{
1561
1562	/* count the work to factorize the ndense-by-ndense submatrix */
1563	f = ndense ;
1564	dmax = MAX (dmax, (double) ndense) ;
1565
1566	/* number of nonzeros in L (excluding the diagonal) */
1567	lnzme = (f-1)*f/2 ;
1568	lnz += lnzme ;
1569
1570	/* number of divide operations for LDL' and for LU */
1571	ndiv += lnzme ;
1572
1573	/* number of multiply-subtract pairs for LU */
1574	s = (f-1)f(2*f-1)/6 ;
1575	nms_lu += s ;
1576
1577	/* number of multiply-subtract pairs for LDL' */
1578	nms_ldl += (s + lnzme)/2 ;
1579
1580	/* number of nz's in L (excl. diagonal) */
1581	Info [AMD_LNZ] = lnz ;
1582
1583	/* number of divide ops for LU and LDL' */
1584	Info [AMD_NDIV] = ndiv ;
1585
1586	/* number of multiply-subtract pairs for LDL' */
1587	Info [AMD_NMULTSUBS_LDL] = nms_ldl ;
1588
1589	/* number of multiply-subtract pairs for LU */
1590	Info [AMD_NMULTSUBS_LU] = nms_lu ;
1591
1592	/* number of "dense" rows/columns */
1593	Info [AMD_NDENSE] = ndense ;
1594
1595	/* largest front is dmax-by-dmax */
1596	Info [AMD_DMAX] = dmax ;
1597
1598	/* number of garbage collections in AMD */
1599	Info [AMD_NCMPA] = ncmpa ;
1600
1601	/* successful ordering */
1602	Info [AMD_STATUS] = AMD_OK ;
1603	}
1604
1605	/* ========================================================================= */
1606	/* POST-ORDERING */
1607	/* ========================================================================= */
1608
1609	/* -------------------------------------------------------------------------
1610	* Variables at this point:
1611	*
1612	* Pe: holds the elimination tree. The parent of j is FLIP (Pe [j]),
1613	* or EMPTY if j is a root. The tree holds both elements and
1614	* non-principal (unordered) variables absorbed into them.
1615	* Dense variables are non-principal and unordered.
1616	*
1617	* Elen: holds the size of each element, including the diagonal part.
1618	* FLIP (Elen [e]) > 0 if e is an element. For unordered
1619	* variables i, Elen [i] is EMPTY.
1620	*
1621	* Nv: Nv [e] > 0 is the number of pivots represented by the element e.
1622	* For unordered variables i, Nv [i] is zero.
1623	*
1624	* Contents no longer needed:
1625	* W, Iw, Len, Degree, Head, Next, Last.
1626	*
1627	* The matrix itself has been destroyed.
1628	*
1629	* n: the size of the matrix.
1630	* No other scalars needed (pfree, iwlen, etc.)
1631	* ------------------------------------------------------------------------- */
1632
1633	/* restore Pe */
1634	for (i = 0 ; i < n ; i++)
1635	{
1636	Pe [i] = FLIP (Pe [i]) ;
1637	}
1638
1639	/* restore Elen, for output information, and for postordering */
1640	for (i = 0 ; i < n ; i++)
1641	{
1642	Elen [i] = FLIP (Elen [i]) ;
1643	}
1644
1645	/* Now the parent of j is Pe [j], or EMPTY if j is a root. Elen [e] > 0
1646	* is the size of element e. Elen [i] is EMPTY for unordered variable i. */
1647
1648	#ifndef NDEBUG
1649	AMD_DEBUG2 (("\nTree:\n")) ;
1650	for (i = 0 ; i < n ; i++)
1651	{
1652	AMD_DEBUG2 ((" "ID" parent: "ID" ", i, Pe [i])) ;
1653	ASSERT (Pe [i] >= EMPTY && Pe [i] < n) ;
1654	if (Nv [i] > 0)
1655	{
1656	/* this is an element */
1657	e = i ;
1658	AMD_DEBUG2 ((" element, size is "ID"\n", Elen [i])) ;
1659	ASSERT (Elen [e] > 0) ;
1660	}
1661	AMD_DEBUG2 (("\n")) ;
1662	}
1663	AMD_DEBUG2 (("\nelements:\n")) ;
1664	for (e = 0 ; e < n ; e++)
1665	{
1666	if (Nv [e] > 0)
1667	{
1668	AMD_DEBUG3 (("Element e= "ID" size "ID" nv "ID" \n", e,
1669	Elen [e], Nv [e])) ;
1670	}
1671	}
1672	AMD_DEBUG2 (("\nvariables:\n")) ;
1673	for (i = 0 ; i < n ; i++)
1674	{
1675	Int cnt ;
1676	if (Nv [i] == 0)
1677	{
1678	AMD_DEBUG3 (("i unordered: "ID"\n", i)) ;
1679	j = Pe [i] ;
1680	cnt = 0 ;
1681	AMD_DEBUG3 ((" j: "ID"\n", j)) ;
1682	if (j == EMPTY)
1683	{
1684	AMD_DEBUG3 ((" i is a dense variable\n")) ;
1685	}
1686	else
1687	{
1688	ASSERT (j >= 0 && j < n) ;
1689	while (Nv [j] == 0)
1690	{
1691	AMD_DEBUG3 ((" j : "ID"\n", j)) ;
1692	j = Pe [j] ;
1693	AMD_DEBUG3 ((" j:: "ID"\n", j)) ;
1694	cnt++ ;
1695	if (cnt > n) break ;
1696	}
1697	e = j ;
1698	AMD_DEBUG3 ((" got to e: "ID"\n", e)) ;
1699	}
1700	}
1701	}
1702	#endif
1703
1704	/* ========================================================================= */
1705	/* compress the paths of the variables */
1706	/* ========================================================================= */
1707
1708	for (i = 0 ; i < n ; i++)
1709	{
1710	if (Nv [i] == 0)
1711	{
1712
1713	/* -------------------------------------------------------------
1714	* i is an un-ordered row. Traverse the tree from i until
1715	* reaching an element, e. The element, e, was the principal
1716	* supervariable of i and all nodes in the path from i to when e
1717	* was selected as pivot.
