SCALAPACK 2.2.2
LAPACK: Linear Algebra PACKage
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◆ pssytd2()

subroutine pssytd2 ( character  uplo,
integer  n,
real, dimension( * )  a,
integer  ia,
integer  ja,
integer, dimension( * )  desca,
real, dimension( * )  d,
real, dimension( * )  e,
real, dimension( * )  tau,
real, dimension( * )  work,
integer  lwork,
integer  info 
)

Definition at line 1 of file pssytd2.f.

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 CHARACTER UPLO
11 INTEGER IA, INFO, JA, LWORK, N
12* ..
13* .. Array Arguments ..
14 INTEGER DESCA( * )
15 REAL A( * ), D( * ), E( * ), TAU( * ), WORK( * )
16* ..
17*
18* Purpose
19* =======
20*
21* PSSYTD2 reduces a real symmetric matrix sub( A ) to symmetric
22* tridiagonal form T by an orthogonal similarity transformation:
23* Q' * sub( A ) * Q = T, where sub( A ) = A(IA:IA+N-1,JA:JA+N-1).
24*
25* Notes
26* =====
27*
28* Each global data object is described by an associated description
29* vector. This vector stores the information required to establish
30* the mapping between an object element and its corresponding process
31* and memory location.
32*
33* Let A be a generic term for any 2D block cyclicly distributed array.
34* Such a global array has an associated description vector DESCA.
35* In the following comments, the character _ should be read as
36* "of the global array".
37*
38* NOTATION STORED IN EXPLANATION
39* --------------- -------------- --------------------------------------
40* DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
41* DTYPE_A = 1.
42* CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
43* the BLACS process grid A is distribu-
44* ted over. The context itself is glo-
45* bal, but the handle (the integer
46* value) may vary.
47* M_A (global) DESCA( M_ ) The number of rows in the global
48* array A.
49* N_A (global) DESCA( N_ ) The number of columns in the global
50* array A.
51* MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
52* the rows of the array.
53* NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
54* the columns of the array.
55* RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
56* row of the array A is distributed.
57* CSRC_A (global) DESCA( CSRC_ ) The process column over which the
58* first column of the array A is
59* distributed.
60* LLD_A (local) DESCA( LLD_ ) The leading dimension of the local
61* array. LLD_A >= MAX(1,LOCr(M_A)).
62*
63* Let K be the number of rows or columns of a distributed matrix,
64* and assume that its process grid has dimension p x q.
65* LOCr( K ) denotes the number of elements of K that a process
66* would receive if K were distributed over the p processes of its
67* process column.
68* Similarly, LOCc( K ) denotes the number of elements of K that a
69* process would receive if K were distributed over the q processes of
70* its process row.
71* The values of LOCr() and LOCc() may be determined via a call to the
72* ScaLAPACK tool function, NUMROC:
73* LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
74* LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).
75* An upper bound for these quantities may be computed by:
76* LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
77* LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
78*
79* Arguments
80* =========
81*
82* UPLO (global input) CHARACTER
83* Specifies whether the upper or lower triangular part of the
84* symmetric matrix sub( A ) is stored:
85* = 'U': Upper triangular
86* = 'L': Lower triangular
87*
88* N (global input) INTEGER
89* The number of rows and columns to be operated on, i.e. the
90* order of the distributed submatrix sub( A ). N >= 0.
91*
92* A (local input/local output) REAL 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* symmetric distributed matrix sub( A ). If UPLO = 'U', the
96* leading N-by-N upper triangular part of sub( A ) contains
97* the upper triangular part of the matrix, and its strictly
98* lower triangular part is not referenced. If UPLO = 'L', the
99* leading N-by-N lower triangular part of sub( A ) contains the
100* lower triangular part of the matrix, and its strictly upper
101* triangular part is not referenced. On exit, if UPLO = 'U',
102* the diagonal and first superdiagonal of sub( A ) are over-
103* written by the corresponding elements of the tridiagonal
104* matrix T, and the elements above the first superdiagonal,
105* with the array TAU, represent the orthogonal matrix Q as a
106* product of elementary reflectors; if UPLO = 'L', the diagonal
107* and first subdiagonal of sub( A ) are overwritten by the
108* corresponding elements of the tridiagonal matrix T, and the
109* elements below the first subdiagonal, with the array TAU,
110* represent the orthogonal matrix Q as a product of elementary
111* reflectors. See Further Details.
112*
113* IA (global input) INTEGER
114* The row index in the global array A indicating the first
115* row of sub( A ).
116*
117* JA (global input) INTEGER
118* The column index in the global array A indicating the
119* first column of sub( A ).
