 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ ctrsyl()

 subroutine ctrsyl ( character TRANA, character TRANB, integer ISGN, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, real SCALE, integer INFO )

CTRSYL

Purpose:
``` CTRSYL solves the complex Sylvester matrix equation:

op(A)*X + X*op(B) = scale*C or
op(A)*X - X*op(B) = scale*C,

where op(A) = A or A**H, and A and B are both upper triangular. A is
M-by-M and B is N-by-N; the right hand side C and the solution X are
M-by-N; and scale is an output scale factor, set <= 1 to avoid
overflow in X.```
Parameters
 [in] TRANA ``` TRANA is CHARACTER*1 Specifies the option op(A): = 'N': op(A) = A (No transpose) = 'C': op(A) = A**H (Conjugate transpose)``` [in] TRANB ``` TRANB is CHARACTER*1 Specifies the option op(B): = 'N': op(B) = B (No transpose) = 'C': op(B) = B**H (Conjugate transpose)``` [in] ISGN ``` ISGN is INTEGER Specifies the sign in the equation: = +1: solve op(A)*X + X*op(B) = scale*C = -1: solve op(A)*X - X*op(B) = scale*C``` [in] M ``` M is INTEGER The order of the matrix A, and the number of rows in the matrices X and C. M >= 0.``` [in] N ``` N is INTEGER The order of the matrix B, and the number of columns in the matrices X and C. N >= 0.``` [in] A ``` A is COMPLEX array, dimension (LDA,M) The upper triangular matrix A.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [in] B ``` B is COMPLEX array, dimension (LDB,N) The upper triangular matrix B.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [in,out] C ``` C is COMPLEX array, dimension (LDC,N) On entry, the M-by-N right hand side matrix C. On exit, C is overwritten by the solution matrix X.``` [in] LDC ``` LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M)``` [out] SCALE ``` SCALE is REAL The scale factor, scale, set <= 1 to avoid overflow in X.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value = 1: A and B have common or very close eigenvalues; perturbed values were used to solve the equation (but the matrices A and B are unchanged).```

Definition at line 155 of file ctrsyl.f.

157 *
158 * -- LAPACK computational routine --
159 * -- LAPACK is a software package provided by Univ. of Tennessee, --
160 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
161 *
162 * .. Scalar Arguments ..
163  CHARACTER TRANA, TRANB
164  INTEGER INFO, ISGN, LDA, LDB, LDC, M, N
165  REAL SCALE
166 * ..
167 * .. Array Arguments ..
168  COMPLEX A( LDA, * ), B( LDB, * ), C( LDC, * )
169 * ..
170 *
171 * =====================================================================
172 *
173 * .. Parameters ..
174  REAL ONE
175  parameter( one = 1.0e+0 )
176 * ..
177 * .. Local Scalars ..
178  LOGICAL NOTRNA, NOTRNB
179  INTEGER J, K, L
180  REAL BIGNUM, DA11, DB, EPS, SCALOC, SGN, SMIN,
181  \$ SMLNUM
182  COMPLEX A11, SUML, SUMR, VEC, X11
183 * ..
184 * .. Local Arrays ..
185  REAL DUM( 1 )
186 * ..
187 * .. External Functions ..
188  LOGICAL LSAME
189  REAL CLANGE, SLAMCH
191  EXTERNAL lsame, clange, slamch, cdotc, cdotu, cladiv
192 * ..
193 * .. External Subroutines ..
195 * ..
196 * .. Intrinsic Functions ..
197  INTRINSIC abs, aimag, cmplx, conjg, max, min, real
198 * ..
199 * .. Executable Statements ..