1718	* ------------------------------------------------------------- */
1719
1720	AMD_DEBUG1 (("Path compression, i unordered: "ID"\n", i)) ;
1721	j = Pe [i] ;
1722	ASSERT (j >= EMPTY && j < n) ;
1723	AMD_DEBUG3 ((" j: "ID"\n", j)) ;
1724	if (j == EMPTY)
1725	{
1726	/* Skip a dense variable. It has no parent. */
1727	AMD_DEBUG3 ((" i is a dense variable\n")) ;
1728	continue ;
1729	}
1730
1731	/* while (j is a variable) */
1732	while (Nv [j] == 0)
1733	{
1734	AMD_DEBUG3 ((" j : "ID"\n", j)) ;
1735	j = Pe [j] ;
1736	AMD_DEBUG3 ((" j:: "ID"\n", j)) ;
1737	ASSERT (j >= 0 && j < n) ;
1738	}
1739	/* got to an element e */
1740	e = j ;
1741	AMD_DEBUG3 (("got to e: "ID"\n", e)) ;
1742
1743	/* -------------------------------------------------------------
1744	* traverse the path again from i to e, and compress the path
1745	* (all nodes point to e). Path compression allows this code to
1746	* compute in O(n) time.
1747	* ------------------------------------------------------------- */
1748
1749	j = i ;
1750	/* while (j is a variable) */
1751	while (Nv [j] == 0)
1752	{
1753	jnext = Pe [j] ;
1754	AMD_DEBUG3 (("j "ID" jnext "ID"\n", j, jnext)) ;
1755	Pe [j] = e ;
1756	j = jnext ;
1757	ASSERT (j >= 0 && j < n) ;
1758	}
1759	}
1760	}
1761
1762	/* ========================================================================= */
1763	/* postorder the assembly tree */
1764	/* ========================================================================= */
1765
1766	AMD_postorder (n, Pe, Nv, Elen,
1767	W, /* output order */
1768	Head, Next, Last) ; /* workspace */
1769
1770	/* ========================================================================= */
1771	/* compute output permutation and inverse permutation */
1772	/* ========================================================================= */
1773
1774	/* W [e] = k means that element e is the kth element in the new
1775	* order. e is in the range 0 to n-1, and k is in the range 0 to
1776	* the number of elements. Use Head for inverse order. */
1777
1778	for (k = 0 ; k < n ; k++)
1779	{
1780	Head [k] = EMPTY ;
1781	Next [k] = EMPTY ;
1782	}
1783	for (e = 0 ; e < n ; e++)
1784	{
1785	k = W [e] ;
1786	ASSERT ((k == EMPTY) == (Nv [e] == 0)) ;
1787	if (k != EMPTY)
1788	{
1789	ASSERT (k >= 0 && k < n) ;
1790	Head [k] = e ;
1791	}
1792	}
1793
1794	/* construct output inverse permutation in Next,
1795	* and permutation in Last */
1796	nel = 0 ;
1797	for (k = 0 ; k < n ; k++)
1798	{
1799	e = Head [k] ;
1800	if (e == EMPTY) break ;
1801	ASSERT (e >= 0 && e < n && Nv [e] > 0) ;
1802	Next [e] = nel ;
1803	nel += Nv [e] ;
1804	}
1805	ASSERT (nel == n - ndense) ;
1806
1807	/* order non-principal variables (dense, & those merged into supervar's) */
1808	for (i = 0 ; i < n ; i++)
1809	{
1810	if (Nv [i] == 0)
1811	{
1812	e = Pe [i] ;
1813	ASSERT (e >= EMPTY && e < n) ;
1814	if (e != EMPTY)
1815	{
1816	/* This is an unordered variable that was merged
1817	* into element e via supernode detection or mass
1818	* elimination of i when e became the pivot element.
1819	* Place i in order just before e. */
1820	ASSERT (Next [i] == EMPTY && Nv [e] > 0) ;
1821	Next [i] = Next [e] ;
1822	Next [e]++ ;
1823	}
1824	else
1825	{
1826	/* This is a dense unordered variable, with no parent.
1827	* Place it last in the output order. */
1828	Next [i] = nel++ ;
1829	}
1830	}
1831	}
1832	ASSERT (nel == n) ;
1833
1834	AMD_DEBUG2 (("\n\nPerm:\n")) ;
1835	for (i = 0 ; i < n ; i++)
1836	{
1837	k = Next [i] ;
1838	ASSERT (k >= 0 && k < n) ;
1839	Last [k] = i ;
1840	AMD_DEBUG2 ((" perm ["ID"] = "ID"\n", k, i)) ;
1841	}
1842	}

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source: glpk-cmake/src/amd/amd_2.c @ 1:c445c931472f

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