120*
121* DESCA (global and local input) INTEGER array of dimension DLEN_.
122* The array descriptor for the distributed matrix A.
123*
124* D (local output) REAL array, dimension LOCc(JA+N-1)
125* The diagonal elements of the tridiagonal matrix T:
126* D(i) = A(i,i). D is tied to the distributed matrix A.
127*
128* E (local output) REAL array, dimension LOCc(JA+N-1)
129* if UPLO = 'U', LOCc(JA+N-2) otherwise. The off-diagonal
130* elements of the tridiagonal matrix T: E(i) = A(i,i+1) if
131* UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. E is tied to the
132* distributed matrix A.
133*
134* TAU (local output) REAL, array, dimension
135* LOCc(JA+N-1). This array contains the scalar factors TAU of
136* the elementary reflectors. TAU is tied to the distributed
137* matrix A.
138*
139* WORK (local workspace/local output) REAL array,
140* dimension (LWORK)
141* On exit, WORK( 1 ) returns the minimal and optimal LWORK.
142*
143* LWORK (local or global input) INTEGER
144* The dimension of the array WORK.
145* LWORK is local input and must be at least
146* LWORK >= 3*N.
147*
148* If LWORK = -1, then LWORK is global input and a workspace
149* query is assumed; the routine only calculates the minimum
150* and optimal size for all work arrays. Each of these
151* values is returned in the first entry of the corresponding
152* work array, and no error message is issued by PXERBLA.
153*
154* INFO (local output) INTEGER
155* = 0: successful exit
156* < 0: If the i-th argument is an array and the j-entry had
157* an illegal value, then INFO = -(i*100+j), if the i-th
158* argument is a scalar and had an illegal value, then
159* INFO = -i.
160*
161* Further Details
162* ===============
163*
164* If UPLO = 'U', the matrix Q is represented as a product of elementary
165* reflectors
166*
167* Q = H(n-1) . . . H(2) H(1).
168*
169* Each H(i) has the form
170*
171* H(i) = I - tau * v * v'
172*
173* where tau is a real scalar, and v is a real vector with
174* v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
175* A(ia:ia+i-2,ja+i), and tau in TAU(ja+i-1).
176*
177* If UPLO = 'L', the matrix Q is represented as a product of elementary
178* reflectors
179*
180* Q = H(1) H(2) . . . H(n-1).
181*
182* Each H(i) has the form
183*
184* H(i) = I - tau * v * v'
185*
186* where tau is a real scalar, and v is a real vector with
187* v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in
188* A(ia+i+1:ia+n-1,ja+i-1), and tau in TAU(ja+i-1).
189*
190* The contents of sub( A ) on exit are illustrated by the following
191* examples with n = 5:
192*
193* if UPLO = 'U': if UPLO = 'L':
194*
195* ( d e v2 v3 v4 ) ( d )
196* ( d e v3 v4 ) ( e d )
197* ( d e v4 ) ( v1 e d )
198* ( d e ) ( v1 v2 e d )
199* ( d ) ( v1 v2 v3 e d )
200*
201* where d and e denote diagonal and off-diagonal elements of T, and vi
202* denotes an element of the vector defining H(i).
203*
204* Alignment requirements
205* ======================
206*
207* The distributed submatrix sub( A ) must verify some alignment proper-
208* ties, namely the following expression should be true:
209* ( MB_A.EQ.NB_A .AND. IROFFA.EQ.ICOFFA ) with
210* IROFFA = MOD( IA-1, MB_A ) and ICOFFA = MOD( JA-1, NB_A ).
211*
212* =====================================================================
213*
214* .. Parameters ..
215 INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
216 $ LLD_, MB_, M_, NB_, N_, RSRC_
217 parameter( block_cyclic_2d = 1, dlen_ = 9, dtype_ = 1,
218 $ ctxt_ = 2, m_ = 3, n_ = 4, mb_ = 5, nb_ = 6,
219 $ rsrc_ = 7, csrc_ = 8, lld_ = 9 )
220 REAL HALF, ONE, ZERO
221 parameter( half = 0.5e+0, one = 1.0e+0, zero = 0.0e+0 )
222* ..
223* .. Local Scalars ..
224 LOGICAL LQUERY, UPPER
225 INTEGER IACOL, IAROW, ICOFFA, ICTXT, II, IK, IROFFA, J,
226 $ JJ, JK, JN, LDA, LWMIN, MYCOL, MYROW, NPCOL,
227 $ NPROW
228 REAL ALPHA, TAUI
229* ..
230* .. External Subroutines ..
