path: root/SRC/dtgsyl.f

      SUBROUTINE DTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
     $                   LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
     $                   IWORK, INFO )
*
*  -- LAPACK routine (version 3.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2006
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF,
     $                   LWORK, M, N
      DOUBLE PRECISION   DIF, SCALE
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
     $                   D( LDD, * ), E( LDE, * ), F( LDF, * ),
     $                   WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DTGSYL 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 real entries. (A, D) and (B, E) must be in
*  generalized (real) Schur canonical form, i.e. A, B are upper quasi
*  triangular and D, E are upper triangular.
*
*  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', Im) ]         (2)
*                 [ kron(In, D)  -kron(E', Im) ].
*
*  Here Ik is the identity matrix of size k and X' is the transpose of
*  X. kron(X, Y) is the Kronecker product between the matrices X and Y.
*
*  If TRANS = 'T', DTGSYL solves the transposed system Z**T*y = scale*b,
*  which is equivalent to solve for R and L in
*
*              A**T * R + D**T * L = scale * C           (3)
*              R * B**T + L * E**T = scale * -F
*
*  This case (TRANS = 'T') 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 DLACON.
*
*  If IJOB >= 1, DTGSYL 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. See [1-2] for more
*  information.
*
*  This is a level 3 BLAS algorithm.
*
*  Arguments
*  =========
*
*  TRANS   (input) CHARACTER*1
*          = 'N', solve the generalized Sylvester equation (1).
*          = 'T', solve the '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 IJOB  = 1 is used).
*           =4: Only an estimate of Dif[(A,D), (B,E)] is computed.
*               ( DGECON on sub-systems is used ).
*          Not referenced if TRANS = 'T'.
*
*  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) DOUBLE PRECISION array, dimension (LDA, M)
*          The upper quasi triangular matrix A.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A. LDA >= max(1, M).
*
*  B       (input) DOUBLE PRECISION array, dimension (LDB, N)
*          The upper quasi triangular matrix B.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B. LDB >= max(1, N).
*
*  C       (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 = 'T', DIF is not touched.
*
*  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, C and F hold the
*          solutions R and L, respectively, to the homogeneous system
*          with C = F = 0. Normally, SCALE = 1.
*
*  WORK    (workspace/output) DOUBLE PRECISION 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+6)
*
*  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 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 DCOPY by calls to DLASET.
*  Sven Hammarling, 1/5/02.
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, NOTRAN
      INTEGER            I, IE, IFUNC, IROUND, IS, ISOLVE, J, JE, JS, K,
     $                   LINFO, LWMIN, MB, NB, P, PPQQ, PQ, Q
      DOUBLE PRECISION   DSCALE, DSUM, SCALE2, SCALOC
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DLACPY, DLASET, DSCAL, DTGSY2, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode and test input parameters
*
      INFO = 0
      NOTRAN = LSAME( TRANS, 'N' )
      LQUERY = ( LWORK.EQ.-1 )
*
      IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
         INFO = -1
      ELSE IF( NOTRAN ) THEN
         IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.4 ) ) 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.EQ.0 ) THEN
         IF( NOTRAN ) THEN
            IF( IJOB.EQ.1 .OR. IJOB.EQ.2 ) THEN
               LWMIN = MAX( 1, 2*M*N )
            ELSE
               LWMIN = 1
            END IF
         ELSE
            LWMIN = 1
         END IF
         WORK( 1 ) = LWMIN
*
         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -20
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTGSYL', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( M.EQ.0 .OR. N.EQ.0 ) THEN
         SCALE = 1
         IF( NOTRAN ) THEN
            IF( IJOB.NE.0 ) THEN
               DIF = 0
            END IF
         END IF
         RETURN
      END IF
*
*     Determine optimal block sizes MB and NB
*
      MB = ILAENV( 2, 'DTGSYL', TRANS, M, N, -1, -1 )
      NB = ILAENV( 5, 'DTGSYL', TRANS, M, N, -1, -1 )
*
      ISOLVE = 1
      IFUNC = 0
      IF( NOTRAN ) THEN
         IF( IJOB.GE.3 ) THEN
            IFUNC = IJOB - 2
            CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
            CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
         ELSE IF( IJOB.GE.1 ) THEN
            ISOLVE = 2
         END IF
      END IF
*
      IF( ( MB.LE.1 .AND. NB.LE.1 ) .OR. ( MB.GE.M .AND. NB.GE.N ) )
     $     THEN
*
         DO 30 IROUND = 1, ISOLVE
*
*           Use unblocked Level 2 solver
*
            DSCALE = ZERO
            DSUM = ONE
            PQ = 0
            CALL DTGSY2( TRANS, IFUNC, M, N, A, LDA, B, LDB, C, LDC, D,
     $                   LDD, E, LDE, F, LDF, SCALE, DSUM, DSCALE,
     $                   IWORK, PQ, INFO )
            IF( DSCALE.NE.ZERO ) THEN
               IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN
                  DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) )
               ELSE
                  DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) )
               END IF
            END IF
*
            IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN
               IF( NOTRAN ) THEN
                  IFUNC = IJOB
               END IF
               SCALE2 = SCALE
               CALL DLACPY( 'F', M, N, C, LDC, WORK, M )
               CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M )
               CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
               CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
            ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN
               CALL DLACPY( 'F', M, N, WORK, M, C, LDC )
               CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF )
               SCALE = SCALE2
            END IF
   30    CONTINUE
*
         RETURN
      END IF
*
*     Determine block structure of A
*
      P = 0
      I = 1
   40 CONTINUE
      IF( I.GT.M )
     $   GO TO 50
      P = P + 1
      IWORK( P ) = I
      I = I + MB
      IF( I.GE.M )
     $   GO TO 50
      IF( A( I, I-1 ).NE.ZERO )
     $   I = I + 1
      GO TO 40
   50 CONTINUE
*
      IWORK( P+1 ) = M + 1
      IF( IWORK( P ).EQ.IWORK( P+1 ) )
     $   P = P - 1
*
*     Determine block structure of B
*
      Q = P + 1
      J = 1
   60 CONTINUE
      IF( J.GT.N )
     $   GO TO 70
      Q = Q + 1
      IWORK( Q ) = J
      J = J + NB
      IF( J.GE.N )
     $   GO TO 70
      IF( B( J, J-1 ).NE.ZERO )
     $   J = J + 1
      GO TO 60
   70 CONTINUE
*
      IWORK( Q+1 ) = N + 1
      IF( IWORK( Q ).EQ.IWORK( Q+1 ) )
     $   Q = Q - 1
*
      IF( NOTRAN ) THEN
*
         DO 150 IROUND = 1, ISOLVE
*
*           Solve (I, J)-subsystem
*               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 = P, P - 1,..., 1; J = 1, 2,..., Q
*
            DSCALE = ZERO
            DSUM = ONE
            PQ = 0
            SCALE = ONE
            DO 130 J = P + 2, Q
               JS = IWORK( J )
               JE = IWORK( J+1 ) - 1
               NB = JE - JS + 1
               DO 120 I = P, 1, -1
                  IS = IWORK( I )
                  IE = IWORK( I+1 ) - 1
                  MB = IE - IS + 1
                  PPQQ = 0
                  CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ), LDA,
     $                         B( JS, JS ), LDB, C( IS, JS ), LDC,
     $                         D( IS, IS ), LDD, E( JS, JS ), LDE,
     $                         F( IS, JS ), LDF, SCALOC, DSUM, DSCALE,
     $                         IWORK( Q+2 ), PPQQ, LINFO )
                  IF( LINFO.GT.0 )
     $               INFO = LINFO
*
                  PQ = PQ + PPQQ
                  IF( SCALOC.NE.ONE ) THEN
                     DO 80 K = 1, JS - 1
                        CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                        CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
   80                CONTINUE
                     DO 90 K = JS, JE
                        CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 )
                        CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 )
   90                CONTINUE
                     DO 100 K = JS, JE
                        CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 )
                        CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 )
  100                CONTINUE
                     DO 110 K = JE + 1, N
                        CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                        CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
  110                CONTINUE
                     SCALE = SCALE*SCALOC
                  END IF
*
*                 Substitute R(I, J) and L(I, J) into remaining
*                 equation.
