LAPACK  3.11.0
LAPACK: Linear Algebra PACKage
ctrsyl3.f
1 *> \brief \b CTRSYL3
2 *
3 * Definition:
4 * ===========
5 *
6 *
7 *> \par Purpose
8 * =============
9 *>
10 *> \verbatim
11 *>
12 *> CTRSYL3 solves the complex Sylvester matrix equation:
13 *>
14 *> op(A)*X + X*op(B) = scale*C or
15 *> op(A)*X - X*op(B) = scale*C,
16 *>
17 *> where op(A) = A or A**H, and A and B are both upper triangular. A is
18 *> M-by-M and B is N-by-N; the right hand side C and the solution X are
19 *> M-by-N; and scale is an output scale factor, set <= 1 to avoid
20 *> overflow in X.
21 *>
22 *> This is the block version of the algorithm.
23 *> \endverbatim
24 *
25 * Arguments
26 * =========
27 *
28 *> \param[in] TRANA
29 *> \verbatim
30 *> TRANA is CHARACTER*1
31 *> Specifies the option op(A):
32 *> = 'N': op(A) = A (No transpose)
33 *> = 'C': op(A) = A**H (Conjugate transpose)
34 *> \endverbatim
35 *>
36 *> \param[in] TRANB
37 *> \verbatim
38 *> TRANB is CHARACTER*1
39 *> Specifies the option op(B):
40 *> = 'N': op(B) = B (No transpose)
41 *> = 'C': op(B) = B**H (Conjugate transpose)
42 *> \endverbatim
43 *>
44 *> \param[in] ISGN
45 *> \verbatim
46 *> ISGN is INTEGER
47 *> Specifies the sign in the equation:
48 *> = +1: solve op(A)*X + X*op(B) = scale*C
49 *> = -1: solve op(A)*X - X*op(B) = scale*C
50 *> \endverbatim
51 *>
52 *> \param[in] M
53 *> \verbatim
54 *> M is INTEGER
55 *> The order of the matrix A, and the number of rows in the
56 *> matrices X and C. M >= 0.
57 *> \endverbatim
58 *>
59 *> \param[in] N
60 *> \verbatim
61 *> N is INTEGER
62 *> The order of the matrix B, and the number of columns in the
63 *> matrices X and C. N >= 0.
64 *> \endverbatim
65 *>
66 *> \param[in] A
67 *> \verbatim
68 *> A is COMPLEX array, dimension (LDA,M)
69 *> The upper triangular matrix A.
70 *> \endverbatim
71 *>
72 *> \param[in] LDA
73 *> \verbatim
74 *> LDA is INTEGER
75 *> The leading dimension of the array A. LDA >= max(1,M).
76 *> \endverbatim
77 *>
78 *> \param[in] B
79 *> \verbatim
80 *> B is COMPLEX array, dimension (LDB,N)
81 *> The upper triangular matrix B.
82 *> \endverbatim
83 *>
84 *> \param[in] LDB
85 *> \verbatim
86 *> LDB is INTEGER
87 *> The leading dimension of the array B. LDB >= max(1,N).
88 *> \endverbatim
89 *>
90 *> \param[in,out] C
91 *> \verbatim
92 *> C is COMPLEX array, dimension (LDC,N)
93 *> On entry, the M-by-N right hand side matrix C.
94 *> On exit, C is overwritten by the solution matrix X.
95 *> \endverbatim
96 *>
97 *> \param[in] LDC
98 *> \verbatim
99 *> LDC is INTEGER
100 *> The leading dimension of the array C. LDC >= max(1,M)
101 *> \endverbatim
102 *>
103 *> \param[out] SCALE
104 *> \verbatim
105 *> SCALE is REAL
106 *> The scale factor, scale, set <= 1 to avoid overflow in X.
107 *> \endverbatim
108 *>
109 *> \param[out] SWORK
110 *> \verbatim
111 *> SWORK is REAL array, dimension (MAX(2, ROWS), MAX(1,COLS)).
112 *> On exit, if INFO = 0, SWORK(1) returns the optimal value ROWS
113 *> and SWORK(2) returns the optimal COLS.
114 *> \endverbatim
115 *>
116 *> \param[in] LDSWORK
117 *> \verbatim
118 *> LDSWORK is INTEGER
119 *> LDSWORK >= MAX(2,ROWS), where ROWS = ((M + NB - 1) / NB + 1)
120 *> and NB is the optimal block size.
121 *>
122 *> If LDSWORK = -1, then a workspace query is assumed; the routine
123 *> only calculates the optimal dimensions of the SWORK matrix,
124 *> returns these values as the first and second entry of the SWORK
125 *> matrix, and no error message related LWORK is issued by XERBLA.
126 *> \endverbatim
127 *>
128 *> \param[out] INFO
129 *> \verbatim
130 *> INFO is INTEGER
131 *> = 0: successful exit
132 *> < 0: if INFO = -i, the i-th argument had an illegal value
133 *> = 1: A and B have common or very close eigenvalues; perturbed
134 *> values were used to solve the equation (but the matrices
135 *> A and B are unchanged).
136 *> \endverbatim
137 *
138 *> \ingroup trsyl3
139 *
140 * =====================================================================
141 * References:
142 * E. S. Quintana-Orti and R. A. Van De Geijn (2003). Formal derivation of
143 * algorithms: The triangular Sylvester equation, ACM Transactions
144 * on Mathematical Software (TOMS), volume 29, pages 218--243.
145 *
146 * A. Schwarz and C. C. Kjelgaard Mikkelsen (2020). Robust Task-Parallel
147 * Solution of the Triangular Sylvester Equation. Lecture Notes in
148 * Computer Science, vol 12043, pages 82--92, Springer.
149 *
150 * Contributor:
151 * Angelika Schwarz, Umea University, Sweden.
152 *
153 * =====================================================================
154  SUBROUTINE ctrsyl3( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
155  $ LDC, SCALE, SWORK, LDSWORK, INFO )
156  IMPLICIT NONE
157 *
158 * .. Scalar Arguments ..