231 EXTERNAL blacs_abort, blacs_gridinfo, chk1mat, infog2l,
232 $ pxerbla, saxpy, sgebr2d, sgebs2d,
233 $ slarfg, ssymv, ssyr2
234* ..
235* .. External Functions ..
236 LOGICAL LSAME
237 REAL SDOT
238 EXTERNAL lsame, sdot
239* ..
240* .. Intrinsic Functions ..
241 INTRINSIC real
242* ..
243* .. Executable Statements ..
244*
245* Get grid parameters
246*
247 ictxt = desca( ctxt_ )
248 CALL blacs_gridinfo( ictxt, nprow, npcol, myrow, mycol )
249*
250* Test the input parameters
251*
252 info = 0
253 IF( nprow.EQ.-1 ) THEN
254 info = -(600+ctxt_)
255 ELSE
256 upper = lsame( uplo, 'U' )
257 CALL chk1mat( n, 2, n, 2, ia, ja, desca, 6, info )
258 lwmin = 3 * n
259*
260 work( 1 ) = real( lwmin )
261 lquery = ( lwork.EQ.-1 )
262 IF( info.EQ.0 ) THEN
263 iroffa = mod( ia-1, desca( mb_ ) )
264 icoffa = mod( ja-1, desca( nb_ ) )
265 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
266 info = -1
267 ELSE IF( iroffa.NE.icoffa ) THEN
268 info = -5
269 ELSE IF( desca( mb_ ).NE.desca( nb_ ) ) THEN
270 info = -(600+nb_)
271 ELSE IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
272 info = -11
273 END IF
274 END IF
275 END IF
276*
277 IF( info.NE.0 ) THEN
278 CALL pxerbla( ictxt, 'PSSYTD2', -info )
279 CALL blacs_abort( ictxt, 1 )
280 RETURN
281 ELSE IF( lquery ) THEN
282 RETURN
283 END IF
284*
285* Quick return if possible
286*
287 IF( n.LE.0 )
288 $ RETURN
289*
290* Compute local information
291*
292 lda = desca( lld_ )
293 CALL infog2l( ia, ja, desca, nprow, npcol, myrow, mycol, ii, jj,
294 $ iarow, iacol )
295*
296 IF( upper ) THEN
297*
298* Process(IAROW, IACOL) owns block to be reduced
299*
300 IF( mycol.EQ.iacol ) THEN
301 IF( myrow.EQ.iarow ) THEN
302*
303* Reduce the upper triangle of sub( A )
304*
305 DO 10 j = n-1, 1, -1
306 ik = ii + j - 1
307 jk = jj + j - 1
308*
309* Generate elementary reflector H(i) = I - tau * v * v'
310* to annihilate A(IA:IA+J-1,JA:JA+J-1)
311*
312 CALL slarfg( j, a( ik+jk*lda ), a( ii+jk*lda ), 1,
313 $ taui )
314 e( jk+1 ) = a( ik+jk*lda )
315*
316 IF( taui.NE.zero ) THEN
317*
318* Apply H(i) from both sides to
319* A(IA:IA+J-1,JA:JA+J-1)
320*
321 a( ik+jk*lda ) = one
322*
323* Compute x := tau * A * v storing x in TAU(1:i)
324*
325 CALL ssymv( uplo, j, taui, a( ii+(jj-1)*lda ),
326 $ lda, a( ii+jk*lda ), 1, zero,
327 $ tau( jj ), 1 )
328*
329* Compute w := x - 1/2 * tau * (x'*v) * v
330*
331 alpha = -half*taui*sdot( j, tau( jj ), 1,
332 $ a( ii+jk*lda ), 1 )
333 CALL saxpy( j, alpha, a( ii+jk*lda ), 1,
334 $ tau( jj ), 1 )
335*
336* Apply the transformation as a rank-2 update:
337* A := A - v * w' - w * v'
338*
339 CALL ssyr2( uplo, j, -one, a( ii+jk*lda ), 1,
340 $ tau( jj ), 1, a( ii+(jj-1)*lda ),
341 $ lda )
342 a( ik+jk*lda ) = e( jk+1 )
343 END IF
344*
345* Copy D, E, TAU to broadcast them columnwise.
