1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
|
SUBROUTINE PZHETD2( UPLO, N, A, IA, JA, DESCA, D, E, TAU, WORK,
$ LWORK, INFO )
*
* -- ScaLAPACK auxiliary routine (version 1.7) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* May 1, 1997
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER IA, INFO, JA, LWORK, N
* ..
* .. Array Arguments ..
INTEGER DESCA( * )
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 A( * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* PZHETD2 reduces a complex Hermitian matrix sub( A ) to Hermitian
* tridiagonal form T by an unitary similarity transformation:
* Q' * sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1).
*
* Notes
* =====
*
* Each global data object is described by an associated description
* vector. This vector stores the information required to establish
* the mapping between an object element and its corresponding process
* and memory location.
*
* Let A be a generic term for any 2D block cyclicly distributed array.
* Such a global array has an associated description vector DESCA.
* In the following comments, the character _ should be read as
* "of the global array".
*
* NOTATION STORED IN EXPLANATION
* --------------- -------------- --------------------------------------
* DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
* DTYPE_A = 1.
* CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
* the BLACS process grid A is distribu-
* ted over. The context itself is glo-
* bal, but the handle (the integer
* value) may vary.
* M_A (global) DESCA( M_ ) The number of rows in the global
* array A.
* N_A (global) DESCA( N_ ) The number of columns in the global
* array A.
* MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
* the rows of the array.
* NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
* the columns of the array.
* RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
* row of the array A is distributed.
* CSRC_A (global) DESCA( CSRC_ ) The process column over which the
* first column of the array A is
* distributed.
* LLD_A (local) DESCA( LLD_ ) The leading dimension of the local
* array. LLD_A >= MAX(1,LOCr(M_A)).
*
* Let K be the number of rows or columns of a distributed matrix,
* and assume that its process grid has dimension p x q.
* LOCr( K ) denotes the number of elements of K that a process
* would receive if K were distributed over the p processes of its
* process column.
* Similarly, LOCc( K ) denotes the number of elements of K that a
* process would receive if K were distributed over the q processes of
* its process row.
* The values of LOCr() and LOCc() may be determined via a call to the
* ScaLAPACK tool function, NUMROC:
* LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
* LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).
* An upper bound for these quantities may be computed by:
* LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
* LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
*
* Arguments
* =========
*
* UPLO (global input) CHARACTER
* Specifies whether the upper or lower triangular part of the
* Hermitian matrix sub( A ) is stored:
* = 'U': Upper triangular
* = 'L': Lower triangular
*
* N (global input) INTEGER
* The number of rows and columns to be operated on, i.e. the
* order of the distributed submatrix sub( A ). N >= 0.
*
* A (local input/local output) COMPLEX*16 pointer into the
* local memory to an array of dimension (LLD_A,LOCc(JA+N-1)).
* On entry, this array contains the local pieces of the
* Hermitian distributed matrix sub( A ). If UPLO = 'U', the
* leading N-by-N upper triangular part of sub( A ) contains
* the upper triangular part of the matrix, and its strictly
* lower triangular part is not referenced. If UPLO = 'L', the
* leading N-by-N lower triangular part of sub( A ) contains the
* lower triangular part of the matrix, and its strictly upper
* triangular part is not referenced. On exit, if UPLO = 'U',
* the diagonal and first superdiagonal of sub( A ) are over-
* written by the corresponding elements of the tridiagonal
* matrix T, and the elements above the first superdiagonal,
* with the array TAU, represent the unitary matrix Q as a
* product of elementary reflectors; if UPLO = 'L', the diagonal
* and first subdiagonal of sub( A ) are overwritten by the
* corresponding elements of the tridiagonal matrix T, and the
* elements below the first subdiagonal, with the array TAU,
* represent the unitary matrix Q as a product of elementary
* reflectors. See Further Details.
*
* IA (global input) INTEGER
* The row index in the global array A indicating the first
* row of sub( A ).
*
* JA (global input) INTEGER
* The column index in the global array A indicating the
* first column of sub( A ).
