ScaLAPACK 2.1  2.1 ScaLAPACK: Scalable Linear Algebra PACKage
pdgebd2.f
Go to the documentation of this file.
1  SUBROUTINE pdgebd2( M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP,
2  \$ WORK, LWORK, INFO )
3 *
4 * -- ScaLAPACK auxiliary routine (version 1.7) --
5 * University of Tennessee, Knoxville, Oak Ridge National Laboratory,
6 * and University of California, Berkeley.
7 * May 1, 1997
8 *
9 * .. Scalar Arguments ..
10  INTEGER IA, INFO, JA, LWORK, M, N
11 * ..
12 * .. Array Arguments ..
13  INTEGER DESCA( * )
14  DOUBLE PRECISION A( * ), D( * ), E( * ), TAUP( * ), TAUQ( * ),
15  \$ work( * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * PDGEBD2 reduces a real general M-by-N distributed matrix
22 * sub( A ) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal
23 * form B by an orthogonal transformation: Q' * sub( A ) * P = B.
24 *
25 * If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal.
26 *
27 * Notes
28 * =====
29 *
30 * Each global data object is described by an associated description
31 * vector. This vector stores the information required to establish
32 * the mapping between an object element and its corresponding process
33 * and memory location.
34 *
35 * Let A be a generic term for any 2D block cyclicly distributed array.
36 * Such a global array has an associated description vector DESCA.
37 * In the following comments, the character _ should be read as
38 * "of the global array".
39 *
40 * NOTATION STORED IN EXPLANATION
41 * --------------- -------------- --------------------------------------
42 * DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
43 * DTYPE_A = 1.
44 * CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
45 * the BLACS process grid A is distribu-
46 * ted over. The context itself is glo-
47 * bal, but the handle (the integer
48 * value) may vary.
49 * M_A (global) DESCA( M_ ) The number of rows in the global
50 * array A.
51 * N_A (global) DESCA( N_ ) The number of columns in the global
52 * array A.
53 * MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
54 * the rows of the array.
55 * NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
56 * the columns of the array.
57 * RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
58 * row of the array A is distributed.
59 * CSRC_A (global) DESCA( CSRC_ ) The process column over which the
60 * first column of the array A is
61 * distributed.
62 * LLD_A (local) DESCA( LLD_ ) The leading dimension of the local
63 * array. LLD_A >= MAX(1,LOCr(M_A)).
64 *
65 * Let K be the number of rows or columns of a distributed matrix,
66 * and assume that its process grid has dimension p x q.
67 * LOCr( K ) denotes the number of elements of K that a process
68 * would receive if K were distributed over the p processes of its
69 * process column.
70 * Similarly, LOCc( K ) denotes the number of elements of K that a
71 * process would receive if K were distributed over the q processes of
72 * its process row.
73 * The values of LOCr() and LOCc() may be determined via a call to the
74 * ScaLAPACK tool function, NUMROC:
75 * LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
76 * LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).
77 * An upper bound for these quantities may be computed by:
78 * LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
79 * LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
80 *
81 * Arguments
82 * =========
83 *
84 * M (global input) INTEGER
85 * The number of rows to be operated on, i.e. the number of rows
86 * of the distributed submatrix sub( A ). M >= 0.
87 *
88 * N (global input) INTEGER
89 * The number of columns to be operated on, i.e. the number of
90 * columns of the distributed submatrix sub( A ). N >= 0.
91 *
92 * A (local input/local output) DOUBLE PRECISION pointer into the
93 * local memory to an array of dimension (LLD_A,LOCc(JA+N-1)).
94 * On entry, this array contains the local pieces of the
95 * general distributed matrix sub( A ). On exit, if M >= N,
96 * the diagonal and the first superdiagonal of sub( A ) are
97 * overwritten with the upper bidiagonal matrix B; the elements
98 * below the diagonal, with the array TAUQ, represent the
99 * orthogonal matrix Q as a product of elementary reflectors,
100 * and the elements above the first superdiagonal, with the
101 * array TAUP, represent the orthogonal matrix P as a product
102 * of elementary reflectors. If M < N, the diagonal and the
103 * first subdiagonal are overwritten with the lower bidiagonal
104 * matrix B; the elements below the first subdiagonal, with the
105 * array TAUQ, represent the orthogonal matrix Q as a product of
106 * elementary reflectors, and the elements above the diagonal,
107 * with the array TAUP, represent the orthogonal matrix P as a
108 * product of elementary reflectors. See Further Details.