159  CHARACTER TRANA, TRANB
160  INTEGER INFO, ISGN, LDA, LDB, LDC, LDSWORK, M, N
161  REAL SCALE
162 * ..
163 * .. Array Arguments ..
164  COMPLEX A( lda, * ), B( ldb, * ), C( ldc, * )
165  REAL SWORK( ldswork, * )
166 * ..
167 * .. Parameters ..
168  REAL ZERO, ONE
169  parameter( zero = 0.0e+0, one = 1.0e+0 )
170  COMPLEX CONE
171  parameter( cone = ( 1.0e+0, 0.0e+0 ) )
172 * ..
173 * .. Local Scalars ..
174  LOGICAL NOTRNA, NOTRNB, LQUERY
175  INTEGER AWRK, BWRK, I, I1, I2, IINFO, J, J1, J2, JJ,
176  $ k, k1, k2, l, l1, l2, ll, nba, nb, nbb
177  REAL ANRM, BIGNUM, BNRM, CNRM, SCAL, SCALOC,
178  $ scamin, sgn, xnrm, buf, smlnum
179  COMPLEX CSGN
180 * ..
181 * .. Local Arrays ..
182  REAL WNRM( max( m, n ) )
183 * ..
184 * .. External Functions ..
185  LOGICAL LSAME
186  INTEGER ILAENV
187  REAL CLANGE, SLAMCH, SLARMM
188  EXTERNAL clange, ilaenv, lsame, slamch, slarmm
189 * ..
190 * .. External Subroutines ..
191  EXTERNAL csscal, cgemm, clascl, ctrsyl, xerbla
192 * ..
193 * .. Intrinsic Functions ..
194  INTRINSIC abs, aimag, exponent, max, min, real
195 * ..
196 * .. Executable Statements ..
197 *
198 * Decode and Test input parameters
199 *
200  notrna = lsame( trana, 'N' )
201  notrnb = lsame( tranb, 'N' )
202 *
203 * Use the same block size for all matrices.
204 *
205  nb = max( 8, ilaenv( 1, 'CTRSYL', '', m, n, -1, -1) )
206 *
207 * Compute number of blocks in A and B
208 *
209  nba = max( 1, (m + nb - 1) / nb )
210  nbb = max( 1, (n + nb - 1) / nb )
211 *
212 * Compute workspace
213 *
214  info = 0
215  lquery = ( ldswork.EQ.-1 )
216  IF( lquery ) THEN
217  ldswork = 2
218  swork(1,1) = max( nba, nbb )
219  swork(2,1) = 2 * nbb + nba
220  END IF
221 *
222 * Test the input arguments
223 *
224  IF( .NOT.notrna .AND. .NOT. lsame( trana, 'C' ) ) THEN
225  info = -1
226  ELSE IF( .NOT.notrnb .AND. .NOT. lsame( tranb, 'C' ) ) THEN
227  info = -2
228  ELSE IF( isgn.NE.1 .AND. isgn.NE.-1 ) THEN
229  info = -3
230  ELSE IF( m.LT.0 ) THEN
231  info = -4
232  ELSE IF( n.LT.0 ) THEN
233  info = -5
234  ELSE IF( lda.LT.max( 1, m ) ) THEN
235  info = -7
236  ELSE IF( ldb.LT.max( 1, n ) ) THEN
237  info = -9
238  ELSE IF( ldc.LT.max( 1, m ) ) THEN
239  info = -11
240  END IF
241  IF( info.NE.0 ) THEN
242  CALL xerbla( 'CTRSYL3', -info )
243  RETURN
244  ELSE IF( lquery ) THEN
245  RETURN
246  END IF
247 *
248 * Quick return if possible
249 *
250  scale = one
251  IF( m.EQ.0 .OR. n.EQ.0 )
252  $ RETURN
253 *
254 * Use unblocked code for small problems or if insufficient
255 * workspace is provided
256 *
257  IF( min( nba, nbb ).EQ.1 .OR. ldswork.LT.max( nba, nbb ) ) THEN
258  CALL ctrsyl( trana, tranb, isgn, m, n, a, lda, b, ldb,
259  $ c, ldc, scale, info )
260  RETURN
261  END IF
262 *
263 * Set constants to control overflow
264 *
265  smlnum = slamch( 'S' )
266  bignum = one / smlnum
267 *
268 * Set local scaling factors.
269 *
270  DO l = 1, nbb
271  DO k = 1, nba
272  swork( k, l ) = one
273  END DO
274  END DO
275 *
276 * Fallback scaling factor to prevent flushing of SWORK( K, L ) to zero.
277 * This scaling is to ensure compatibility with TRSYL and may get flushed.
278 *
279  buf = one
280 *
281 * Compute upper bounds of blocks of A and B
282 *
283  awrk = nbb
284  DO k = 1, nba
285  k1 = (k - 1) * nb + 1
286  k2 = min( k * nb, m ) + 1
287  DO l = k, nba
288  l1 = (l - 1) * nb + 1
289  l2 = min( l * nb, m ) + 1
290  IF( notrna ) THEN
291  swork( k, awrk + l ) = clange( 'I', k2-k1, l2-l1,
292  $ a( k1, l1 ), lda, wnrm )
293  ELSE
294  swork( l, awrk + k ) = clange( '1', k2-k1, l2-l1,
295  $ a( k1, l1 ), lda, wnrm )
296  END IF
297  END DO
298  END DO
299  bwrk = nbb + nba
300  DO k = 1, nbb
301  k1 = (k - 1) * nb + 1
302  k2 = min( k * nb, n ) + 1
303  DO l = k, nbb
304  l1 = (l - 1) * nb + 1
305  l2 = min( l * nb, n ) + 1
306  IF( notrnb ) THEN
307  swork( k, bwrk + l ) = clange( 'I', k2-k1, l2-l1,
308  $ b( k1, l1 ), ldb, wnrm )
309  ELSE
310  swork( l, bwrk + k ) = clange( '1', k2-k1, l2-l1,
311  $ b( k1, l1 ), ldb, wnrm )
312  END IF
313  END DO
314  END DO
315 *
316  sgn = REAL( isgn )
317  csgn = cmplx( sgn, zero )
318 *
319  IF( notrna .AND. notrnb ) THEN
320 *
321 * Solve A*X + ISGN*X*B = scale*C.
