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version 1.8, 2011/07/22 07:38:21 version 1.9, 2011/11/21 20:43:22
Line 1 Line 1
   *> \brief \b ZTGSYL
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZTGSYL + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ztgsyl.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ztgsyl.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztgsyl.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
   *                          LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
   *                          IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          TRANS
   *       INTEGER            IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF,
   *      $                   LWORK, M, N
   *       DOUBLE PRECISION   DIF, SCALE
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IWORK( * )
   *       COMPLEX*16         A( LDA, * ), B( LDB, * ), C( LDC, * ),
   *      $                   D( LDD, * ), E( LDE, * ), F( LDF, * ),
   *      $                   WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZTGSYL solves the generalized Sylvester equation:
   *>
   *>             A * R - L * B = scale * C            (1)
   *>             D * R - L * E = scale * F
   *>
   *> 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, with complex entries. 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 (1) is equivalent 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) ],
   *>
   *> Here Ix is the identity matrix of size x and X**H is the conjugate
   *> 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 (TRANS = 'C') is used to compute an one-norm-based estimate
   *> of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
   *> and (B,E), using ZLACON.
   *>
   *> If IJOB >= 1, ZTGSYL computes a Frobenius norm-based estimate of
   *> Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
   *> reciprocal of the smallest singular value of Z.
   *>
   *> This is a level-3 BLAS algorithm.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] TRANS
   *> \verbatim
   *>          TRANS is CHARACTER*1
   *>          = 'N': solve the generalized sylvester equation (1).
   *>          = 'C': solve the "conjugate transposed" system (3).
   *> \endverbatim
   *>
   *> \param[in] IJOB
   *> \verbatim
   *>          IJOB is INTEGER
   *>          Specifies what kind of functionality to be performed.
   *>          =0: solve (1) only.
   *>          =1: The functionality of 0 and 3.
   *>          =2: The functionality of 0 and 4.
   *>          =3: Only an estimate of Dif[(A,D), (B,E)] is computed.
   *>              (look ahead strategy is used).
   *>          =4: Only an estimate of Dif[(A,D), (B,E)] is computed.
   *>              (ZGECON on sub-systems is used).
   *>          Not referenced if TRANS = 'C'.
   *> \endverbatim
   *>
   *> \param[in] M
   *> \verbatim
   *>          M is INTEGER
   *>          The order of the matrices A and D, and the row dimension of
   *>          the matrices C, F, R and L.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrices B and E, and the column dimension
   *>          of the matrices C, F, R and L.
   *> \endverbatim
   *>
   *> \param[in] A
   *> \verbatim
   *>          A is COMPLEX*16 array, dimension (LDA, M)
   *>          The upper triangular matrix A.
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A. LDA >= max(1, M).
   *> \endverbatim
   *>
   *> \param[in] B
   *> \verbatim
   *>          B is COMPLEX*16 array, dimension (LDB, N)
   *>          The upper triangular matrix B.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array 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) or (3).
   *>          On exit, if IJOB = 0, 1 or 2, C has been overwritten by
   *>          the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
   *>          the solution achieved during the computation of the
   *>          Dif-estimate.
   *> \endverbatim
   *>
   *> \param[in] LDC
   *> \verbatim
   *>          LDC is INTEGER
   *>          The leading dimension of the array C. LDC >= max(1, M).
   *> \endverbatim
   *>
   *> \param[in] D
   *> \verbatim
   *>          D is COMPLEX*16 array, dimension (LDD, M)
   *>          The upper triangular matrix D.
   *> \endverbatim
   *>
   *> \param[in] LDD
   *> \verbatim
   *>          LDD is INTEGER
   *>          The leading dimension of the array D. LDD >= max(1, M).
   *> \endverbatim
   *>
   *> \param[in] E
   *> \verbatim
   *>          E is COMPLEX*16 array, dimension (LDE, N)
   *>          The upper triangular matrix E.
   *> \endverbatim
   *>
   *> \param[in] LDE
   *> \verbatim
   *>          LDE is INTEGER
   *>          The leading dimension of the array 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) or (3).
   *>          On exit, if IJOB = 0, 1 or 2, F has been overwritten by
   *>          the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
   *>          the solution achieved during the computation of the
   *>          Dif-estimate.