109 *
110 * IA (global input) INTEGER
111 * The row index in the global array A indicating the first
112 * row of sub( A ).
113 *
114 * JA (global input) INTEGER
115 * The column index in the global array A indicating the
116 * first column of sub( A ).
117 *
118 * DESCA (global and local input) INTEGER array of dimension DLEN_.
119 * The array descriptor for the distributed matrix A.
120 *
121 * D (local output) DOUBLE PRECISION array, dimension
122 * LOCc(JA+MIN(M,N)-1) if M >= N; LOCr(IA+MIN(M,N)-1) otherwise.
123 * The distributed diagonal elements of the bidiagonal matrix
124 * B: D(i) = A(i,i). D is tied to the distributed matrix A.
125 *
126 * E (local output) DOUBLE PRECISION array, dimension
127 * LOCr(IA+MIN(M,N)-1) if M >= N; LOCc(JA+MIN(M,N)-2) otherwise.
128 * The distributed off-diagonal elements of the bidiagonal
129 * distributed matrix B:
130 * if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
131 * if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
132 * E is tied to the distributed matrix A.
133 *
134 * TAUQ (local output) DOUBLE PRECISION array dimension
135 * LOCc(JA+MIN(M,N)-1). The scalar factors of the elementary
136 * reflectors which represent the orthogonal matrix Q. TAUQ
137 * is tied to the distributed matrix A. See Further Details.
138 *
139 * TAUP (local output) DOUBLE PRECISION array, dimension
140 * LOCr(IA+MIN(M,N)-1). The scalar factors of the elementary
141 * reflectors which represent the orthogonal matrix P. TAUP
142 * is tied to the distributed matrix A. See Further Details.
143 *
144 * WORK (local workspace/local output) DOUBLE PRECISION array,
145 * dimension (LWORK)
146 * On exit, WORK(1) returns the minimal and optimal LWORK.
147 *
148 * LWORK (local or global input) INTEGER
149 * The dimension of the array WORK.
150 * LWORK is local input and must be at least
151 * LWORK >= MAX( MpA0, NqA0 )
152 *
153 * where NB = MB_A = NB_A, IROFFA = MOD( IA-1, NB )
154 * IAROW = INDXG2P( IA, NB, MYROW, RSRC_A, NPROW ),
155 * IACOL = INDXG2P( JA, NB, MYCOL, CSRC_A, NPCOL ),
156 * MpA0 = NUMROC( M+IROFFA, NB, MYROW, IAROW, NPROW ),
157 * NqA0 = NUMROC( N+IROFFA, NB, MYCOL, IACOL, NPCOL ).
158 *
159 * INDXG2P and NUMROC are ScaLAPACK tool functions;
160 * MYROW, MYCOL, NPROW and NPCOL can be determined by calling
161 * the subroutine BLACS_GRIDINFO.
162 *
163 * If LWORK = -1, then LWORK is global input and a workspace
164 * query is assumed; the routine only calculates the minimum
165 * and optimal size for all work arrays. Each of these
166 * values is returned in the first entry of the corresponding
167 * work array, and no error message is issued by PXERBLA.
168 *
169 * INFO (local output) INTEGER
170 * = 0: successful exit
171 * < 0: If the i-th argument is an array and the j-entry had
172 * an illegal value, then INFO = -(i*100+j), if the i-th
173 * argument is a scalar and had an illegal value, then
174 * INFO = -i.
175 *
176 * Further Details
177 * ===============
178 *
179 * The matrices Q and P are represented as products of elementary
180 * reflectors:
181 *
182 * If m >= n,
183 *
184 * Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
185 *
186 * Each H(i) and G(i) has the form:
187 *
188 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
189 *
190 * where tauq and taup are real scalars, and v and u are real vectors;
191 * v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
192 * A(ia+i:ia+m-1,ja+i-1);
193 * u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
194 * A(ia+i-1,ja+i+1:ja+n-1);
195 * tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1).