200 *
201 * Decode and Test input parameters
202 *
203  notrna = lsame( trana, 'N' )
204  notrnb = lsame( tranb, 'N' )
205 *
206  info = 0
207  IF( .NOT.notrna .AND. .NOT.lsame( trana, 'C' ) ) THEN
208  info = -1
209  ELSE IF( .NOT.notrnb .AND. .NOT.lsame( tranb, 'C' ) ) THEN
210  info = -2
211  ELSE IF( isgn.NE.1 .AND. isgn.NE.-1 ) THEN
212  info = -3
213  ELSE IF( m.LT.0 ) THEN
214  info = -4
215  ELSE IF( n.LT.0 ) THEN
216  info = -5
217  ELSE IF( lda.LT.max( 1, m ) ) THEN
218  info = -7
219  ELSE IF( ldb.LT.max( 1, n ) ) THEN
220  info = -9
221  ELSE IF( ldc.LT.max( 1, m ) ) THEN
222  info = -11
223  END IF
224  IF( info.NE.0 ) THEN
225  CALL xerbla( 'CTRSYL', -info )
226  RETURN
227  END IF
228 *
229 * Quick return if possible
230 *
231  scale = one
232  IF( m.EQ.0 .OR. n.EQ.0 )
233  \$ RETURN
234 *
235 * Set constants to control overflow
236 *
237  eps = slamch( 'P' )
238  smlnum = slamch( 'S' )
239  bignum = one / smlnum
240  CALL slabad( smlnum, bignum )
241  smlnum = smlnum*real( m*n ) / eps
242  bignum = one / smlnum
243  smin = max( smlnum, eps*clange( 'M', m, m, a, lda, dum ),
244  \$ eps*clange( 'M', n, n, b, ldb, dum ) )
245  sgn = isgn
246 *
247  IF( notrna .AND. notrnb ) THEN
248 *
249 * Solve A*X + ISGN*X*B = scale*C.
250 *
251 * The (K,L)th block of X is determined starting from
252 * bottom-left corner column by column by
253 *
254 * A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
255 *
256 * Where
257 * M L-1
258 * R(K,L) = SUM [A(K,I)*X(I,L)] +ISGN*SUM [X(K,J)*B(J,L)].
259 * I=K+1 J=1
260 *
261  DO 30 l = 1, n
262  DO 20 k = m, 1, -1
263 *
264  suml = cdotu( m-k, a( k, min( k+1, m ) ), lda,
265  \$ c( min( k+1, m ), l ), 1 )
266  sumr = cdotu( l-1, c( k, 1 ), ldc, b( 1, l ), 1 )
267  vec = c( k, l ) - ( suml+sgn*sumr )
268 *
269  scaloc = one
270  a11 = a( k, k ) + sgn*b( l, l )
271  da11 = abs( real( a11 ) ) + abs( aimag( a11 ) )
272  IF( da11.LE.smin ) THEN
273  a11 = smin
274  da11 = smin
275  info = 1
276  END IF
277  db = abs( real( vec ) ) + abs( aimag( vec ) )
278  IF( da11.LT.one .AND. db.GT.one ) THEN
279  IF( db.GT.bignum*da11 )
280  \$ scaloc = one / db
281  END IF
282  x11 = cladiv( vec*cmplx( scaloc ), a11 )
283 *
284  IF( scaloc.NE.one ) THEN
285  DO 10 j = 1, n
286  CALL csscal( m, scaloc, c( 1, j ), 1 )
287  10 CONTINUE
288  scale = scale*scaloc
289  END IF
290  c( k, l ) = x11
291 *
292  20 CONTINUE
293  30 CONTINUE
294 *
295  ELSE IF( .NOT.notrna .AND. notrnb ) THEN
296 *
297 * Solve A**H *X + ISGN*X*B = scale*C.
298 *
299 * The (K,L)th block of X is determined starting from
300 * upper-left corner column by column by
301 *
302 * A**H(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
303 *
304 * Where
305 * K-1 L-1
306 * R(K,L) = SUM [A**H(I,K)*X(I,L)] + ISGN*SUM [X(K,J)*B(J,L)]
307 * I=1 J=1
308 *
309  DO 60 l = 1, n
310  DO 50 k = 1, m
311 *
312  suml = cdotc( k-1, a( 1, k ), 1, c( 1, l ), 1 )
313  sumr = cdotu( l-1, c( k, 1 ), ldc, b( 1, l ), 1 )
314  vec = c( k, l ) - ( suml+sgn*sumr )
315 *
316  scaloc = one
317  a11 = conjg( a( k, k ) ) + sgn*b( l, l )
318  da11 = abs( real( a11 ) ) + abs( aimag( a11 ) )
319  IF( da11.LE.smin ) THEN
320  a11 = smin
321  da11 = smin
322  info = 1
323  END IF
324  db = abs( real( vec ) ) + abs( aimag( vec ) )
325  IF( da11.LT.one .AND. db.GT.one ) THEN
326  IF( db.GT.bignum*da11 )
327  \$ scaloc = one / db
328  END IF
329 *
330  x11 = cladiv( vec*cmplx( scaloc ), a11 )
331 *
332  IF( scaloc.NE.one ) THEN
333  DO 40 j = 1, n
334  CALL csscal( m, scaloc, c( 1, j ), 1 )
335  40 CONTINUE
336  scale = scale*scaloc
337  END IF
338  c( k, l ) = x11
339 *
340  50 CONTINUE
341  60 CONTINUE
342 *
343  ELSE IF( .NOT.notrna .AND. .NOT.notrnb ) THEN
344 *
345 * Solve A**H*X + ISGN*X*B**H = C.