   *> \endverbatim
   *>
   *> \param[in] LDF
   *> \verbatim
   *>          LDF is INTEGER
   *>          The leading dimension of the array F. LDF >= max(1, M).
   *> \endverbatim
   *>
   *> \param[out] DIF
   *> \verbatim
   *>          DIF is DOUBLE PRECISION
   *>          On exit DIF is the reciprocal of a lower bound of the
   *>          reciprocal of the Dif-function, i.e. DIF is an upper bound of
   *>          Dif[(A,D), (B,E)] = sigma-min(Z), where Z as in (2).
   *>          IF IJOB = 0 or TRANS = 'C', DIF is not referenced.
   *> \endverbatim
   *>
   *> \param[out] SCALE
   *> \verbatim
   *>          SCALE is DOUBLE PRECISION
   *>          On exit SCALE is the scaling factor in (1) or (3).
   *>          If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
   *>          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 homogenious system with C = F = 0.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
   *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          LWORK is INTEGER
   *>          The dimension of the array WORK. LWORK > = 1.
   *>          If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).
   *>
   *>          If LWORK = -1, then a workspace query is assumed; the routine
   *>          only calculates the optimal size of the WORK array, returns
   *>          this value as the first entry of the WORK array, and no error
   *>          message related to LWORK is issued by XERBLA.
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (M+N+2)
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>            =0: successful exit
   *>            <0: If INFO = -i, the i-th argument had an illegal value.
   *>            >0: (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 November 2011
   *
   *> \ingroup complex16SYcomputational
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
   *>     Umea University, S-901 87 Umea, Sweden.
   *
   *> \par References:
   *  ================
   *>
   *>  [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
   *>      for Solving the Generalized Sylvester Equation and Estimating the
   *>      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
   *>      Department of Computing Science, Umea University, S-901 87 Umea,
   *>      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
   *>      Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
   *>      No 1, 1996.
   *> \n
   *>  [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
   *>      Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
   *>      Appl., 15(4):1045-1060, 1994.
   *> \n
   *>  [3] B. Kagstrom and L. Westin, Generalized Schur Methods with
   *>      Condition Estimators for Solving the Generalized Sylvester
   *>      Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
   *>      July 1989, pp 745-751.
   *>
   *  =====================================================================
       SUBROUTINE ZTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,        SUBROUTINE ZTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
      $                   LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,       $                   LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
      $                   IWORK, INFO )       $                   IWORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.3.1) --  *  -- LAPACK computational routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *  -- April 2011                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          TRANS        CHARACTER          TRANS
Line 20 Line 313
      $                   WORK( * )       $                   WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZTGSYL solves the generalized Sylvester equation:  
 *  
 *              A * R - L * B = scale * C            (1)  
 *              D * R - L * E = scale * F  
 *  
 *  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, with complex entries. 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 (1) is equivalent 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) ],  
 *  
 *  Here Ix is the identity matrix of size x and X**H is the conjugate  
 *  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 (TRANS = 'C') is used to compute an one-norm-based estimate  
 *  of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)  
 *  and (B,E), using ZLACON.  
 *  
 *  If IJOB >= 1, ZTGSYL computes a Frobenius norm-based estimate of  
 *  Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the  
 *  reciprocal of the smallest singular value of Z.  
 *  
 *  This is a level-3 BLAS algorithm.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  TRANS   (input) CHARACTER*1  
 *          = 'N': solve the generalized sylvester equation (1).  
 *          = 'C': solve the "conjugate transposed" system (3).  
 *  
 *  IJOB    (input) INTEGER  
 *          Specifies what kind of functionality to be performed.  
 *          =0: solve (1) only.  
 *          =1: The functionality of 0 and 3.  
 *          =2: The functionality of 0 and 4.  
 *          =3: Only an estimate of Dif[(A,D), (B,E)] is computed.  
 *              (look ahead strategy is used).  
 *          =4: Only an estimate of Dif[(A,D), (B,E)] is computed.  
 *              (ZGECON on sub-systems is used).  
 *          Not referenced if TRANS = 'C'.  
 *  
 *  M       (input) INTEGER  
 *          The order of the matrices A and D, and the row dimension of  
 *          the matrices C, F, R and L.  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrices B and E, and the column dimension  
 *          of the matrices C, F, R and L.  
 *  
 *  A       (input) COMPLEX*16 array, dimension (LDA, M)  
 *          The upper triangular matrix A.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A. LDA >= max(1, M).  
