 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ ctgsy2()

 subroutine ctgsy2 ( character TRANS, integer IJOB, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( ldd, * ) D, integer LDD, complex, dimension( lde, * ) E, integer LDE, complex, dimension( ldf, * ) F, integer LDF, real SCALE, real RDSUM, real RDSCAL, integer INFO )

CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Purpose:
``` CTGSY2 solves the generalized Sylvester equation

A * R - L * B = scale *  C               (1)
D * R - L * E = scale * F

using Level 1 and 2 BLAS, where R and L are unknown M-by-N matrices,
(A, D), (B, E) and (C, F) are given matrix pairs of size M-by-M,
N-by-N and M-by-N, respectively. A, B, D and E are upper triangular
(i.e., (A,D) and (B,E) in generalized Schur form).

The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
scaling factor chosen to avoid overflow.

In matrix notation solving equation (1) corresponds to solve
Zx = scale * b, where Z is defined as

Z = [ kron(In, A)  -kron(B**H, Im) ]             (2)
[ kron(In, D)  -kron(E**H, Im) ],

Ik is the identity matrix of size k and X**H is the transpose of X.
kron(X, Y) is the Kronecker product between the matrices X and Y.

If TRANS = 'C', y in the conjugate transposed system Z**H*y = scale*b
is solved for, which is equivalent to solve for R and L in

A**H * R  + D**H * L   = scale * C           (3)
R  * B**H + L  * E**H  = scale * -F

This case is used to compute an estimate of Dif[(A, D), (B, E)] =
= sigma_min(Z) using reverse communication with CLACON.

CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL
of an upper bound on the separation between to matrix pairs. Then
the input (A, D), (B, E) are sub-pencils of two matrix pairs in
CTGSYL.```
Parameters
 [in] TRANS ``` TRANS is CHARACTER*1 = 'N': solve the generalized Sylvester equation (1). = 'T': solve the 'transposed' system (3).``` [in] IJOB ``` IJOB is INTEGER Specifies what kind of functionality to be performed. = 0: solve (1) only. = 1: A contribution from this subsystem to a Frobenius norm-based estimate of the separation between two matrix pairs is computed. (look ahead strategy is used). = 2: A contribution from this subsystem to a Frobenius norm-based estimate of the separation between two matrix pairs is computed. (SGECON on sub-systems is used.) Not referenced if TRANS = 'T'.``` [in] M ``` M is INTEGER On entry, M specifies the order of A and D, and the row dimension of C, F, R and L.``` [in] N ``` N is INTEGER On entry, N specifies the order of B and E, and the column dimension of C, F, R and L.``` [in] A ``` A is COMPLEX array, dimension (LDA, M) On entry, A contains an upper triangular matrix.``` [in] LDA ``` LDA is INTEGER The leading dimension of the matrix A. LDA >= max(1, M).``` [in] B ``` B is COMPLEX array, dimension (LDB, N) On entry, B contains an upper triangular matrix.``` [in] LDB ``` LDB is INTEGER The leading dimension of the matrix B. LDB >= max(1, N).``` [in,out] C ``` C is COMPLEX array, dimension (LDC, N) On entry, C contains the right-hand-side of the first matrix equation in (1). On exit, if IJOB = 0, C has been overwritten by the solution R.``` [in] LDC ``` LDC is INTEGER The leading dimension of the matrix C. LDC >= max(1, M).``` [in] D ``` D is COMPLEX array, dimension (LDD, M) On entry, D contains an upper triangular matrix.``` [in] LDD ``` LDD is INTEGER The leading dimension of the matrix D. LDD >= max(1, M).``` [in] E ``` E is COMPLEX array, dimension (LDE, N) On entry, E contains an upper triangular matrix.``` [in] LDE ``` LDE is INTEGER The leading dimension of the matrix E. LDE >= max(1, N).``` [in,out] F ``` F is COMPLEX array, dimension (LDF, N) On entry, F contains the right-hand-side of the second matrix equation in (1). On exit, if IJOB = 0, F has been overwritten by the solution L.``` [in] LDF ``` LDF is INTEGER The leading dimension of the matrix F. LDF >= max(1, M).``` [out] SCALE ``` SCALE is REAL On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions R and L (C and F on entry) will hold the solutions to a slightly perturbed system but the input matrices A, B, D and E have not been changed. If SCALE = 0, R and L will hold the solutions to the homogeneous system with C = F = 0. Normally, SCALE = 1.``` [in,out] RDSUM ``` RDSUM is REAL On entry, the sum of squares of computed contributions to the Dif-estimate under computation by CTGSYL, where the scaling factor RDSCAL (see below) has been factored out. On exit, the corresponding sum of squares updated with the contributions from the current sub-system. If TRANS = 'T' RDSUM is not touched. NOTE: RDSUM only makes sense when CTGSY2 is called by CTGSYL.``` [in,out] RDSCAL ``` RDSCAL is REAL On entry, scaling factor used to prevent overflow in RDSUM. On exit, RDSCAL is updated w.r.t. the current contributions in RDSUM. If TRANS = 'T', RDSCAL is not touched. NOTE: RDSCAL only makes sense when CTGSY2 is called by CTGSYL.``` [out] INFO ``` INFO is INTEGER On exit, if INFO is set to =0: Successful exit <0: If INFO = -i, input argument number i is illegal. >0: The matrix pairs (A, D) and (B, E) have common or very close eigenvalues.```
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

Definition at line 256 of file ctgsy2.f.

