*> \brief \b ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm). * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZTGSY2 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZTGSY2( 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 * DOUBLE PRECISION RDSCAL, RDSUM, SCALE * .. * .. Array Arguments .. * COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ), * $ D( LDD, * ), E( LDE, * ), F( LDF, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZTGSY2 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 conjuguate 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 communicaton with ZLACON. *> *> ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL *> 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 *> ZTGSYL. *> \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. (DGECON 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*16 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*16 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*16 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*16 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*16 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*16 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 DOUBLE PRECISION *> 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 DOUBLE PRECISION *> On entry, the sum of squares of computed contributions to *> the Dif-estimate under computation by ZTGSYL, 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 ZTGSY2 is called by *> ZTGSYL. *> \endverbatim *> *> \param[in,out] RDSCAL *> \verbatim *> RDSCAL is DOUBLE PRECISION *> 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 ZTGSY2 is called by *> ZTGSYL. *> \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. * *> \date December 2016 * *> \ingroup complex16SYauxiliary * *> \par Contributors: * ================== *> *> Bo Kagstrom and Peter Poromaa, Department of Computing Science, *> Umea University, S-901 87 Umea, Sweden. * * ===================================================================== SUBROUTINE ZTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, $ LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, $ INFO ) * * -- LAPACK auxiliary routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. CHARACTER TRANS INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N DOUBLE PRECISION RDSCAL, RDSUM, SCALE * .. * .. Array Arguments .. COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ), $ D( LDD, * ), E( LDE, * ), F( LDF, * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE INTEGER LDZ PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, LDZ = 2 ) * .. * .. Local Scalars .. LOGICAL NOTRAN INTEGER I, IERR, J, K DOUBLE PRECISION SCALOC COMPLEX*16 ALPHA * .. * .. Local Arrays .. INTEGER IPIV( LDZ ), JPIV( LDZ ) COMPLEX*16 RHS( LDZ ), Z( LDZ, LDZ ) * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL XERBLA, ZAXPY, ZGESC2, ZGETC2, ZLATDF, ZSCAL * .. * .. Intrinsic Functions .. INTRINSIC DCMPLX, DCONJG, 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( 'ZTGSY2', -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 ZGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) $ INFO = IERR IF( IJOB.EQ.0 ) THEN CALL ZGESC2( LDZ, Z, LDZ, RHS, IPIV, JPIV, SCALOC ) IF( SCALOC.NE.ONE ) THEN DO 10 K = 1, N CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ), $ C( 1, K ), 1 ) CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ), $ F( 1, K ), 1 ) 10 CONTINUE SCALE = SCALE*SCALOC END IF ELSE CALL ZLATDF( 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 ZAXPY( I-1, ALPHA, A( 1, I ), 1, C( 1, J ), 1 ) CALL ZAXPY( I-1, ALPHA, D( 1, I ), 1, F( 1, J ), 1 ) END IF IF( J.LT.N ) THEN CALL ZAXPY( N-J, RHS( 2 ), B( J, J+1 ), LDB, $ C( I, J+1 ), LDC ) CALL ZAXPY( 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 ) = DCONJG( A( I, I ) ) Z( 2, 1 ) = -DCONJG( B( J, J ) ) Z( 1, 2 ) = DCONJG( D( I, I ) ) Z( 2, 2 ) = -DCONJG( 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 ZGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR ) IF( IERR.GT.0 ) $ INFO = IERR CALL ZGESC2( LDZ, Z, LDZ, RHS, IPIV, JPIV, SCALOC ) IF( SCALOC.NE.ONE ) THEN DO 40 K = 1, N CALL ZSCAL( M, DCMPLX( SCALOC, ZERO ), C( 1, K ), $ 1 ) CALL ZSCAL( M, DCMPLX( 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 )*DCONJG( B( K, J ) ) + $ RHS( 2 )*DCONJG( E( K, J ) ) 50 CONTINUE DO 60 K = I + 1, M C( K, J ) = C( K, J ) - DCONJG( A( I, K ) )*RHS( 1 ) - $ DCONJG( D( I, K ) )*RHS( 2 ) 60 CONTINUE * 70 CONTINUE 80 CONTINUE END IF RETURN * * End of ZTGSY2 * END