 *  
 *  B       (input) COMPLEX*16 array, dimension (LDB, N)  
 *          The upper triangular matrix B.  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of the array B. LDB >= max(1, N).  
 *  
 *  C       (input/output) COMPLEX*16 array, dimension (LDC, N)  
 *          On entry, C contains the right-hand-side of the first matrix  
 *          equation in (1) or (3).  
 *          On exit, if IJOB = 0, 1 or 2, C has been overwritten by  
 *          the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,  
 *          the solution achieved during the computation of the  
 *          Dif-estimate.  
 *  
 *  LDC     (input) INTEGER  
 *          The leading dimension of the array C. LDC >= max(1, M).  
 *  
 *  D       (input) COMPLEX*16 array, dimension (LDD, M)  
 *          The upper triangular matrix D.  
 *  
 *  LDD     (input) INTEGER  
 *          The leading dimension of the array D. LDD >= max(1, M).  
 *  
 *  E       (input) COMPLEX*16 array, dimension (LDE, N)  
 *          The upper triangular matrix E.  
 *  
 *  LDE     (input) INTEGER  
 *          The leading dimension of the array E. LDE >= max(1, N).  
 *  
 *  F       (input/output) COMPLEX*16 array, dimension (LDF, N)  
 *          On entry, F contains the right-hand-side of the second matrix  
 *          equation in (1) or (3).  
 *          On exit, if IJOB = 0, 1 or 2, F has been overwritten by  
 *          the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,  
 *          the solution achieved during the computation of the  
 *          Dif-estimate.  
 *  
 *  LDF     (input) INTEGER  
 *          The leading dimension of the array F. LDF >= max(1, M).  
 *  
 *  DIF     (output) DOUBLE PRECISION  
 *          On exit DIF is the reciprocal of a lower bound of the  
 *          reciprocal of the Dif-function, i.e. DIF is an upper bound of  
 *          Dif[(A,D), (B,E)] = sigma-min(Z), where Z as in (2).  
 *          IF IJOB = 0 or TRANS = 'C', DIF is not referenced.  
 *  
 *  SCALE   (output) DOUBLE PRECISION  
 *          On exit SCALE is the scaling factor in (1) or (3).  
 *          If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,  
 *          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 homogenious system with C = F = 0.  
 *  
 *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))  
 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.  
 *  
 *  LWORK   (input) INTEGER  
 *          The dimension of the array WORK. LWORK > = 1.  
 *          If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).  
 *  
 *          If LWORK = -1, then a workspace query is assumed; the routine  
 *          only calculates the optimal size of the WORK array, returns  
 *          this value as the first entry of the WORK array, and no error  
 *          message related to LWORK is issued by XERBLA.  
 *  
 *  IWORK   (workspace) INTEGER array, dimension (M+N+2)  
 *  
 *  INFO    (output) INTEGER  
 *            =0: successful exit  
 *            <0: If INFO = -i, the i-th argument had an illegal value.  
 *            >0: (A, D) and (B, E) have common or very close  
 *                eigenvalues.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  Based on contributions by  
 *     Bo Kagstrom and Peter Poromaa, Department of Computing Science,  
 *     Umea University, S-901 87 Umea, Sweden.  
 *  
 *  [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software  
 *      for Solving the Generalized Sylvester Equation and Estimating the  
 *      Separation between Regular Matrix Pairs, Report UMINF - 93.23,  
 *      Department of Computing Science, Umea University, S-901 87 Umea,  
 *      Sweden, December 1993, Revised April 1994, Also as LAPACK Working  
 *      Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,  
 *      No 1, 1996.  
 *  
 *  [2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester  
 *      Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.  
 *      Appl., 15(4):1045-1060, 1994.  
 *  
 *  [3] B. Kagstrom and L. Westin, Generalized Schur Methods with  
 *      Condition Estimators for Solving the Generalized Sylvester  
 *      Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,  
 *      July 1989, pp 745-751.  
 *  
 *  =====================================================================  *  =====================================================================
 *  Replaced various illegal calls to CCOPY by calls to CLASET.  *  Replaced various illegal calls to CCOPY by calls to CLASET.
 *  Sven Hammarling, 1/5/02.  *  Sven Hammarling, 1/5/02.
Removed from v.1.8  
changed lines
  Added in v.1.9

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