322 *
323 * The (K,L)th block of X is determined starting from
324 * bottom-left corner column by column by
325 *
326 * A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
327 *
328 * Where
329 * M L-1
330 * R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B(J,L)].
331 * I=K+1 J=1
332 *
333 * Start loop over block rows (index = K) and block columns (index = L)
334 *
335  DO k = nba, 1, -1
336 *
337 * K1: row index of the first row in X( K, L )
338 * K2: row index of the first row in X( K+1, L )
339 * so the K2 - K1 is the column count of the block X( K, L )
340 *
341  k1 = (k - 1) * nb + 1
342  k2 = min( k * nb, m ) + 1
343  DO l = 1, nbb
344 *
345 * L1: column index of the first column in X( K, L )
346 * L2: column index of the first column in X( K, L + 1)
347 * so that L2 - L1 is the row count of the block X( K, L )
348 *
349  l1 = (l - 1) * nb + 1
350  l2 = min( l * nb, n ) + 1
351 *
352  CALL ctrsyl( trana, tranb, isgn, k2-k1, l2-l1,
353  $ a( k1, k1 ), lda,
354  $ b( l1, l1 ), ldb,
355  $ c( k1, l1 ), ldc, scaloc, iinfo )
356  info = max( info, iinfo )
357 *
358  IF( scaloc * swork( k, l ) .EQ. zero ) THEN
359  IF( scaloc .EQ. zero ) THEN
360 * The magnitude of the largest entry of X(K1:K2-1, L1:L2-1)
361 * is larger than the product of BIGNUM**2 and cannot be
362 * represented in the form (1/SCALE)*X(K1:K2-1, L1:L2-1).
363 * Mark the computation as pointless.
364  buf = zero
365  ELSE
366 * Use second scaling factor to prevent flushing to zero.
367  buf = buf*2.e0**exponent( scaloc )
368  END IF
369  DO jj = 1, nbb
370  DO ll = 1, nba
371 * Bound by BIGNUM to not introduce Inf. The value
372 * is irrelevant; corresponding entries of the
373 * solution will be flushed in consistency scaling.
374  swork( ll, jj ) = min( bignum,
375  $ swork( ll, jj ) / 2.e0**exponent( scaloc ) )
376  END DO
377  END DO
378  END IF
379  swork( k, l ) = scaloc * swork( k, l )
380  xnrm = clange( 'I', k2-k1, l2-l1, c( k1, l1 ), ldc,
381  $ wnrm )
382 *
383  DO i = k - 1, 1, -1
384 *
385 * C( I, L ) := C( I, L ) - A( I, K ) * C( K, L )
386 *
387  i1 = (i - 1) * nb + 1
388  i2 = min( i * nb, m ) + 1
389 *
390 * Compute scaling factor to survive the linear update
391 * simulating consistent scaling.
392 *
393  cnrm = clange( 'I', i2-i1, l2-l1, c( i1, l1 ),
394  $ ldc, wnrm )
395  scamin = min( swork( i, l ), swork( k, l ) )
396  cnrm = cnrm * ( scamin / swork( i, l ) )
397  xnrm = xnrm * ( scamin / swork( k, l ) )
398  anrm = swork( i, awrk + k )
399  scaloc = slarmm( anrm, xnrm, cnrm )
400  IF( scaloc * scamin .EQ. zero ) THEN
401 * Use second scaling factor to prevent flushing to zero.
402  buf = buf*2.e0**exponent( scaloc )
403  DO jj = 1, nbb
404  DO ll = 1, nba
405  swork( ll, jj ) = min( bignum,
406  $ swork( ll, jj ) / 2.e0**exponent( scaloc ) )
407  END DO
408  END DO
409  scamin = scamin / 2.e0**exponent( scaloc )
410  scaloc = scaloc / 2.e0**exponent( scaloc )
411  END IF
412  cnrm = cnrm * scaloc
413  xnrm = xnrm * scaloc
414 *
415 * Simultaneously apply the robust update factor and the
416 * consistency scaling factor to C( I, L ) and C( K, L ).
417 *
418  scal = ( scamin / swork( k, l ) ) * scaloc
419  IF( scal.NE.one ) THEN
420  DO jj = l1, l2-1
421  CALL csscal( k2-k1, scal, c( k1, jj ), 1)
422  END DO
423  ENDIF
424 *
425  scal = ( scamin / swork( i, l ) ) * scaloc
426  IF( scal.NE.one ) THEN
427  DO ll = l1, l2-1
428  CALL csscal( i2-i1, scal, c( i1, ll ), 1)
429  END DO
430  ENDIF
431 *
432 * Record current scaling factor
433 *
434  swork( k, l ) = scamin * scaloc
435  swork( i, l ) = scamin * scaloc
436 *
437  CALL cgemm( 'N', 'N', i2-i1, l2-l1, k2-k1, -cone,
438  $ a( i1, k1 ), lda, c( k1, l1 ), ldc,
439  $ cone, c( i1, l1 ), ldc )
440 *
441  END DO
442 *
443  DO j = l + 1, nbb
444 *
445 * C( K, J ) := C( K, J ) - SGN * C( K, L ) * B( L, J )
446 *
447  j1 = (j - 1) * nb + 1
448  j2 = min( j * nb, n ) + 1
449 *
450 * Compute scaling factor to survive the linear update
451 * simulating consistent scaling.