259 *
260 * -- LAPACK auxiliary routine --
261 * -- LAPACK is a software package provided by Univ. of Tennessee, --
262 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
263 *
264 * .. Scalar Arguments ..
265  CHARACTER TRANS
266  INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
267  REAL RDSCAL, RDSUM, SCALE
268 * ..
269 * .. Array Arguments ..
270  COMPLEX A( LDA, * ), B( LDB, * ), C( LDC, * ),
271  \$ D( LDD, * ), E( LDE, * ), F( LDF, * )
272 * ..
273 *
274 * =====================================================================
275 *
276 * .. Parameters ..
277  REAL ZERO, ONE
278  INTEGER LDZ
279  parameter( zero = 0.0e+0, one = 1.0e+0, ldz = 2 )
280 * ..
281 * .. Local Scalars ..
282  LOGICAL NOTRAN
283  INTEGER I, IERR, J, K
284  REAL SCALOC
285  COMPLEX ALPHA
286 * ..
287 * .. Local Arrays ..
288  INTEGER IPIV( LDZ ), JPIV( LDZ )
289  COMPLEX RHS( LDZ ), Z( LDZ, LDZ )
290 * ..
291 * .. External Functions ..
292  LOGICAL LSAME
293  EXTERNAL lsame
294 * ..
295 * .. External Subroutines ..
296  EXTERNAL caxpy, cgesc2, cgetc2, cscal, clatdf, xerbla
297 * ..
298 * .. Intrinsic Functions ..
299  INTRINSIC cmplx, conjg, max
300 * ..
301 * .. Executable Statements ..
302 *
303 * Decode and test input parameters
304 *
305  info = 0
306  ierr = 0
307  notran = lsame( trans, 'N' )
308  IF( .NOT.notran .AND. .NOT.lsame( trans, 'C' ) ) THEN
309  info = -1
310  ELSE IF( notran ) THEN
311  IF( ( ijob.LT.0 ) .OR. ( ijob.GT.2 ) ) THEN
312  info = -2
313  END IF
314  END IF
315  IF( info.EQ.0 ) THEN
316  IF( m.LE.0 ) THEN
317  info = -3
318  ELSE IF( n.LE.0 ) THEN
319  info = -4
320  ELSE IF( lda.LT.max( 1, m ) ) THEN
321  info = -6
322  ELSE IF( ldb.LT.max( 1, n ) ) THEN
323  info = -8
324  ELSE IF( ldc.LT.max( 1, m ) ) THEN
325  info = -10
326  ELSE IF( ldd.LT.max( 1, m ) ) THEN
327  info = -12
328  ELSE IF( lde.LT.max( 1, n ) ) THEN
329  info = -14
330  ELSE IF( ldf.LT.max( 1, m ) ) THEN
331  info = -16
332  END IF
333  END IF
334  IF( info.NE.0 ) THEN
335  CALL xerbla( 'CTGSY2', -info )
336  RETURN
337  END IF
338 *
339  IF( notran ) THEN
340 *
341 * Solve (I, J) - system
342 * A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J)
343 * D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J)
344 * for I = M, M - 1, ..., 1; J = 1, 2, ..., N
345 *
346  scale = one
347  scaloc = one
348  DO 30 j = 1, n
349  DO 20 i = m, 1, -1
350 *
351 * Build 2 by 2 system
352 *
353  z( 1, 1 ) = a( i, i )
354  z( 2, 1 ) = d( i, i )
355  z( 1, 2 ) = -b( j, j )
356  z( 2, 2 ) = -e( j, j )
357 *
358 * Set up right hand side(s)
359 *
360  rhs( 1 ) = c( i, j )
361  rhs( 2 ) = f( i, j )
362 *
363 * Solve Z * x = RHS
364 *
365  CALL cgetc2( ldz, z, ldz, ipiv, jpiv, ierr )
366  IF( ierr.GT.0 )
367  \$ info = ierr
368  IF( ijob.EQ.0 ) THEN
369  CALL cgesc2( ldz, z, ldz, rhs, ipiv, jpiv, scaloc )
370  IF( scaloc.NE.one ) THEN
371  DO 10 k = 1, n
372  CALL cscal( m, cmplx( scaloc, zero ), c( 1, k ),
373  \$ 1 )
374  CALL cscal( m, cmplx( scaloc, zero ), f( 1, k ),
375  \$ 1 )
376  10 CONTINUE
377  scale = scale*scaloc
378  END IF
379  ELSE
380  CALL clatdf( ijob, ldz, z, ldz, rhs, rdsum, rdscal,
381  \$ ipiv, jpiv )
382  END IF
383 *
384 * Unpack solution vector(s)
385 *
386  c( i, j ) = rhs( 1 )
387  f( i, j ) = rhs( 2 )
388 *
389 * Substitute R(I, J) and L(I, J) into remaining equation.