*
* DESCA (global and local input) INTEGER array of dimension DLEN_.
* The array descriptor for the distributed matrix A.
*
* D (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1)
* The diagonal elements of the tridiagonal matrix T:
* D(i) = A(i,i). D is tied to the distributed matrix A.
*
* E (local output) DOUBLE PRECISION array, dimension LOCc(JA+N-1)
* if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal
* elements of the tridiagonal matrix T: E(i) = A(i,i+1) if
* UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the
* distributed matrix A.
*
* TAU (local output) COMPLEX*16, array, dimension
* LOCc(JA+N-1). This array contains the scalar factors TAU of
* the elementary reflectors. TAU is tied to the distributed
* matrix A.
*
* WORK (local workspace/local output) COMPLEX*16 array,
* dimension (LWORK)
* On exit, WORK( 1 ) returns the minimal and optimal LWORK.
*
* LWORK (local or global input) INTEGER
* The dimension of the array WORK.
* LWORK is local input and must be at least
* LWORK >= 3*N.
*
* If LWORK = -1, then LWORK is global input and a workspace
* query is assumed; the routine only calculates the minimum
* and optimal size for all work arrays. Each of these
* values is returned in the first entry of the corresponding
* work array, and no error message is issued by PXERBLA.
*
* INFO (local output) INTEGER
* = 0: successful exit
* < 0: If the i-th argument is an array and the j-entry had
* an illegal value, then INFO = -(i*100+j), if the i-th
* argument is a scalar and had an illegal value, then
* INFO = -i.
*
* Further Details
* ===============
*
* If UPLO = 'U', the matrix Q is represented as a product of elementary
* reflectors
*
* Q = H(n-1) . . . H(2) H(1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
* A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1).
*
* If UPLO = 'L', the matrix Q is represented as a product of elementary
* reflectors
*
* Q = H(1) H(2) . . . H(n-1).
*
* Each H(i) has the form
*
* H(i) = I - tau * v * v'
*
* where tau is a complex scalar, and v is a complex vector with
* v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in
* A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1).
*
* The contents of sub( A ) on exit are illustrated by the following
* examples with n = 5:
*
* if UPLO = 'U': if UPLO = 'L':
*
* ( d e v2 v3 v4 ) ( d )
* ( d e v3 v4 ) ( e d )
* ( d e v4 ) ( v1 e d )
* ( d e ) ( v1 v2 e d )
* ( d ) ( v1 v2 v3 e d )
*
* where d and e denote diagonal and off-diagonal elements of T, and vi
* denotes an element of the vector defining H(i).
*
* Alignment requirements
* ======================
*
* The distributed submatrix sub( A ) must verify some alignment proper-
* ties, namely the following expression should be true:
* ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) with
* IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ).
*
* =====================================================================
*
* .. Parameters ..
INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
$ LLD_, MB_, M_, NB_, N_, RSRC_
PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1,
$ CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6,
$ RSRC_ = 7, CSRC_ = 8, LLD_ = 9 )
COMPLEX*16 HALF, ONE, ZERO
PARAMETER ( HALF = ( 0.5D+0, 0.0D+0 ),
$ ONE = ( 1.0D+0, 0.0D+0 ),
$ ZERO = ( 0.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, UPPER
INTEGER IACOL, IAROW, ICOFFA, ICTXT, II, IK, IROFFA, J,
$ JJ, JK, JN, LDA, LWMIN, MYCOL, MYROW, NPCOL,
$ NPROW
COMPLEX*16 ALPHA, TAUI, DOTC
* ..
* .. External Subroutines ..
EXTERNAL BLACS_ABORT, BLACS_GRIDINFO, CHK1MAT, INFOG2L,
$ PXERBLA, ZAXPY, ZGEBR2D, ZGEBS2D,
$ ZHEMV, ZHER2, ZLARFG, ZZDOTC
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, DCMPLX
* ..
* .. Executable Statements ..