452 *
453  cnrm = clange( 'I', k2-k1, j2-j1, c( k1, j1 ),
454  $ ldc, wnrm )
455  scamin = min( swork( k, j ), swork( k, l ) )
456  cnrm = cnrm * ( scamin / swork( k, j ) )
457  xnrm = xnrm * ( scamin / swork( k, l ) )
458  bnrm = swork(l, bwrk + j)
459  scaloc = slarmm( bnrm, xnrm, cnrm )
460  IF( scaloc * scamin .EQ. zero ) THEN
461 * Use second scaling factor to prevent flushing to zero.
462  buf = buf*2.e0**exponent( scaloc )
463  DO jj = 1, nbb
464  DO ll = 1, nba
465  swork( ll, jj ) = min( bignum,
466  $ swork( ll, jj ) / 2.e0**exponent( scaloc ) )
467  END DO
468  END DO
469  scamin = scamin / 2.e0**exponent( scaloc )
470  scaloc = scaloc / 2.e0**exponent( scaloc )
471  END IF
472  cnrm = cnrm * scaloc
473  xnrm = xnrm * scaloc
474 *
475 * Simultaneously apply the robust update factor and the
476 * consistency scaling factor to C( K, J ) and C( K, L).
477 *
478  scal = ( scamin / swork( k, l ) ) * scaloc
479  IF( scal .NE. one ) THEN
480  DO ll = l1, l2-1
481  CALL csscal( k2-k1, scal, c( k1, ll ), 1 )
482  END DO
483  ENDIF
484 *
485  scal = ( scamin / swork( k, j ) ) * scaloc
486  IF( scal .NE. one ) THEN
487  DO jj = j1, j2-1
488  CALL csscal( k2-k1, scal, c( k1, jj ), 1 )
489  END DO
490  ENDIF
491 *
492 * Record current scaling factor
493 *
494  swork( k, l ) = scamin * scaloc
495  swork( k, j ) = scamin * scaloc
496 *
497  CALL cgemm( 'N', 'N', k2-k1, j2-j1, l2-l1, -csgn,
498  $ c( k1, l1 ), ldc, b( l1, j1 ), ldb,
499  $ cone, c( k1, j1 ), ldc )
500  END DO
501  END DO
502  END DO
503  ELSE IF( .NOT.notrna .AND. notrnb ) THEN
504 *
505 * Solve A**H *X + ISGN*X*B = scale*C.
506 *
507 * The (K,L)th block of X is determined starting from
508 * upper-left corner column by column by
509 *
510 * A(K,K)**H*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
511 *
512 * Where
513 * K-1 L-1
514 * R(K,L) = SUM [A(I,K)**H*X(I,L)] +ISGN*SUM [X(K,J)*B(J,L)]
515 * I=1 J=1
516 *
517 * Start loop over block rows (index = K) and block columns (index = L)
518 *
519  DO k = 1, nba
520 *
521 * K1: row index of the first row in X( K, L )
522 * K2: row index of the first row in X( K+1, L )
523 * so the K2 - K1 is the column count of the block X( K, L )
524 *
525  k1 = (k - 1) * nb + 1
526  k2 = min( k * nb, m ) + 1
527  DO l = 1, nbb
528 *
529 * L1: column index of the first column in X( K, L )
530 * L2: column index of the first column in X( K, L + 1)
531 * so that L2 - L1 is the row count of the block X( K, L )
532 *
533  l1 = (l - 1) * nb + 1
534  l2 = min( l * nb, n ) + 1
535 *
536  CALL ctrsyl( trana, tranb, isgn, k2-k1, l2-l1,
537  $ a( k1, k1 ), lda,
538  $ b( l1, l1 ), ldb,
539  $ c( k1, l1 ), ldc, scaloc, iinfo )
540  info = max( info, iinfo )
541 *
542  IF( scaloc * swork( k, l ) .EQ. zero ) THEN
543  IF( scaloc .EQ. zero ) THEN
544 * The magnitude of the largest entry of X(K1:K2-1, L1:L2-1)
545 * is larger than the product of BIGNUM**2 and cannot be
546 * represented in the form (1/SCALE)*X(K1:K2-1, L1:L2-1).
547 * Mark the computation as pointless.
548  buf = zero
549  ELSE
550 * Use second scaling factor to prevent flushing to zero.
551  buf = buf*2.e0**exponent( scaloc )
552  END IF
553  DO jj = 1, nbb
554  DO ll = 1, nba
555 * Bound by BIGNUM to not introduce Inf. The value
556 * is irrelevant; corresponding entries of the
557 * solution will be flushed in consistency scaling.
558  swork( ll, jj ) = min( bignum,
559  $ swork( ll, jj ) / 2.e0**exponent( scaloc ) )
560  END DO
561  END DO
562  END IF
563  swork( k, l ) = scaloc * swork( k, l )
564  xnrm = clange( 'I', k2-k1, l2-l1, c( k1, l1 ), ldc,
565  $ wnrm )
566 *
567  DO i = k + 1, nba
568 *
569 * C( I, L ) := C( I, L ) - A( K, I )**H * C( K, L )
570 *
571  i1 = (i - 1) * nb + 1
572  i2 = min( i * nb, m ) + 1
573 *
574 * Compute scaling factor to survive the linear update
575 * simulating consistent scaling.
576 *
577  cnrm = clange( 'I', i2-i1, l2-l1, c( i1, l1 ),
578  $ ldc, wnrm )
579  scamin = min( swork( i, l ), swork( k, l ) )
580  cnrm = cnrm * ( scamin / swork( i, l ) )
581  xnrm = xnrm * ( scamin / swork( k, l ) )
582  anrm = swork( i, awrk + k )
583  scaloc = slarmm( anrm, xnrm, cnrm )
584  IF( scaloc * scamin .EQ. zero ) THEN
585 * Use second scaling factor to prevent flushing to zero.
