*> \brief \b CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE CTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
* LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER TRANS
* INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
* REAL RDSCAL, RDSUM, SCALE
* ..
* .. Array Arguments ..
* COMPLEX A( LDA, * ), B( LDB, * ), C( LDC, * ),
* $ D( LDD, * ), E( LDE, * ), F( LDF, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> = 'N': solve the generalized Sylvester equation (1).
*> = 'T': solve the 'transposed' system (3).
*> \endverbatim
*>
*> \param[in] IJOB
*> \verbatim
*> 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'.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> On entry, M specifies the order of A and D, and the row
*> dimension of C, F, R and L.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, N specifies the order of B and E, and the column
*> dimension of C, F, R and L.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA, M)
*> On entry, A contains an upper triangular matrix.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the matrix A. LDA >= max(1, M).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is COMPLEX array, dimension (LDB, N)
*> On entry, B contains an upper triangular matrix.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the matrix B. LDB >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] C
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the matrix C. LDC >= max(1, M).
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is COMPLEX array, dimension (LDD, M)
*> On entry, D contains an upper triangular matrix.
*> \endverbatim
*>
*> \param[in] LDD
*> \verbatim
*> LDD is INTEGER
*> The leading dimension of the matrix D. LDD >= max(1, M).
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is COMPLEX array, dimension (LDE, N)
*> On entry, E contains an upper triangular matrix.
*> \endverbatim
*>
*> \param[in] LDE
*> \verbatim
*> LDE is INTEGER
*> The leading dimension of the matrix E. LDE >= max(1, N).
*> \endverbatim
*>
*> \param[in,out] F
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in] LDF
*> \verbatim
*> LDF is INTEGER
*> The leading dimension of the matrix F. LDF >= max(1, M).
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in,out] RDSUM
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in,out] RDSCAL
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> 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.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup tgsy2
*
*> \par Contributors:
* ==================
*>
*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*> Umea University, S-901 87 Umea, Sweden.
*
* =====================================================================
SUBROUTINE CTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC,
$ D,
$ LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
$ INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
REAL RDSCAL, RDSUM, SCALE
* ..
* .. Array Arguments ..
COMPLEX A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), E( LDE, * ), F( LDF, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
INTEGER LDZ
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, LDZ = 2 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN
INTEGER I, IERR, J, K
REAL SCALOC
COMPLEX ALPHA
* ..
* .. Local Arrays ..
INTEGER IPIV( LDZ ), JPIV( LDZ )
COMPLEX RHS( LDZ ), Z( LDZ, LDZ )
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL CAXPY, CGESC2, CGETC2, CSCAL, CLATDF,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC CMPLX, CONJG, MAX
* ..
* .. Executable Statements ..
*
* Decode and test input parameters
*
INFO = 0
IERR = 0
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'C' ) ) THEN
INFO = -1
ELSE IF( NOTRAN ) THEN
IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.2 ) ) THEN
INFO = -2
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( M.LE.0 ) THEN
INFO = -3
ELSE IF( N.LE.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
ELSE IF( LDD.LT.MAX( 1, M ) ) THEN
INFO = -12
ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
INFO = -14
ELSE IF( LDF.LT.MAX( 1, M ) ) THEN
INFO = -16
END IF
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CTGSY2', -INFO )
RETURN
END IF
*
IF( NOTRAN ) THEN
*
* Solve (I, J) - system
* A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J)
* D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J)
* for I = M, M - 1, ..., 1; J = 1, 2, ..., N
*
SCALE = ONE
SCALOC = ONE
DO 30 J = 1, N
DO 20 I = M, 1, -1
*
* Build 2 by 2 system
*
Z( 1, 1 ) = A( I, I )
Z( 2, 1 ) = D( I, I )
Z( 1, 2 ) = -B( J, J )
Z( 2, 2 ) = -E( J, J )
*
* Set up right hand side(s)
*
RHS( 1 ) = C( I, J )
RHS( 2 ) = F( I, J )
*
* Solve Z * x = RHS
*
CALL CGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
IF( IJOB.EQ.0 ) THEN
CALL CGESC2( LDZ, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 10 K = 1, N
CALL CSCAL( M, CMPLX( SCALOC, ZERO ), C( 1,
$ K ),
$ 1 )
CALL CSCAL( M, CMPLX( SCALOC, ZERO ), F( 1,
$ K ),
$ 1 )
10 CONTINUE
SCALE = SCALE*SCALOC
END IF
ELSE
CALL CLATDF( IJOB, LDZ, Z, LDZ, RHS, RDSUM, RDSCAL,
$ IPIV, JPIV )
END IF
*
* Unpack solution vector(s)
*
C( I, J ) = RHS( 1 )
F( I, J ) = RHS( 2 )
*
* Substitute R(I, J) and L(I, J) into remaining equation.
*
IF( I.GT.1 ) THEN
ALPHA = -RHS( 1 )
CALL CAXPY( I-1, ALPHA, A( 1, I ), 1, C( 1, J ),
$ 1 )
CALL CAXPY( I-1, ALPHA, D( 1, I ), 1, F( 1, J ),
$ 1 )
END IF
IF( J.LT.N ) THEN
CALL CAXPY( N-J, RHS( 2 ), B( J, J+1 ), LDB,
$ C( I, J+1 ), LDC )
CALL CAXPY( N-J, RHS( 2 ), E( J, J+1 ), LDE,
$ F( I, J+1 ), LDF )
END IF
*
20 CONTINUE
30 CONTINUE
ELSE
*
* Solve transposed (I, J) - system:
* A(I, I)**H * R(I, J) + D(I, I)**H * L(J, J) = C(I, J)
* R(I, I) * B(J, J) + L(I, J) * E(J, J) = -F(I, J)
* for I = 1, 2, ..., M, J = N, N - 1, ..., 1
*
SCALE = ONE
SCALOC = ONE
DO 80 I = 1, M
DO 70 J = N, 1, -1
*
* Build 2 by 2 system Z**H
*
Z( 1, 1 ) = CONJG( A( I, I ) )
Z( 2, 1 ) = -CONJG( B( J, J ) )
Z( 1, 2 ) = CONJG( D( I, I ) )
Z( 2, 2 ) = -CONJG( E( J, J ) )
*
*
* Set up right hand side(s)
*
RHS( 1 ) = C( I, J )
RHS( 2 ) = F( I, J )
*
* Solve Z**H * x = RHS
*
CALL CGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR )
IF( IERR.GT.0 )
$ INFO = IERR
CALL CGESC2( LDZ, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
IF( SCALOC.NE.ONE ) THEN
DO 40 K = 1, N
CALL CSCAL( M, CMPLX( SCALOC, ZERO ), C( 1, K ),
$ 1 )
CALL CSCAL( M, CMPLX( SCALOC, ZERO ), F( 1, K ),
$ 1 )
40 CONTINUE
SCALE = SCALE*SCALOC
END IF
*
* Unpack solution vector(s)
*
C( I, J ) = RHS( 1 )
F( I, J ) = RHS( 2 )
*
* Substitute R(I, J) and L(I, J) into remaining equation.
*
DO 50 K = 1, J - 1
F( I, K ) = F( I, K ) + RHS( 1 )*CONJG( B( K, J ) ) +
$ RHS( 2 )*CONJG( E( K, J ) )
50 CONTINUE
DO 60 K = I + 1, M
C( K, J ) = C( K, J ) - CONJG( A( I, K ) )*RHS( 1 ) -
$ CONJG( D( I, K ) )*RHS( 2 )
60 CONTINUE
*
70 CONTINUE
80 CONTINUE
END IF
RETURN
*
* End of CTGSY2
*
END