196 *
197 * If m < n,
198 *
199 * Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
200 *
201 * Each H(i) and G(i) has the form:
202 *
203 * H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
204 *
205 * where tauq and taup are real scalars, and v and u are real vectors;
206 * v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in
207 * A(ia+i+1:ia+m-1,ja+i-1);
208 * u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in
209 * A(ia+i-1,ja+i:ja+n-1);
210 * tauq is stored in TAUQ(ja+i-1) and taup in TAUP(ia+i-1).
211 *
212 * The contents of sub( A ) on exit are illustrated by the following
213 * examples:
214 *
215 * m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
216 *
217 * ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
218 * ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
219 * ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
220 * ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
221 * ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
222 * ( v1 v2 v3 v4 v5 )
223 *
224 * where d and e denote diagonal and off-diagonal elements of B, vi
225 * denotes an element of the vector defining H(i), and ui an element of
226 * the vector defining G(i).
227 *
228 * Alignment requirements
229 * ======================
230 *
231 * The distributed submatrix sub( A ) must verify some alignment proper-
232 * ties, namely the following expressions should be true:
233 * ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA )
234 *
235 * =====================================================================
236 *
237 * .. Parameters ..
238  INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
239  \$ lld_, mb_, m_, nb_, n_, rsrc_
240  parameter( block_cyclic_2d = 1, dlen_ = 9, dtype_ = 1,
241  \$ ctxt_ = 2, m_ = 3, n_ = 4, mb_ = 5, nb_ = 6,
242  \$ rsrc_ = 7, csrc_ = 8, lld_ = 9 )
243  DOUBLE PRECISION ONE, ZERO
244  parameter( one = 1.0d+0, zero = 0.0d+0 )
245 * ..
246 * .. Local Scalars ..
247  LOGICAL LQUERY
248  INTEGER I, IACOL, IAROW, ICOFFA, ICTXT, II, IROFFA, J,
249  \$ jj, k, lwmin, mpa0, mycol, myrow, npcol, nprow,
250  \$ nqa0
251  DOUBLE PRECISION ALPHA
252 * ..
253 * .. Local Arrays ..
254  INTEGER DESCD( DLEN_ ), DESCE( DLEN_ )
255 * ..
256 * .. External Subroutines ..
257  EXTERNAL blacs_abort, blacs_gridinfo, chk1mat, descset,
258  \$ dgebr2d, dgebs2d, dlarfg, infog2l,
260 * ..
261 * .. External Functions ..
262  INTEGER INDXG2P, NUMROC
263  EXTERNAL indxg2p, numroc
264 * ..
265 * .. Intrinsic Functions ..
266  INTRINSIC dble, max, min, mod
267 * ..
268 * .. Executable Statements ..