*
* Get grid parameters
*
ICTXT = DESCA( CTXT_ )
CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
*
* Test the input parameters
*
INFO = 0
IF( NPROW.EQ.-1 ) THEN
INFO = -(600+CTXT_)
ELSE
UPPER = LSAME( UPLO, 'U' )
CALL CHK1MAT( N, 2, N, 2, IA, JA, DESCA, 6, INFO )
LWMIN = 3 * N
*
WORK( 1 ) = DCMPLX( DBLE( LWMIN ) )
LQUERY = ( LWORK.EQ.-1 )
IF( INFO.EQ.0 ) THEN
IROFFA = MOD( IA-1, DESCA( MB_ ) )
ICOFFA = MOD( JA-1, DESCA( NB_ ) )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( IROFFA.NE.ICOFFA ) THEN
INFO = -5
ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN
INFO = -(600+NB_)
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -11
END IF
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL PXERBLA( ICTXT, 'PZHETD2', -INFO )
CALL BLACS_ABORT( ICTXT, 1 )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
* Compute local information
*
LDA = DESCA( LLD_ )
CALL INFOG2L( IA, JA, DESCA, NPROW, NPCOL, MYROW, MYCOL, II, JJ,
$ IAROW, IACOL )
*
IF( UPPER ) THEN
*
* Process(IAROW, IACOL) owns block to be reduced
*
IF( MYCOL.EQ.IACOL ) THEN
IF( MYROW.EQ.IAROW ) THEN
*
* Reduce the upper triangle of sub( A )
*
IK = II+N-1+(JJ+N-2)*LDA
A( IK ) = DBLE( A( IK ) )
DO 10 J = N-1, 1, -1
IK = II + J - 1
JK = JJ + J - 1
*
* Generate elementary reflector H(i) = I - tau * v * v'
* to annihilate A(IA:IA+J-1,JA:JA+J-1)
*
ALPHA = A( IK+JK*LDA )
CALL ZLARFG( J, ALPHA, A( II+JK*LDA ), 1, TAUI )
E( JK+1 ) = DBLE( ALPHA )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to
* A(IA:IA+J-1,JA:JA+J-1)
*
A( IK+JK*LDA ) = ONE
*
* Compute x := tau * A * v storing x in TAU(1:i)
*
CALL ZHEMV( UPLO, J, TAUI, A( II+(JJ-1)*LDA ),
$ LDA, A( II+JK*LDA ), 1, ZERO,
$ TAU( JJ ), 1 )
*
* Compute w := x - 1/2 * tau * (x'*v) * v
*
CALL ZZDOTC( J, DOTC, TAU( JJ ), 1, A( II+JK*LDA ),
$ 1 )
ALPHA = -HALF*TAUI*DOTC
CALL ZAXPY( J, ALPHA, A( II+JK*LDA ), 1,
$ TAU( JJ ), 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w' - w * v'
*
CALL ZHER2( UPLO, J, -ONE, A( II+JK*LDA ), 1,
$ TAU( JJ ), 1, A( II+(JJ-1)*LDA ),
$ LDA )
END IF
*
* Copy D, E, TAU to broadcast them columnwise.