346*
347 d( jk+1 ) = a( ik+1+jk*lda )
348 work( j+1 ) = d( jk+1 )
349 work( n+j+1 ) = e( jk+1 )
350 tau( jk+1 ) = taui
351 work( 2*n+j+1 ) = tau( jk+1 )
352*
353 10 CONTINUE
354 d( jj ) = a( ii+(jj-1)*lda )
355 work( 1 ) = d( jj )
356 work( n+1 ) = zero
357 work( 2*n+1 ) = zero
358*
359 CALL sgebs2d( ictxt, 'Columnwise', ' ', 1, 3*n, work, 1 )
360*
361 ELSE
362 CALL sgebr2d( ictxt, 'Columnwise', ' ', 1, 3*n, work, 1,
363 $ iarow, iacol )
364 DO 20 j = 2, n
365 jn = jj + j - 1
366 d( jn ) = work( j )
367 e( jn ) = work( n+j )
368 tau( jn ) = work( 2*n+j )
369 20 CONTINUE
370 d( jj ) = work( 1 )
371 END IF
372 END IF
373*
374 ELSE
375*
376* Process (IAROW, IACOL) owns block to be factorized
377*
378 IF( mycol.EQ.iacol ) THEN
379 IF( myrow.EQ.iarow ) THEN
380*
381* Reduce the lower triangle of sub( A )
382*
383 DO 30 j = 1, n - 1
384 ik = ii + j - 1
385 jk = jj + j - 1
386*
387* Generate elementary reflector H(i) = I - tau * v * v'
388* to annihilate A(IA+J-JA+2:IA+N-1,JA+J-1)
389*
390 CALL slarfg( n-j, a( ik+1+(jk-1)*lda ),
391 $ a( ik+2+(jk-1)*lda ), 1, taui )
392 e( jk ) = a( ik+1+(jk-1)*lda )
393*
394 IF( taui.NE.zero ) THEN
395*
396* Apply H(i) from both sides to
397* A(IA+J-JA+1:IA+N-1,JA+J+1:JA+N-1)
398*
399 a( ik+1+(jk-1)*lda ) = one
400*
401* Compute x := tau * A * v storing y in TAU(i:n-1)
402*
403 CALL ssymv( uplo, n-j, taui, a( ik+1+jk*lda ),
404 $ lda, a( ik+1+(jk-1)*lda ), 1,
405 $ zero, tau( jk ), 1 )
406*
407* Compute w := x - 1/2 * tau * (x'*v) * v
408*
409 alpha = -half*taui*sdot( n-j, tau( jk ), 1,
410 $ a( ik+1+(jk-1)*lda ), 1 )
411 CALL saxpy( n-j, alpha, a( ik+1+(jk-1)*lda ),
412 $ 1, tau( jk ), 1 )
413*
414* Apply the transformation as a rank-2 update:
415* A := A - v * w' - w * v'
416*
417 CALL ssyr2( uplo, n-j, -one,
418 $ a( ik+1+(jk-1)*lda ), 1,
419 $ tau( jk ), 1, a( ik+1+jk*lda ),
420 $ lda )
421 a( ik+1+(jk-1)*lda ) = e( jk )
422 END IF
423*
424* Copy D(JK), E(JK), TAU(JK) to broadcast them
425* columnwise.
426*
427 d( jk ) = a( ik+(jk-1)*lda )
428 work( j ) = d( jk )
429 work( n+j ) = e( jk )
430 tau( jk ) = taui
431 work( 2*n+j ) = tau( jk )
432 30 CONTINUE
433 jn = jj + n - 1
434 d( jn ) = a( ii+n-1+(jn-1)*lda )
435 work( n ) = d( jn )
436 tau( jn ) = zero
437 work( 2*n ) = zero
438*
439 CALL sgebs2d( ictxt, 'Columnwise', ' ', 1, 3*n-1, work,
440 $ 1 )
441*
442 ELSE
443 CALL sgebr2d( ictxt, 'Columnwise', ' ', 1, 3*n-1, work,
444 $ 1, iarow, iacol )
445 DO 40 j = 1, n - 1
446 jn = jj + j - 1
447 d( jn ) = work( j )
448 e( jn ) = work( n+j )
449 tau( jn ) = work( 2*n+j )
450 40 CONTINUE
451 jn = jj + n - 1
452 d( jn ) = work( n )
453 tau( jn ) = zero
454 END IF
455 END IF
456 END IF
457*
458 work( 1 ) = real( lwmin )
459*
460 RETURN
461*
462* End of PSSYTD2
463*
chk1mat
subroutine chk1mat(ma, mapos0, na, napos0, ia, ja, desca, descapos0, info)
Definition chk1mat.f:3
infog2l
subroutine infog2l(grindx, gcindx, desc, nprow, npcol, myrow, mycol, lrindx, lcindx, rsrc, csrc)
Definition infog2l.f:3
pxerbla
subroutine pxerbla(ictxt, srname, info)
Definition pxerbla.f:2
lsame
logical function lsame(ca, cb)
Definition tools.f:1724
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