lemon-project-template-glpk
view deps/glpk/src/amd/amd_2.c @ 9:33de93886c88
Import GLPK 4.47
| author | Alpar Juttner <alpar@cs.elte.hu> |
|---|---|
| date | Sun, 06 Nov 2011 20:59:10 +0100 |
| parents | |
| children |
line source
1 /* ========================================================================= */
2 /* === AMD_2 =============================================================== */
3 /* ========================================================================= */
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 /* ------------------------------------------------------------------------- */
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 */
17 #include "amd_internal.h"
19 /* ========================================================================= */
20 /* === clear_flag ========================================================== */
21 /* ========================================================================= */
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 }
39 /* ========================================================================= */
40 /* === AMD_2 =============================================================== */
41 /* ========================================================================= */
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 */
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 [ ],
62 /* control parameters and output statistics */
63 double Control [ ], /* array of size AMD_CONTROL */
64 double Info [ ] /* array of size AMD_INFO */
65 )
66 {
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.
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 **********************************************************************
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.
196 * ----------------------------------------------------------------------------
197 * INPUT ARGUMENTS (unaltered):
198 * ----------------------------------------------------------------------------
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).
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.
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.
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).
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).
456 * ----------------------------------------------------------------------------
457 * LOCAL INTEGERS:
458 * ----------------------------------------------------------------------------
459 */
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 ;
466 unsigned Int hash ; /* unsigned, so that hash % n is well defined.*/
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
511 * ----------------------------------------------------------------------------
512 * LOCAL DOUBLES, used for statistical output only (except for alpha):
513 * ----------------------------------------------------------------------------
514 */
516 double f, r, ndiv, s, nms_lu, nms_ldl, dmax, alpha, lnz, lnzme ;
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
533 * ----------------------------------------------------------------------------
534 * LOCAL "POINTERS" (indices into the Iw array)
535 * ----------------------------------------------------------------------------
536 */
538 Int p, p1, p2, p3, p4, pdst, pend, pj, pme, pme1, pme2, pn, psrc ;
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 */
560 /* ========================================================================= */
561 /* INITIALIZATIONS */
562 /* ========================================================================= */
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) ;
571 /* initialize output statistics */
572 lnz = 0 ;
573 ndiv = 0 ;
574 nms_lu = 0 ;
575 nms_ldl = 0 ;
576 dmax = 1 ;
577 me = EMPTY ;
579 mindeg = 0 ;
580 ncmpa = 0 ;
581 nel = 0 ;
582 lemax = 0 ;
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)) ;
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 }
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
630 /* initialize wflg */
631 wbig = Int_MAX - n ;
632 wflg = clear_flag (0, wbig, W, n) ;
634 /* --------------------------------------------------------------------- */
635 /* initialize degree lists and eliminate dense and empty rows */
636 /* --------------------------------------------------------------------- */
638 ndense = 0 ;
640 for (i = 0 ; i < n ; i++)
641 {
642 deg = Degree [i] ;
643 ASSERT (deg >= 0 && deg < n) ;
644 if (deg == 0)
645 {
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 * ------------------------------------------------------------- */
654 Elen [i] = FLIP (1) ;
655 nel++ ;
656 Pe [i] = EMPTY ;
657 W [i] = 0 ;
659 }
660 else if (deg > dense)
661 {
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 * ------------------------------------------------------------- */
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 ;
677 }
678 else
679 {
681 /* -------------------------------------------------------------
682 * place i in the degree list corresponding to its degree
683 * ------------------------------------------------------------- */
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 ;
691 }
692 }
694 /* ========================================================================= */
695 /* WHILE (selecting pivots) DO */
696 /* ========================================================================= */
698 while (nel < n)
699 {
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