586  buf = buf*2.e0**exponent( scaloc )
587  DO jj = 1, nbb
588  DO ll = 1, nba
589  swork( ll, jj ) = min( bignum,
590  $ swork( ll, jj ) / 2.e0**exponent( scaloc ) )
591  END DO
592  END DO
593  scamin = scamin / 2.e0**exponent( scaloc )
594  scaloc = scaloc / 2.e0**exponent( scaloc )
595  END IF
596  cnrm = cnrm * scaloc
597  xnrm = xnrm * scaloc
598 *
599 * Simultaneously apply the robust update factor and the
600 * consistency scaling factor to to C( I, L ) and C( K, L).
601 *
602  scal = ( scamin / swork( k, l ) ) * scaloc
603  IF( scal .NE. one ) THEN
604  DO ll = l1, l2-1
605  CALL csscal( k2-k1, scal, c( k1, ll ), 1 )
606  END DO
607  ENDIF
608 *
609  scal = ( scamin / swork( i, l ) ) * scaloc
610  IF( scal .NE. one ) THEN
611  DO ll = l1, l2-1
612  CALL csscal( i2-i1, scal, c( i1, ll ), 1 )
613  END DO
614  ENDIF
615 *
616 * Record current scaling factor
617 *
618  swork( k, l ) = scamin * scaloc
619  swork( i, l ) = scamin * scaloc
620 *
621  CALL cgemm( 'C', 'N', i2-i1, l2-l1, k2-k1, -cone,
622  $ a( k1, i1 ), lda, c( k1, l1 ), ldc,
623  $ cone, c( i1, l1 ), ldc )
624  END DO
625 *
626  DO j = l + 1, nbb
627 *
628 * C( K, J ) := C( K, J ) - SGN * C( K, L ) * B( L, J )
629 *
630  j1 = (j - 1) * nb + 1
631  j2 = min( j * nb, n ) + 1
632 *
633 * Compute scaling factor to survive the linear update
634 * simulating consistent scaling.
635 *
636  cnrm = clange( 'I', k2-k1, j2-j1, c( k1, j1 ),
637  $ ldc, wnrm )
638  scamin = min( swork( k, j ), swork( k, l ) )
639  cnrm = cnrm * ( scamin / swork( k, j ) )
640  xnrm = xnrm * ( scamin / swork( k, l ) )
641  bnrm = swork( l, bwrk + j )
642  scaloc = slarmm( bnrm, xnrm, cnrm )
643  IF( scaloc * scamin .EQ. zero ) THEN
644 * Use second scaling factor to prevent flushing to zero.
645  buf = buf*2.e0**exponent( scaloc )
646  DO jj = 1, nbb
647  DO ll = 1, nba
648  swork( ll, jj ) = min( bignum,
649  $ swork( ll, jj ) / 2.e0**exponent( scaloc ) )
650  END DO
651  END DO
652  scamin = scamin / 2.e0**exponent( scaloc )
653  scaloc = scaloc / 2.e0**exponent( scaloc )
654  END IF
655  cnrm = cnrm * scaloc
656  xnrm = xnrm * scaloc
657 *
658 * Simultaneously apply the robust update factor and the
659 * consistency scaling factor to to C( K, J ) and C( K, L).
660 *
661  scal = ( scamin / swork( k, l ) ) * scaloc
662  IF( scal .NE. one ) THEN
663  DO ll = l1, l2-1
664  CALL csscal( k2-k1, scal, c( k1, ll ), 1 )
665  END DO
666  ENDIF
667 *
668  scal = ( scamin / swork( k, j ) ) * scaloc
669  IF( scal .NE. one ) THEN
670  DO jj = j1, j2-1
671  CALL csscal( k2-k1, scal, c( k1, jj ), 1 )
672  END DO
673  ENDIF
674 *
675 * Record current scaling factor
676 *
677  swork( k, l ) = scamin * scaloc
678  swork( k, j ) = scamin * scaloc
679 *
680  CALL cgemm( 'N', 'N', k2-k1, j2-j1, l2-l1, -csgn,
681  $ c( k1, l1 ), ldc, b( l1, j1 ), ldb,
682  $ cone, c( k1, j1 ), ldc )
683  END DO
684  END DO
685  END DO
686  ELSE IF( .NOT.notrna .AND. .NOT.notrnb ) THEN
687 *
688 * Solve A**H *X + ISGN*X*B**H = scale*C.
689 *
690 * The (K,L)th block of X is determined starting from
691 * top-right corner column by column by
692 *
693 * A(K,K)**H*X(K,L) + ISGN*X(K,L)*B(L,L)**H = C(K,L) - R(K,L)
694 *
695 * Where
696 * K-1 N
697 * R(K,L) = SUM [A(I,K)**H*X(I,L)] + ISGN*SUM [X(K,J)*B(L,J)**H].
698 * I=1 J=L+1
699 *
700 * Start loop over block rows (index = K) and block columns (index = L)
701 *
702  DO k = 1, nba
703 *
704 * K1: row index of the first row in X( K, L )
705 * K2: row index of the first row in X( K+1, L )
706 * so the K2 - K1 is the column count of the block X( K, L )
707 *
708  k1 = (k - 1) * nb + 1
709  k2 = min( k * nb, m ) + 1
710  DO l = nbb, 1, -1
711 *
712 * L1: column index of the first column in X( K, L )
713 * L2: column index of the first column in X( K, L + 1)
714 * so that L2 - L1 is the row count of the block X( K, L )
715 *
716  l1 = (l - 1) * nb + 1
717  l2 = min( l * nb, n ) + 1
718 *
719  CALL ctrsyl( trana, tranb, isgn, k2-k1, l2-l1,
720  $ a( k1, k1 ), lda,
721  $ b( l1, l1 ), ldb,
722  $ c( k1, l1 ), ldc, scaloc, iinfo )
723  info = max( info, iinfo )
724 *
725  IF( scaloc * swork( k, l ) .EQ. zero ) THEN
726  IF( scaloc .EQ. zero ) THEN
727 * The magnitude of the largest entry of X(K1:K2-1, L1:L2-1)
728 * is larger than the product of BIGNUM**2 and cannot be
729 * represented in the form (1/SCALE)*X(K1:K2-1, L1:L2-1).