390 *
391  IF( i.GT.1 ) THEN
392  alpha = -rhs( 1 )
393  CALL caxpy( i-1, alpha, a( 1, i ), 1, c( 1, j ), 1 )
394  CALL caxpy( i-1, alpha, d( 1, i ), 1, f( 1, j ), 1 )
395  END IF
396  IF( j.LT.n ) THEN
397  CALL caxpy( n-j, rhs( 2 ), b( j, j+1 ), ldb,
398  \$ c( i, j+1 ), ldc )
399  CALL caxpy( n-j, rhs( 2 ), e( j, j+1 ), lde,
400  \$ f( i, j+1 ), ldf )
401  END IF
402 *
403  20 CONTINUE
404  30 CONTINUE
405  ELSE
406 *
407 * Solve transposed (I, J) - system:
408 * A(I, I)**H * R(I, J) + D(I, I)**H * L(J, J) = C(I, J)
409 * R(I, I) * B(J, J) + L(I, J) * E(J, J) = -F(I, J)
410 * for I = 1, 2, ..., M, J = N, N - 1, ..., 1
411 *
412  scale = one
413  scaloc = one
414  DO 80 i = 1, m
415  DO 70 j = n, 1, -1
416 *
417 * Build 2 by 2 system Z**H
418 *
419  z( 1, 1 ) = conjg( a( i, i ) )
420  z( 2, 1 ) = -conjg( b( j, j ) )
421  z( 1, 2 ) = conjg( d( i, i ) )
422  z( 2, 2 ) = -conjg( e( j, j ) )
423 *
424 *
425 * Set up right hand side(s)
426 *
427  rhs( 1 ) = c( i, j )
428  rhs( 2 ) = f( i, j )
429 *
430 * Solve Z**H * x = RHS
431 *
432  CALL cgetc2( ldz, z, ldz, ipiv, jpiv, ierr )
433  IF( ierr.GT.0 )
434  \$ info = ierr
435  CALL cgesc2( ldz, z, ldz, rhs, ipiv, jpiv, scaloc )
436  IF( scaloc.NE.one ) THEN
437  DO 40 k = 1, n
438  CALL cscal( m, cmplx( scaloc, zero ), c( 1, k ),
439  \$ 1 )
440  CALL cscal( m, cmplx( scaloc, zero ), f( 1, k ),
441  \$ 1 )
442  40 CONTINUE
443  scale = scale*scaloc
444  END IF
445 *
446 * Unpack solution vector(s)
447 *
448  c( i, j ) = rhs( 1 )
449  f( i, j ) = rhs( 2 )
450 *
451 * Substitute R(I, J) and L(I, J) into remaining equation.
452 *
453  DO 50 k = 1, j - 1
454  f( i, k ) = f( i, k ) + rhs( 1 )*conjg( b( k, j ) ) +
455  \$ rhs( 2 )*conjg( e( k, j ) )
456  50 CONTINUE
457  DO 60 k = i + 1, m
458  c( k, j ) = c( k, j ) - conjg( a( i, k ) )*rhs( 1 ) -
459  \$ conjg( d( i, k ) )*rhs( 2 )
460  60 CONTINUE
461 *
462  70 CONTINUE
463  80 CONTINUE
464  END IF
465  RETURN
466 *
467 * End of CTGSY2
468 *
logical function lde(RI, RJ, LR)
Definition: dblat2.f:2942
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:88
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:78
subroutine cgetc2(N, A, LDA, IPIV, JPIV, INFO)
CGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.
Definition: cgetc2.f:111
subroutine cgesc2(N, A, LDA, RHS, IPIV, JPIV, SCALE)
CGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed...
Definition: cgesc2.f:115
subroutine clatdf(IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, JPIV)
CLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution ...
Definition: clatdf.f:169
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