346 *
347 * The (K,L)th block of X is determined starting from
348 * upper-right corner column by column by
349 *
350 * A**H(K,K)*X(K,L) + ISGN*X(K,L)*B**H(L,L) = C(K,L) - R(K,L)
351 *
352 * Where
353 * K-1
354 * R(K,L) = SUM [A**H(I,K)*X(I,L)] +
355 * I=1
356 * N
357 * ISGN*SUM [X(K,J)*B**H(L,J)].
358 * J=L+1
359 *
360  DO 90 l = n, 1, -1
361  DO 80 k = 1, m
362 *
363  suml = cdotc( k-1, a( 1, k ), 1, c( 1, l ), 1 )
364  sumr = cdotc( n-l, c( k, min( l+1, n ) ), ldc,
365  \$ b( l, min( l+1, n ) ), ldb )
366  vec = c( k, l ) - ( suml+sgn*conjg( sumr ) )
367 *
368  scaloc = one
369  a11 = conjg( a( k, k )+sgn*b( l, l ) )
370  da11 = abs( real( a11 ) ) + abs( aimag( a11 ) )
371  IF( da11.LE.smin ) THEN
372  a11 = smin
373  da11 = smin
374  info = 1
375  END IF
376  db = abs( real( vec ) ) + abs( aimag( vec ) )
377  IF( da11.LT.one .AND. db.GT.one ) THEN
378  IF( db.GT.bignum*da11 )
379  \$ scaloc = one / db
380  END IF
381 *
382  x11 = cladiv( vec*cmplx( scaloc ), a11 )
383 *
384  IF( scaloc.NE.one ) THEN
385  DO 70 j = 1, n
386  CALL csscal( m, scaloc, c( 1, j ), 1 )
387  70 CONTINUE
388  scale = scale*scaloc
389  END IF
390  c( k, l ) = x11
391 *
392  80 CONTINUE
393  90 CONTINUE
394 *
395  ELSE IF( notrna .AND. .NOT.notrnb ) THEN
396 *
397 * Solve A*X + ISGN*X*B**H = C.
398 *
399 * The (K,L)th block of X is determined starting from
400 * bottom-left corner column by column by
401 *
402 * A(K,K)*X(K,L) + ISGN*X(K,L)*B**H(L,L) = C(K,L) - R(K,L)
403 *
404 * Where
405 * M N
406 * R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B**H(L,J)]
407 * I=K+1 J=L+1
408 *
409  DO 120 l = n, 1, -1
410  DO 110 k = m, 1, -1
411 *
412  suml = cdotu( m-k, a( k, min( k+1, m ) ), lda,
413  \$ c( min( k+1, m ), l ), 1 )
414  sumr = cdotc( n-l, c( k, min( l+1, n ) ), ldc,
415  \$ b( l, min( l+1, n ) ), ldb )
416  vec = c( k, l ) - ( suml+sgn*conjg( sumr ) )
417 *
418  scaloc = one
419  a11 = a( k, k ) + sgn*conjg( b( l, l ) )
420  da11 = abs( real( a11 ) ) + abs( aimag( a11 ) )
421  IF( da11.LE.smin ) THEN
422  a11 = smin
423  da11 = smin
424  info = 1
425  END IF
426  db = abs( real( vec ) ) + abs( aimag( vec ) )
427  IF( da11.LT.one .AND. db.GT.one ) THEN
428  IF( db.GT.bignum*da11 )
429  \$ scaloc = one / db
430  END IF
431 *
432  x11 = cladiv( vec*cmplx( scaloc ), a11 )
433 *
434  IF( scaloc.NE.one ) THEN
435  DO 100 j = 1, n
436  CALL csscal( m, scaloc, c( 1, j ), 1 )
437  100 CONTINUE
438  scale = scale*scaloc
439  END IF
440  c( k, l ) = x11
441 *
442  110 CONTINUE
443  120 CONTINUE
444 *
445  END IF
446 *
447  RETURN
448 *
449 * End of CTRSYL
450 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
complex function cdotc(N, CX, INCX, CY, INCY)
CDOTC
Definition: cdotc.f:83
subroutine csscal(N, SA, CX, INCX)
CSSCAL
Definition: csscal.f:78
complex function cdotu(N, CX, INCX, CY, INCY)
CDOTU
Definition: cdotu.f:83
real function clange(NORM, M, N, A, LDA, WORK)
CLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: clange.f:115