730 * Mark the computation as pointless.
731  buf = zero
732  ELSE
733 * Use second scaling factor to prevent flushing to zero.
734  buf = buf*2.e0**exponent( scaloc )
735  END IF
736  DO jj = 1, nbb
737  DO ll = 1, nba
738 * Bound by BIGNUM to not introduce Inf. The value
739 * is irrelevant; corresponding entries of the
740 * solution will be flushed in consistency scaling.
741  swork( ll, jj ) = min( bignum,
742  $ swork( ll, jj ) / 2.e0**exponent( scaloc ) )
743  END DO
744  END DO
745  END IF
746  swork( k, l ) = scaloc * swork( k, l )
747  xnrm = clange( 'I', k2-k1, l2-l1, c( k1, l1 ), ldc,
748  $ wnrm )
749 *
750  DO i = k + 1, nba
751 *
752 * C( I, L ) := C( I, L ) - A( K, I )**H * C( K, L )
753 *
754  i1 = (i - 1) * nb + 1
755  i2 = min( i * nb, m ) + 1
756 *
757 * Compute scaling factor to survive the linear update
758 * simulating consistent scaling.
759 *
760  cnrm = clange( 'I', i2-i1, l2-l1, c( i1, l1 ),
761  $ ldc, wnrm )
762  scamin = min( swork( i, l ), swork( k, l ) )
763  cnrm = cnrm * ( scamin / swork( i, l ) )
764  xnrm = xnrm * ( scamin / swork( k, l ) )
765  anrm = swork( i, awrk + k )
766  scaloc = slarmm( anrm, xnrm, cnrm )
767  IF( scaloc * scamin .EQ. zero ) THEN
768 * Use second scaling factor to prevent flushing to zero.
769  buf = buf*2.e0**exponent( scaloc )
770  DO jj = 1, nbb
771  DO ll = 1, nba
772  swork( ll, jj ) = min( bignum,
773  $ swork( ll, jj ) / 2.e0**exponent( scaloc ) )
774  END DO
775  END DO
776  scamin = scamin / 2.e0**exponent( scaloc )
777  scaloc = scaloc / 2.e0**exponent( scaloc )
778  END IF
779  cnrm = cnrm * scaloc
780  xnrm = xnrm * scaloc
781 *
782 * Simultaneously apply the robust update factor and the
783 * consistency scaling factor to C( I, L ) and C( K, L).
784 *
785  scal = ( scamin / swork( k, l ) ) * scaloc
786  IF( scal .NE. one ) THEN
787  DO ll = l1, l2-1
788  CALL csscal( k2-k1, scal, c( k1, ll ), 1 )
789  END DO
790  ENDIF
791 *
792  scal = ( scamin / swork( i, l ) ) * scaloc
793  IF( scal .NE. one ) THEN
794  DO ll = l1, l2-1
795  CALL csscal( i2-i1, scal, c( i1, ll ), 1 )
796  END DO
797  ENDIF
798 *
799 * Record current scaling factor
800 *
801  swork( k, l ) = scamin * scaloc
802  swork( i, l ) = scamin * scaloc
803 *
804  CALL cgemm( 'C', 'N', i2-i1, l2-l1, k2-k1, -cone,
805  $ a( k1, i1 ), lda, c( k1, l1 ), ldc,
806  $ cone, c( i1, l1 ), ldc )
807  END DO
808 *
809  DO j = 1, l - 1
810 *
811 * C( K, J ) := C( K, J ) - SGN * C( K, L ) * B( J, L )**H
812 *
813  j1 = (j - 1) * nb + 1
814  j2 = min( j * nb, n ) + 1
815 *
816 * Compute scaling factor to survive the linear update
817 * simulating consistent scaling.
818 *
819  cnrm = clange( 'I', k2-k1, j2-j1, c( k1, j1 ),
820  $ ldc, wnrm )
821  scamin = min( swork( k, j ), swork( k, l ) )
822  cnrm = cnrm * ( scamin / swork( k, j ) )
823  xnrm = xnrm * ( scamin / swork( k, l ) )
824  bnrm = swork( l, bwrk + j )
825  scaloc = slarmm( bnrm, xnrm, cnrm )
826  IF( scaloc * scamin .EQ. zero ) THEN
827 * Use second scaling factor to prevent flushing to zero.
828  buf = buf*2.e0**exponent( scaloc )
829  DO jj = 1, nbb
830  DO ll = 1, nba
831  swork( ll, jj ) = min( bignum,
832  $ swork( ll, jj ) / 2.e0**exponent( scaloc ) )
833  END DO
834  END DO
835  scamin = scamin / 2.e0**exponent( scaloc )
836  scaloc = scaloc / 2.e0**exponent( scaloc )
837  END IF
838  cnrm = cnrm * scaloc
839  xnrm = xnrm * scaloc
840 *
841 * Simultaneously apply the robust update factor and the
842 * consistency scaling factor to C( K, J ) and C( K, L).
843 *
844  scal = ( scamin / swork( k, l ) ) * scaloc
845  IF( scal .NE. one ) THEN
846  DO ll = l1, l2-1
847  CALL csscal( k2-k1, scal, c( k1, ll ), 1)
848  END DO
849  ENDIF
850 *
851  scal = ( scamin / swork( k, j ) ) * scaloc
852  IF( scal .NE. one ) THEN
853  DO jj = j1, j2-1
854  CALL csscal( k2-k1, scal, c( k1, jj ), 1 )
855  END DO
856  ENDIF
857 *
858 * Record current scaling factor
859 *
860  swork( k, l ) = scamin * scaloc
861  swork( k, j ) = scamin * scaloc
862 *
863  CALL cgemm( 'N', 'C', k2-k1, j2-j1, l2-l1, -csgn,
864  $ c( k1, l1 ), ldc, b( j1, l1 ), ldb,
865  $ cone, c( k1, j1 ), ldc )
866  END DO
867  END DO
868  END DO
869  ELSE IF( notrna .AND. .NOT.notrnb ) THEN
870 *
871 * Solve A*X + ISGN*X*B**H = scale*C.