*
                  IF( I.GT.1 ) THEN
                     CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
     $                           A( 1, IS ), LDA, C( IS, JS ), LDC, ONE,
     $                           C( 1, JS ), LDC )
                     CALL DGEMM( 'N', 'N', IS-1, NB, MB, -ONE,
     $                           D( 1, IS ), LDD, C( IS, JS ), LDC, ONE,
     $                           F( 1, JS ), LDF )
                  END IF
                  IF( J.LT.Q ) THEN
                     CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
     $                           F( IS, JS ), LDF, B( JS, JE+1 ), LDB,
     $                           ONE, C( IS, JE+1 ), LDC )
                     CALL DGEMM( 'N', 'N', MB, N-JE, NB, ONE,
     $                           F( IS, JS ), LDF, E( JS, JE+1 ), LDE,
     $                           ONE, F( IS, JE+1 ), LDF )
                  END IF
  120          CONTINUE
  130       CONTINUE
            IF( DSCALE.NE.ZERO ) THEN
               IF( IJOB.EQ.1 .OR. IJOB.EQ.3 ) THEN
                  DIF = SQRT( DBLE( 2*M*N ) ) / ( DSCALE*SQRT( DSUM ) )
               ELSE
                  DIF = SQRT( DBLE( PQ ) ) / ( DSCALE*SQRT( DSUM ) )
               END IF
            END IF
            IF( ISOLVE.EQ.2 .AND. IROUND.EQ.1 ) THEN
               IF( NOTRAN ) THEN
                  IFUNC = IJOB
               END IF
               SCALE2 = SCALE
               CALL DLACPY( 'F', M, N, C, LDC, WORK, M )
               CALL DLACPY( 'F', M, N, F, LDF, WORK( M*N+1 ), M )
               CALL DLASET( 'F', M, N, ZERO, ZERO, C, LDC )
               CALL DLASET( 'F', M, N, ZERO, ZERO, F, LDF )
            ELSE IF( ISOLVE.EQ.2 .AND. IROUND.EQ.2 ) THEN
               CALL DLACPY( 'F', M, N, WORK, M, C, LDC )
               CALL DLACPY( 'F', M, N, WORK( M*N+1 ), M, F, LDF )
               SCALE = SCALE2
            END IF
  150    CONTINUE
*
      ELSE
*
*        Solve transposed (I, J)-subsystem
*             A(I, I)**T * R(I, J)  + D(I, I)**T * L(I, J)  =  C(I, J)
*             R(I, J)  * B(J, J)**T + L(I, J)  * E(J, J)**T = -F(I, J)
*        for I = 1,2,..., P; J = Q, Q-1,..., 1
*
         SCALE = ONE
         DO 210 I = 1, P
            IS = IWORK( I )
            IE = IWORK( I+1 ) - 1
            MB = IE - IS + 1
            DO 200 J = Q, P + 2, -1
               JS = IWORK( J )
               JE = IWORK( J+1 ) - 1
               NB = JE - JS + 1
               CALL DTGSY2( TRANS, IFUNC, MB, NB, A( IS, IS ), LDA,
     $                      B( JS, JS ), LDB, C( IS, JS ), LDC,
     $                      D( IS, IS ), LDD, E( JS, JS ), LDE,
     $                      F( IS, JS ), LDF, SCALOC, DSUM, DSCALE,
     $                      IWORK( Q+2 ), PPQQ, LINFO )
               IF( LINFO.GT.0 )
     $            INFO = LINFO
               IF( SCALOC.NE.ONE ) THEN
                  DO 160 K = 1, JS - 1
                     CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                     CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
  160             CONTINUE
                  DO 170 K = JS, JE
                     CALL DSCAL( IS-1, SCALOC, C( 1, K ), 1 )
                     CALL DSCAL( IS-1, SCALOC, F( 1, K ), 1 )
  170             CONTINUE
                  DO 180 K = JS, JE
                     CALL DSCAL( M-IE, SCALOC, C( IE+1, K ), 1 )
                     CALL DSCAL( M-IE, SCALOC, F( IE+1, K ), 1 )
  180             CONTINUE
                  DO 190 K = JE + 1, N
                     CALL DSCAL( M, SCALOC, C( 1, K ), 1 )
                     CALL DSCAL( M, SCALOC, F( 1, K ), 1 )
  190             CONTINUE
                  SCALE = SCALE*SCALOC
               END IF
*
*              Substitute R(I, J) and L(I, J) into remaining equation.
*
               IF( J.GT.P+2 ) THEN
                  CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, C( IS, JS ),
     $                        LDC, B( 1, JS ), LDB, ONE, F( IS, 1 ),
     $                        LDF )
                  CALL DGEMM( 'N', 'T', MB, JS-1, NB, ONE, F( IS, JS ),
     $                        LDF, E( 1, JS ), LDE, ONE, F( IS, 1 ),
     $                        LDF )
               END IF
               IF( I.LT.P ) THEN
                  CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
     $                        A( IS, IE+1 ), LDA, C( IS, JS ), LDC, ONE,
     $                        C( IE+1, JS ), LDC )
                  CALL DGEMM( 'T', 'N', M-IE, NB, MB, -ONE,
     $                        D( IS, IE+1 ), LDD, F( IS, JS ), LDF, ONE,
     $                        C( IE+1, JS ), LDC )
               END IF
  200       CONTINUE
  210    CONTINUE
*
      END IF
*
      WORK( 1 ) = LWMIN
*
      RETURN
*
*     End of DTGSYL
*
      END