710 /* ========================================================================= */
711 /* GET PIVOT OF MINIMUM DEGREE */
712 /* ========================================================================= */
714 /* ----------------------------------------------------------------- */
715 /* find next supervariable for elimination */
716 /* ----------------------------------------------------------------- */
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)) ;
728 /* ----------------------------------------------------------------- */
729 /* remove chosen variable from link list */
730 /* ----------------------------------------------------------------- */
732 inext = Next [me] ;
733 ASSERT (inext >= EMPTY && inext < n) ;
734 if (inext != EMPTY) Last [inext] = EMPTY ;
735 Head [deg] = inext ;
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 /* ----------------------------------------------------------------- */
742 elenme = Elen [me] ;
743 nvpiv = Nv [me] ;
744 ASSERT (nvpiv > 0) ;
745 nel += nvpiv ;
747 /* ========================================================================= */
748 /* CONSTRUCT NEW ELEMENT */
749 /* ========================================================================= */
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 * ----------------------------------------------------------------- */
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) ;
764 if (elenme == 0)
765 {
767 /* ------------------------------------------------------------- */
768 /* construct the new element in place */
769 /* ------------------------------------------------------------- */
771 pme1 = Pe [me] ;
772 pme2 = pme1 - 1 ;
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 {
782 /* ----------------------------------------------------- */
783 /* i is a principal variable not yet placed in Lme. */
784 /* store i in new list */
785 /* ----------------------------------------------------- */
787 /* flag i as being in Lme by negating Nv [i] */
788 degme += nvi ;
789 Nv [i] = -nvi ;
790 Iw [++pme2] = i ;
792 /* ----------------------------------------------------- */
793 /* remove variable i from degree list. */
794 /* ----------------------------------------------------- */
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 {
817 /* ------------------------------------------------------------- */
818 /* construct the new element in empty space, Iw [pfree ...] */
819 /* ------------------------------------------------------------- */
821 p = Pe [me] ;
822 pme1 = pfree ;
823 slenme = Len [me] - elenme ;
825 for (knt1 = 1 ; knt1 <= elenme + 1 ; knt1++)
826 {
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))) ;
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 * --------------------------------------------------------- */
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)) ;
863 if (nvi > 0)
864 {
866 /* ------------------------------------------------- */
867 /* compress Iw, if necessary */
868 /* ------------------------------------------------- */
870 if (pfree >= iwlen)
871 {
873 AMD_DEBUG1 (("GARBAGE COLLECTION\n")) ;
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. */
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 ;
889 ncmpa++ ; /* one more garbage collection */
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 }
904 /* psrc/pdst point to source/destination */
905 psrc = 0 ;
906 pdst = 0 ;
907 pend = pme1 - 1 ;
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 }
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] ;
938 }
940 /* ------------------------------------------------- */
941 /* i is a principal variable not yet placed in Lme */
942 /* store i in new list */
943 /* ------------------------------------------------- */
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]));
951 /* ------------------------------------------------- */
952 /* remove variable i from degree link list */
953 /* ------------------------------------------------- */
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 }
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 }
983 pme2 = pfree - 1 ;
984 }
986 /* ----------------------------------------------------------------- */
987 /* me has now been converted into an element in Iw [pme1..pme2] */
988 /* ----------------------------------------------------------------- */
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) ;
996 Elen [me] = FLIP (nvpiv + degme) ;
997 /* FLIP (Elen (me)) is now the degree of pivot (including
998 * diagonal part). */
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
1006 /* ----------------------------------------------------------------- */
1007 /* make sure that wflg is not too large. */
1008 /* ----------------------------------------------------------------- */
1010 /* With the current value of wflg, wflg+n must not cause integer
1011 * overflow */
1013 wflg = clear_flag (wflg, wbig, W, n) ;
1015 /* ========================================================================= */
1016 /* COMPUTE (W [e] - wflg) = |Le\Lme| FOR ALL ELEMENTS */
1017 /* ========================================================================= */
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 * ----------------------------------------------------------------- */
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")) ;
1072 /* ========================================================================= */
1073 /* DEGREE UPDATE AND ELEMENT ABSORPTION */