872 *
873 * The (K,L)th block of X is determined starting from
874 * bottom-right corner column by column by
875 *
876 * A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L)**H = C(K,L) - R(K,L)
877 *
878 * Where
879 * M N
880 * R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B(L,J)**H].
881 * I=K+1 J=L+1
882 *
883 * Start loop over block rows (index = K) and block columns (index = L)
884 *
885  DO k = nba, 1, -1
886 *
887 * K1: row index of the first row in X( K, L )
888 * K2: row index of the first row in X( K+1, L )
889 * so the K2 - K1 is the column count of the block X( K, L )
890 *
891  k1 = (k - 1) * nb + 1
892  k2 = min( k * nb, m ) + 1
893  DO l = nbb, 1, -1
894 *
895 * L1: column index of the first column in X( K, L )
896 * L2: column index of the first column in X( K, L + 1)
897 * so that L2 - L1 is the row count of the block X( K, L )
898 *
899  l1 = (l - 1) * nb + 1
900  l2 = min( l * nb, n ) + 1
901 *
902  CALL ctrsyl( trana, tranb, isgn, k2-k1, l2-l1,
903  $ a( k1, k1 ), lda,
904  $ b( l1, l1 ), ldb,
905  $ c( k1, l1 ), ldc, scaloc, iinfo )
906  info = max( info, iinfo )
907 *
908  IF( scaloc * swork( k, l ) .EQ. zero ) THEN
909  IF( scaloc .EQ. zero ) THEN
910 * The magnitude of the largest entry of X(K1:K2-1, L1:L2-1)
911 * is larger than the product of BIGNUM**2 and cannot be
912 * represented in the form (1/SCALE)*X(K1:K2-1, L1:L2-1).
913 * Mark the computation as pointless.
914  buf = zero
915  ELSE
916 * Use second scaling factor to prevent flushing to zero.
917  buf = buf*2.e0**exponent( scaloc )
918  END IF
919  DO jj = 1, nbb
920  DO ll = 1, nba
921 * Bound by BIGNUM to not introduce Inf. The value
922 * is irrelevant; corresponding entries of the
923 * solution will be flushed in consistency scaling.
924  swork( ll, jj ) = min( bignum,
925  $ swork( ll, jj ) / 2.e0**exponent( scaloc ) )
926  END DO
927  END DO
928  END IF
929  swork( k, l ) = scaloc * swork( k, l )
930  xnrm = clange( 'I', k2-k1, l2-l1, c( k1, l1 ), ldc,
931  $ wnrm )
932 *
933  DO i = 1, k - 1
934 *
935 * C( I, L ) := C( I, L ) - A( I, K ) * C( K, L )
936 *
937  i1 = (i - 1) * nb + 1
938  i2 = min( i * nb, m ) + 1
939 *
940 * Compute scaling factor to survive the linear update
941 * simulating consistent scaling.
942 *
943  cnrm = clange( 'I', i2-i1, l2-l1, c( i1, l1 ),
944  $ ldc, wnrm )
945  scamin = min( swork( i, l ), swork( k, l ) )
946  cnrm = cnrm * ( scamin / swork( i, l ) )
947  xnrm = xnrm * ( scamin / swork( k, l ) )
948  anrm = swork( i, awrk + k )
949  scaloc = slarmm( anrm, xnrm, cnrm )
950  IF( scaloc * scamin .EQ. zero ) THEN
951 * Use second scaling factor to prevent flushing to zero.
952  buf = buf*2.e0**exponent( scaloc )
953  DO jj = 1, nbb
954  DO ll = 1, nba
955  swork( ll, jj ) = min( bignum,
956  $ swork( ll, jj ) / 2.e0**exponent( scaloc ) )
957  END DO
958  END DO
959  scamin = scamin / 2.e0**exponent( scaloc )
960  scaloc = scaloc / 2.e0**exponent( scaloc )
961  END IF
962  cnrm = cnrm * scaloc
963  xnrm = xnrm * scaloc
964 *
965 * Simultaneously apply the robust update factor and the
966 * consistency scaling factor to C( I, L ) and C( K, L).
967 *
968  scal = ( scamin / swork( k, l ) ) * scaloc
969  IF( scal .NE. one ) THEN
970  DO ll = l1, l2-1
971  CALL csscal( k2-k1, scal, c( k1, ll ), 1 )
972  END DO
973  ENDIF
974 *
975  scal = ( scamin / swork( i, l ) ) * scaloc
976  IF( scal .NE. one ) THEN
977  DO ll = l1, l2-1
978  CALL csscal( i2-i1, scal, c( i1, ll ), 1 )
979  END DO
980  ENDIF
981 *
982 * Record current scaling factor
983 *
984  swork( k, l ) = scamin * scaloc
985  swork( i, l ) = scamin * scaloc
986 *
987  CALL cgemm( 'N', 'N', i2-i1, l2-l1, k2-k1, -cone,
988  $ a( i1, k1 ), lda, c( k1, l1 ), ldc,
989  $ cone, c( i1, l1 ), ldc )
990 *
991  END DO
992 *
993  DO j = 1, l - 1
994 *
995 * C( K, J ) := C( K, J ) - SGN * C( K, L ) * B( J, L )**H
996 *
997  j1 = (j - 1) * nb + 1
998  j2 = min( j * nb, n ) + 1
999 *
1000 * Compute scaling factor to survive the linear update
1001 * simulating consistent scaling.