269 *
270 * Test the input parameters
271 *
272  ictxt = desca( ctxt_ )
273  CALL blacs_gridinfo( ictxt, nprow, npcol, myrow, mycol )
274 *
275 * Test the input parameters
276 *
277  info = 0
278  IF( nprow.EQ.-1 ) THEN
279  info = -(600+ctxt_)
280  ELSE
281  CALL chk1mat( m, 1, n, 2, ia, ja, desca, 6, info )
282  IF( info.EQ.0 ) THEN
283  iroffa = mod( ia-1, desca( mb_ ) )
284  icoffa = mod( ja-1, desca( nb_ ) )
285  iarow = indxg2p( ia, desca( mb_ ), myrow, desca( rsrc_ ),
286  \$ nprow )
287  iacol = indxg2p( ja, desca( nb_ ), mycol, desca( csrc_ ),
288  \$ npcol )
289  mpa0 = numroc( m+iroffa, desca( mb_ ), myrow, iarow, nprow )
290  nqa0 = numroc( n+icoffa, desca( nb_ ), mycol, iacol, npcol )
291  lwmin = max( mpa0, nqa0 )
292 *
293  work( 1 ) = dble( lwmin )
294  lquery = ( lwork.EQ.-1 )
295  IF( iroffa.NE.icoffa ) THEN
296  info = -5
297  ELSE IF( desca( mb_ ).NE.desca( nb_ ) ) THEN
298  info = -(600+nb_)
299  ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
300  info = -12
301  END IF
302  END IF
303  END IF
304 *
305  IF( info.LT.0 ) THEN
306  CALL pxerbla( ictxt, 'PDGEBD2', -info )
307  CALL blacs_abort( ictxt, 1 )
308  RETURN
309  ELSE IF( lquery ) THEN
310  RETURN
311  END IF
312 *
313  CALL infog2l( ia, ja, desca, nprow, npcol, myrow, mycol, ii, jj,
314  \$ iarow, iacol )
315 *
316  IF( m.EQ.1 .AND. n.EQ.1 ) THEN
317  IF( mycol.EQ.iacol ) THEN
318  IF( myrow.EQ.iarow ) THEN
319  i = ii+(jj-1)*desca( lld_ )
320  CALL dlarfg( 1, a( i ), a( i ), 1, tauq( jj ) )
321  d( jj ) = a( i )
322  CALL dgebs2d( ictxt, 'Columnwise', ' ', 1, 1, d( jj ),
323  \$ 1 )
324  CALL dgebs2d( ictxt, 'Columnwise', ' ', 1, 1, tauq( jj ),
325  \$ 1 )
326  ELSE
327  CALL dgebr2d( ictxt, 'Columnwise', ' ', 1, 1, d( jj ),
328  \$ 1, iarow, iacol )
329  CALL dgebr2d( ictxt, 'Columnwise', ' ', 1, 1, tauq( jj ),
330  \$ 1, iarow, iacol )
331  END IF
332  END IF
333  IF( myrow.EQ.iarow )
334  \$ taup( ii ) = zero
335  RETURN
336  END IF
337 *
338  alpha = zero
339 *
340  IF( m.GE.n ) THEN
341 *
342 * Reduce to upper bidiagonal form
343 *
344  CALL descset( descd, 1, ja+min(m,n)-1, 1, desca( nb_ ), myrow,
345  \$ desca( csrc_ ), desca( ctxt_ ), 1 )
346  CALL descset( desce, ia+min(m,n)-1, 1, desca( mb_ ), 1,
347  \$ desca( rsrc_ ), mycol, desca( ctxt_ ),
348  \$ desca( lld_ ) )
349  DO 10 k = 1, n
350  i = ia + k - 1
351  j = ja + k - 1
352 *
353 * Generate elementary reflector H(j) to annihilate
354 * A(ia+i:ia+m-1,j)
355 *
356  CALL pdlarfg( m-k+1, alpha, i, j, a, min( i+1, m+ia-1 ),
357  \$ j, desca, 1, tauq )
358  CALL pdelset( d, 1, j, descd, alpha )
359  CALL pdelset( a, i, j, desca, one )
360 *
361 * Apply H(i) to A(i:ia+m-1,i+1:ja+n-1) from the left
362 *
363  CALL pdlarf( 'Left', m-k+1, n-k, a, i, j, desca, 1, tauq, a,
364  \$ i, j+1, desca, work )
365  CALL pdelset( a, i, j, desca, alpha )
366 *
367  IF( k.LT.n ) THEN
368 *
369 * Generate elementary reflector G(i) to annihilate
370 * A(i,ja+j+1:ja+n-1)
371 *