1074 /* ========================================================================= */
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 * ----------------------------------------------------------------- */
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) ;
1095 /* ------------------------------------------------------------- */
1096 /* scan the element list associated with supervariable i */
1097 /* ------------------------------------------------------------- */
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 }
1151 /* count the number of elements in i (including me): */
1152 Elen [i] = pn - p1 + 1 ;
1154 /* ------------------------------------------------------------- */
1155 /* scan the supervariables in the list associated with i */
1156 /* ------------------------------------------------------------- */
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 }
1180 /* ------------------------------------------------------------- */
1181 /* update the degree and check for mass elimination */
1182 /* ------------------------------------------------------------- */
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))) ;
1188 if (Elen [i] == 1 && p3 == pn)
1189 {
1191 /* --------------------------------------------------------- */
1192 /* mass elimination */
1193 /* --------------------------------------------------------- */
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. */
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 ;
1224 }
1225 else
1226 {
1228 /* --------------------------------------------------------- */
1229 /* update the upper-bound degree of i */
1230 /* --------------------------------------------------------- */
1232 /* the following degree does not yet include the size
1233 * of the current element, which is added later: */
1235 Degree [i] = MIN (Degree [i], deg) ;
1237 /* --------------------------------------------------------- */
1238 /* add me to the list for i */
1239 /* --------------------------------------------------------- */
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 ;
1250 /* --------------------------------------------------------- */
1251 /* place in hash bucket. Save hash key of i in Last [i]. */
1252 /* --------------------------------------------------------- */
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) ;
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 }
1277 /* if a separate Hhead array is used: *
1278 Next [i] = Hhead [hash] ;
1279 Hhead [hash] = i ;
1280 */
1282 Last [i] = hash ;
1283 }
1284 }
1286 Degree [me] = degme ;
1288 /* ----------------------------------------------------------------- */
1289 /* Clear the counter array, W [...], by incrementing wflg. */
1290 /* ----------------------------------------------------------------- */
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 */
1298 /* ========================================================================= */
1299 /* SUPERVARIABLE DETECTION */
1300 /* ========================================================================= */
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 */
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 * --------------------------------------------------------- */
1318 /* let i = head of hash bucket, and empty the hash bucket */
1319 ASSERT (Last [i] >= 0 && Last [i] < n) ;
1320 hash = Last [i] ;
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 }
1342 /* if separate Hhead array is used: *
1343 i = Hhead [hash] ;
1344 Hhead [hash] = EMPTY ;
1345 */
1347 ASSERT (i >= EMPTY && i < n) ;
1348 AMD_DEBUG2 (("----i "ID" hash "ID"\n", i, hash)) ;
1350 while (i != EMPTY && Next [i] != EMPTY)
1351 {
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 * ----------------------------------------------------- */
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 }
1370 /* ----------------------------------------------------- */
1371 /* scan every other entry j following i in bucket */
1372 /* ----------------------------------------------------- */
1374 jlast = i ;
1375 j = Next [i] ;
1376 ASSERT (j >= EMPTY && j < n) ;
1378 while (j != EMPTY)
1379 {
1380 /* ------------------------------------------------- */
1381 /* check if j and i have identical nonzero pattern */
1382 /* ------------------------------------------------- */
1384 AMD_DEBUG3 (("compare i "ID" and j "ID"\n", i,j)) ;
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 /* --------------------------------------------- */
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 ;
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 }
1426 /* -----------------------------------------------------
1427 * no more variables can be absorbed into i
1428 * go to next i in bucket and clear flag array
1429 * ----------------------------------------------------- */
1431 wflg++ ;
1432 i = Next [i] ;
1433 ASSERT (i >= EMPTY && i < n) ;
1435 }
1436 }
1437 }
1438 AMD_DEBUG2 (("detect done\n")) ;
1440 /* ========================================================================= */
1441 /* RESTORE DEGREE LISTS AND REMOVE NONPRINCIPAL SUPERVARIABLES FROM ELEMENT */
1442 /* ========================================================================= */
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 ;
1458 /* --------------------------------------------------------- */
1459 /* compute the external degree (add size of current element) */
1460 /* --------------------------------------------------------- */