1002 *
1003  cnrm = clange( 'I', k2-k1, j2-j1, c( k1, j1 ),
1004  $ ldc, wnrm )
1005  scamin = min( swork( k, j ), swork( k, l ) )
1006  cnrm = cnrm * ( scamin / swork( k, j ) )
1007  xnrm = xnrm * ( scamin / swork( k, l ) )
1008  bnrm = swork( l, bwrk + j )
1009  scaloc = slarmm( bnrm, xnrm, cnrm )
1010  IF( scaloc * scamin .EQ. zero ) THEN
1011 * Use second scaling factor to prevent flushing to zero.
1012  buf = buf*2.e0**exponent( scaloc )
1013  DO jj = 1, nbb
1014  DO ll = 1, nba
1015  swork( ll, jj ) = min( bignum,
1016  $ swork( ll, jj ) / 2.e0**exponent( scaloc ) )
1017  END DO
1018  END DO
1019  scamin = scamin / 2.e0**exponent( scaloc )
1020  scaloc = scaloc / 2.e0**exponent( scaloc )
1021  END IF
1022  cnrm = cnrm * scaloc
1023  xnrm = xnrm * scaloc
1024 *
1025 * Simultaneously apply the robust update factor and the
1026 * consistency scaling factor to C( K, J ) and C( K, L).
1027 *
1028  scal = ( scamin / swork( k, l ) ) * scaloc
1029  IF( scal .NE. one ) THEN
1030  DO jj = l1, l2-1
1031  CALL csscal( k2-k1, scal, c( k1, jj ), 1 )
1032  END DO
1033  ENDIF
1034 *
1035  scal = ( scamin / swork( k, j ) ) * scaloc
1036  IF( scal .NE. one ) THEN
1037  DO jj = j1, j2-1
1038  CALL csscal( k2-k1, scal, c( k1, jj ), 1 )
1039  END DO
1040  ENDIF
1041 *
1042 * Record current scaling factor
1043 *
1044  swork( k, l ) = scamin * scaloc
1045  swork( k, j ) = scamin * scaloc
1046 *
1047  CALL cgemm( 'N', 'C', k2-k1, j2-j1, l2-l1, -csgn,
1048  $ c( k1, l1 ), ldc, b( j1, l1 ), ldb,
1049  $ cone, c( k1, j1 ), ldc )
1050  END DO
1051  END DO
1052  END DO
1053 *
1054  END IF
1055 *
1056 * Reduce local scaling factors
1057 *
1058  scale = swork( 1, 1 )
1059  DO k = 1, nba
1060  DO l = 1, nbb
1061  scale = min( scale, swork( k, l ) )
1062  END DO
1063  END DO
1064  IF( scale .EQ. zero ) THEN
1065 *
1066 * The magnitude of the largest entry of the solution is larger
1067 * than the product of BIGNUM**2 and cannot be represented in the
1068 * form (1/SCALE)*X if SCALE is REAL. Set SCALE to
1069 * zero and give up.
1070 *
1071  swork(1,1) = max( nba, nbb )
1072  swork(2,1) = 2 * nbb + nba
1073  RETURN
1074  END IF
1075 *
1076 * Realize consistent scaling
1077 *
1078  DO k = 1, nba
1079  k1 = (k - 1) * nb + 1
1080  k2 = min( k * nb, m ) + 1
1081  DO l = 1, nbb
1082  l1 = (l - 1) * nb + 1
1083  l2 = min( l * nb, n ) + 1
1084  scal = scale / swork( k, l )
1085  IF( scal .NE. one ) THEN
1086  DO ll = l1, l2-1
1087  CALL csscal( k2-k1, scal, c( k1, ll ), 1 )
1088  END DO
1089  ENDIF
1090  END DO
1091  END DO
1092 *
1093  IF( buf .NE. one .AND. buf.GT.zero ) THEN
1094 *
1095 * Decrease SCALE as much as possible.
1096 *
1097  scaloc = min( scale / smlnum, one / buf )
1098  buf = buf * scaloc
1099  scale = scale / scaloc
1100  END IF
1101 *
1102  IF( buf.NE.one .AND. buf.GT.zero ) THEN
1103 *
1104 * In case of overly aggressive scaling during the computation,
1105 * flushing of the global scale factor may be prevented by
1106 * undoing some of the scaling. This step is to ensure that
1107 * this routine flushes only scale factors that TRSYL also
1108 * flushes and be usable as a drop-in replacement.
1109 *
1110 * How much can the normwise largest entry be upscaled?
1111 *
1112  scal = max( abs( REAL( C( 1, 1 ) ) ),
1113  $ abs( aimag( c( 1, 1 ) ) ) )
1114  DO k = 1, m
1115  DO l = 1, n
1116  scal = max( scal, abs( REAL ( C( K, L ) ) ),
1117  $ abs( aimag( c( k, l ) ) ) )
1118  END DO
1119  END DO
1120 *
1121 * Increase BUF as close to 1 as possible and apply scaling.
1122 *
1123  scaloc = min( bignum / scal, one / buf )
1124  buf = buf * scaloc
1125  CALL clascl( 'G', -1, -1, one, scaloc, m, n, c, ldc, iinfo )
1126  END IF
1127 *
1128 * Combine with buffer scaling factor. SCALE will be flushed if
1129 * BUF is less than one here.
1130 *
1131  scale = scale * buf
1132 *
1133 * Restore workspace dimensions
1134 *
1135  swork(1,1) = max( nba, nbb )
1136  swork(2,1) = 2 * nbb + nba
1137 *
1138  RETURN
1139 *
1140 * End of CTRSYL3
1141 *
1142  END
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
csscal
subroutine csscal(N, SA, CX, INCX)
CSSCAL
Definition: csscal.f:78
ctrsyl3
subroutine ctrsyl3(TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, SWORK, LDSWORK, INFO)
CTRSYL3
Definition: ctrsyl3.f:156
cgemm
subroutine cgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CGEMM
Definition: cgemm.f:187
ctrsyl
subroutine ctrsyl(TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO)
CTRSYL
Definition: ctrsyl.f:157
clascl
subroutine clascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: clascl.f:143