372  CALL pdlarfg( n-k, alpha, i, j+1, a, i,
373  \$ min( j+2, ja+n-1 ), desca, desca( m_ ),
374  \$ taup )
375  CALL pdelset( e, i, 1, desce, alpha )
376  CALL pdelset( a, i, j+1, desca, one )
377 *
378 * Apply G(i) to A(i+1:ia+m-1,i+1:ja+n-1) from the right
379 *
380  CALL pdlarf( 'Right', m-k, n-k, a, i, j+1, desca,
381  \$ desca( m_ ), taup, a, i+1, j+1, desca,
382  \$ work )
383  CALL pdelset( a, i, j+1, desca, alpha )
384  ELSE
385  CALL pdelset( taup, i, 1, desce, zero )
386  END IF
387  10 CONTINUE
388 *
389  ELSE
390 *
391 * Reduce to lower bidiagonal form
392 *
393  CALL descset( descd, ia+min(m,n)-1, 1, desca( mb_ ), 1,
394  \$ desca( rsrc_ ), mycol, desca( ctxt_ ),
395  \$ desca( lld_ ) )
396  CALL descset( desce, 1, ja+min(m,n)-1, 1, desca( nb_ ), myrow,
397  \$ desca( csrc_ ), desca( ctxt_ ), 1 )
398  DO 20 k = 1, m
399  i = ia + k - 1
400  j = ja + k - 1
401 *
402 * Generate elementary reflector G(i) to annihilate
403 * A(i,ja+j:ja+n-1)
404 *
405  CALL pdlarfg( n-k+1, alpha, i, j, a, i,
406  \$ min( j+1, ja+n-1 ), desca, desca( m_ ), taup )
407  CALL pdelset( d, i, 1, descd, alpha )
408  CALL pdelset( a, i, j, desca, one )
409 *
410 * Apply G(i) to A(i:ia+m-1,j:ja+n-1) from the right
411 *
412  CALL pdlarf( 'Right', m-k, n-k+1, a, i, j, desca,
413  \$ desca( m_ ), taup, a, min( i+1, ia+m-1 ), j,
414  \$ desca, work )
415  CALL pdelset( a, i, j, desca, alpha )
416 *
417  IF( k.LT.m ) THEN
418 *
419 * Generate elementary reflector H(i) to annihilate
420 * A(i+2:ia+m-1,j)
421 *
422  CALL pdlarfg( m-k, alpha, i+1, j, a,
423  \$ min( i+2, ia+m-1 ), j, desca, 1, tauq )
424  CALL pdelset( e, 1, j, desce, alpha )
425  CALL pdelset( a, i+1, j, desca, one )
426 *
427 * Apply H(i) to A(i+1:ia+m-1,j+1:ja+n-1) from the left
428 *
429  CALL pdlarf( 'Left', m-k, n-k, a, i+1, j, desca, 1, tauq,
430  \$ a, i+1, j+1, desca, work )
431  CALL pdelset( a, i+1, j, desca, alpha )
432  ELSE
433  CALL pdelset( tauq, 1, j, desce, zero )
434  END IF
435  20 CONTINUE
436  END IF
437 *
438  work( 1 ) = dble( lwmin )
439 *
440  RETURN
441 *
442 * End of PDGEBD2
443 *
444  END
pdgebd2
subroutine pdgebd2(M, N, A, IA, JA, DESCA, D, E, TAUQ, TAUP, WORK, LWORK, INFO)
Definition: pdgebd2.f:3
pdlarf
subroutine pdlarf(SIDE, M, N, V, IV, JV, DESCV, INCV, TAU, C, IC, JC, DESCC, WORK)
Definition: pdlarf.f:3
max
#define max(A, B)
Definition: pcgemr.c:180
infog2l
subroutine infog2l(GRINDX, GCINDX, DESC, NPROW, NPCOL, MYROW, MYCOL, LRINDX, LCINDX, RSRC, CSRC)
Definition: infog2l.f:3
descset
subroutine descset(DESC, M, N, MB, NB, IRSRC, ICSRC, ICTXT, LLD)
Definition: descset.f:3
chk1mat
subroutine chk1mat(MA, MAPOS0, NA, NAPOS0, IA, JA, DESCA, DESCAPOS0, INFO)
Definition: chk1mat.f:3
pxerbla
subroutine pxerbla(ICTXT, SRNAME, INFO)
Definition: pxerbla.f:2
pdlarfg
subroutine pdlarfg(N, ALPHA, IAX, JAX, X, IX, JX, DESCX, INCX, TAU)
Definition: pdlarfg.f:3
pdelset
subroutine pdelset(A, IA, JA, DESCA, ALPHA)
Definition: pdelset.f:2
min
#define min(A, B)
Definition: pcgemr.c:181