*
A( IK+JK*LDA ) = DCMPLX( E( JK+1 ) )
D( JK+1 ) = DBLE( A( IK+1+JK*LDA ) )
WORK( J+1 ) = DCMPLX( D( JK+1 ) )
WORK( N+J+1 ) = DCMPLX( E( JK+1 ) )
TAU( JK+1 ) = TAUI
WORK( 2*N+J+1 ) = TAU( JK+1 )
*
10 CONTINUE
D( JJ ) = DBLE( A( II+(JJ-1)*LDA ) )
WORK( 1 ) = DCMPLX( D( JJ ) )
WORK( N+1 ) = ZERO
WORK( 2*N+1 ) = ZERO
*
CALL ZGEBS2D( ICTXT, 'Columnwise', ' ', 1, 3*N, WORK, 1 )
*
ELSE
CALL ZGEBR2D( ICTXT, 'Columnwise', ' ', 1, 3*N, WORK, 1,
$ IAROW, IACOL )
DO 20 J = 2, N
JN = JJ + J - 1
D( JN ) = DBLE( WORK( J ) )
E( JN ) = DBLE( WORK( N+J ) )
TAU( JN ) = WORK( 2*N+J )
20 CONTINUE
D( JJ ) = DBLE( WORK( 1 ) )
END IF
END IF
*
ELSE
*
* Process (IAROW, IACOL) owns block to be factorized
*
IF( MYCOL.EQ.IACOL ) THEN
IF( MYROW.EQ.IAROW ) THEN
*
* Reduce the lower triangle of sub( A )
*
A( II+(JJ-1)*LDA ) = DBLE( A( II+(JJ-1)*LDA ) )
DO 30 J = 1, N - 1
IK = II + J - 1
JK = JJ + J - 1
*
* Generate elementary reflector H(i) = I - tau * v * v'
* to annihilate A(IA+J-JA+2:IA+N-1,JA+J-1)
*
ALPHA = A( IK+1+(JK-1)*LDA )
CALL ZLARFG( N-J, ALPHA, A( IK+2+(JK-1)*LDA ), 1,
$ TAUI )
E( JK ) = DBLE( ALPHA )
*
IF( TAUI.NE.ZERO ) THEN
*
* Apply H(i) from both sides to
* A(IA+J-JA+1:IA+N-1,JA+J+1:JA+N-1)
*
A( IK+1+(JK-1)*LDA ) = ONE
*
* Compute x := tau * A * v storing y in TAU(i:n-1)
*
CALL ZHEMV( UPLO, N-J, TAUI, A( IK+1+JK*LDA ),
$ LDA, A( IK+1+(JK-1)*LDA ), 1,
$ ZERO, TAU( JK ), 1 )
*
* Compute w := x - 1/2 * tau * (x'*v) * v
*
CALL ZZDOTC( N-J, DOTC, TAU( JK ), 1,
$ A( IK+1+(JK-1)*LDA ), 1 )
ALPHA = -HALF*TAUI*DOTC
CALL ZAXPY( N-J, ALPHA, A( IK+1+(JK-1)*LDA ),
$ 1, TAU( JK ), 1 )
*
* Apply the transformation as a rank-2 update:
* A := A - v * w' - w * v'
*
CALL ZHER2( UPLO, N-J, -ONE,
$ A( IK+1+(JK-1)*LDA ), 1,
$ TAU( JK ), 1, A( IK+1+JK*LDA ),
$ LDA )
END IF
*
* Copy D(JK), E(JK), TAU(JK) to broadcast them
* columnwise.
*
A( IK+1+(JK-1)*LDA ) = DCMPLX( E( JK ) )
D( JK ) = DBLE( A( IK+(JK-1)*LDA ) )
WORK( J ) = DCMPLX( D( JK ) )
WORK( N+J ) = DCMPLX( E( JK ) )
TAU( JK ) = TAUI
WORK( 2*N+J ) = TAU( JK )
30 CONTINUE
JN = JJ + N - 1
D( JN ) = DBLE( A( II+N-1+(JN-1)*LDA ) )
WORK( N ) = DCMPLX( D( JN ) )
TAU( JN ) = ZERO
WORK( 2*N ) = ZERO
*
CALL ZGEBS2D( ICTXT, 'Columnwise', ' ', 1, 3*N-1, WORK,
$ 1 )
*
ELSE
CALL ZGEBR2D( ICTXT, 'Columnwise', ' ', 1, 3*N-1, WORK,
$ 1, IAROW, IACOL )
DO 40 J = 1, N - 1
JN = JJ + J - 1
D( JN ) = DBLE( WORK( J ) )
E( JN ) = DBLE( WORK( N+J ) )
TAU( JN ) = WORK( 2*N+J )
40 CONTINUE
JN = JJ + N - 1
D( JN ) = DBLE( WORK( N ) )
TAU( JN ) = ZERO
END IF
END IF
END IF
*
WORK( 1 ) = DCMPLX( DBLE( LWMIN ) )
*
RETURN
*
* End of PZHETD2
*
END
|