1462 deg = Degree [i] + degme - nvi ;
1463 deg = MIN (deg, nleft - nvi) ;
1464 ASSERT (IMPLIES (aggressive, deg > 0) && deg >= 0 && deg < n) ;
1466 /* --------------------------------------------------------- */
1467 /* place the supervariable at the head of the degree list */
1468 /* --------------------------------------------------------- */
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 ;
1477 /* --------------------------------------------------------- */
1478 /* save the new degree, and find the minimum degree */
1479 /* --------------------------------------------------------- */
1481 mindeg = MIN (mindeg, deg) ;
1482 Degree [i] = deg ;
1484 /* --------------------------------------------------------- */
1485 /* place the supervariable in the element pattern */
1486 /* --------------------------------------------------------- */
1488 Iw [p++] = i ;
1490 }
1491 }
1492 AMD_DEBUG2 (("restore done\n")) ;
1494 /* ========================================================================= */
1495 /* FINALIZE THE NEW ELEMENT */
1496 /* ========================================================================= */
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 }
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 */
1523 if (Info != (double *) NULL)
1524 {
1525 f = nvpiv ;
1526 r = degme + ndense ;
1527 dmax = MAX (dmax, f + r) ;
1529 /* number of nonzeros in L (excluding the diagonal) */
1530 lnzme = f*r + (f-1)*f/2 ;
1531 lnz += lnzme ;
1533 /* number of divide operations for LDL' and for LU */
1534 ndiv += lnzme ;
1536 /* number of multiply-subtract pairs for LU */
1537 s = f*r*r + r*(f-1)*f + (f-1)*f*(2*f-1)/6 ;
1538 nms_lu += s ;
1540 /* number of multiply-subtract pairs for LDL' */
1541 nms_ldl += (s + lnzme)/2 ;
1542 }
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
1553 }
1555 /* ========================================================================= */
1556 /* DONE SELECTING PIVOTS */
1557 /* ========================================================================= */
1559 if (Info != (double *) NULL)
1560 {
1562 /* count the work to factorize the ndense-by-ndense submatrix */
1563 f = ndense ;
1564 dmax = MAX (dmax, (double) ndense) ;
1566 /* number of nonzeros in L (excluding the diagonal) */
1567 lnzme = (f-1)*f/2 ;
1568 lnz += lnzme ;
1570 /* number of divide operations for LDL' and for LU */
1571 ndiv += lnzme ;
1573 /* number of multiply-subtract pairs for LU */
1574 s = (f-1)*f*(2*f-1)/6 ;
1575 nms_lu += s ;
1577 /* number of multiply-subtract pairs for LDL' */
1578 nms_ldl += (s + lnzme)/2 ;
1580 /* number of nz's in L (excl. diagonal) */
1581 Info [AMD_LNZ] = lnz ;
1583 /* number of divide ops for LU and LDL' */
1584 Info [AMD_NDIV] = ndiv ;
1586 /* number of multiply-subtract pairs for LDL' */
1587 Info [AMD_NMULTSUBS_LDL] = nms_ldl ;
1589 /* number of multiply-subtract pairs for LU */
1590 Info [AMD_NMULTSUBS_LU] = nms_lu ;
1592 /* number of "dense" rows/columns */
1593 Info [AMD_NDENSE] = ndense ;
1595 /* largest front is dmax-by-dmax */
1596 Info [AMD_DMAX] = dmax ;
1598 /* number of garbage collections in AMD */
1599 Info [AMD_NCMPA] = ncmpa ;
1601 /* successful ordering */
1602 Info [AMD_STATUS] = AMD_OK ;
1603 }
1605 /* ========================================================================= */
1606 /* POST-ORDERING */
1607 /* ========================================================================= */
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 * ------------------------------------------------------------------------- */
1633 /* restore Pe */
1634 for (i = 0 ; i < n ; i++)
1635 {
1636 Pe [i] = FLIP (Pe [i]) ;
1637 }
1639 /* restore Elen, for output information, and for postordering */
1640 for (i = 0 ; i < n ; i++)
1641 {
1642 Elen [i] = FLIP (Elen [i]) ;
1643 }
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. */
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
1704 /* ========================================================================= */
1705 /* compress the paths of the variables */
1706 /* ========================================================================= */
1708 for (i = 0 ; i < n ; i++)
1709 {
1710 if (Nv [i] == 0)
1711 {
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 * ------------------------------------------------------------- */
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 }
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)) ;
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 * ------------------------------------------------------------- */
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 }
1762 /* ========================================================================= */
1763 /* postorder the assembly tree */
1764 /* ========================================================================= */
1766 AMD_postorder (n, Pe, Nv, Elen,
1767 W, /* output order */
1768 Head, Next, Last) ; /* workspace */
1770 /* ========================================================================= */
1771 /* compute output permutation and inverse permutation */
1772 /* ========================================================================= */
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. */
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 }
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) ;
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) ;
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 }