ztrsyl.f

          SUBROUTINE ZTRSYL( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
         $                   LDC, SCALE, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          TRANA, TRANB
          INTEGER            INFO, ISGN, LDA, LDB, LDC, M, N
          DOUBLE PRECISION   SCALE
    *     ..
    *     .. Array Arguments ..
          COMPLEX*16         A( LDA, * ), B( LDB, * ), C( LDC, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  ZTRSYL solves the complex Sylvester matrix equation:
    *
    *     op(A)*X + X*op(B) = scale*C or
    *     op(A)*X - X*op(B) = scale*C,
    *
    *  where op(A) = A or A**H, and A and B are both upper triangular. A is
    *  M-by-M and B is N-by-N; the right hand side C and the solution X are
    *  M-by-N; and scale is an output scale factor, set <= 1 to avoid
    *  overflow in X.
    *
    *  Arguments
    *  =========
    *
    *  TRANA   (input) CHARACTER*1
    *          Specifies the option op(A):
    *          = 'N': op(A) = A    (No transpose)
    *          = 'C': op(A) = A**H (Conjugate transpose)
    *
    *  TRANB   (input) CHARACTER*1
    *          Specifies the option op(B):
    *          = 'N': op(B) = B    (No transpose)
    *          = 'C': op(B) = B**H (Conjugate transpose)
    *
    *  ISGN    (input) INTEGER
    *          Specifies the sign in the equation:
    *          = +1: solve op(A)*X + X*op(B) = scale*C
    *          = -1: solve op(A)*X - X*op(B) = scale*C
    *
    *  M       (input) INTEGER
    *          The order of the matrix A, and the number of rows in the
    *          matrices X and C. M >= 0.
    *
    *  N       (input) INTEGER
    *          The order of the matrix B, and the number of columns in the
    *          matrices X and C. N >= 0.
    *
    *  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, the M-by-N right hand side matrix C.
    *          On exit, C is overwritten by the solution matrix X.
    *
    *  LDC     (input) INTEGER
    *          The leading dimension of the array C. LDC >= max(1,M)
    *
    *  SCALE   (output) DOUBLE PRECISION
    *          The scale factor, scale, set <= 1 to avoid overflow in X.
    *
    *  INFO    (output) INTEGER
    *          = 0: successful exit
    *          < 0: if INFO = -i, the i-th argument had an illegal value
    *          = 1: A and B have common or very close eigenvalues; perturbed
    *               values were used to solve the equation (but the matrices
    *               A and B are unchanged).
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ONE
          PARAMETER          ( ONE = 1.0D+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            NOTRNA, NOTRNB
          INTEGER            J, K, L
          DOUBLE PRECISION   BIGNUM, DA11, DB, EPS, SCALOC, SGN, SMIN,
         $                   SMLNUM
          COMPLEX*16         A11, SUML, SUMR, VEC, X11
    *     ..
    *     .. Local Arrays ..
          DOUBLE PRECISION   DUM( 1 )
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DLAMCH, ZLANGE
          COMPLEX*16         ZDOTC, ZDOTU, ZLADIV
          EXTERNAL           LSAME, DLAMCH, ZLANGE, ZDOTC, ZDOTU, ZLADIV
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DLABAD, XERBLA, ZDSCAL
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN
    *     ..
    *     .. Executable Statements ..
    *
    *     Decode and Test input parameters
    *
          NOTRNA = LSAME( TRANA, 'N' )
          NOTRNB = LSAME( TRANB, 'N' )
    *
          INFO = 0
          IF( .NOT.NOTRNA .AND. .NOT.LSAME( TRANA, 'C' ) ) THEN
             INFO = -1
          ELSE IF( .NOT.NOTRNB .AND. .NOT.LSAME( TRANB, 'C' ) ) THEN
             INFO = -2
          ELSE IF( ISGN.NE.1 .AND. ISGN.NE.-1 ) THEN
             INFO = -3
          ELSE IF( M.LT.0 ) THEN
             INFO = -4
          ELSE IF( N.LT.0 ) THEN
             INFO = -5
          ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
             INFO = -7
          ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
             INFO = -9
          ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
             INFO = -11
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'ZTRSYL', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( M.EQ.0 .OR. N.EQ.0 )
         $   RETURN
    *
    *     Set constants to control overflow
    *
          EPS = DLAMCH( 'P' )
          SMLNUM = DLAMCH( 'S' )
          BIGNUM = ONE / SMLNUM
          CALL DLABAD( SMLNUM, BIGNUM )
          SMLNUM = SMLNUM*DBLE( M*N ) / EPS
          BIGNUM = ONE / SMLNUM
          SMIN = MAX( SMLNUM, EPS*ZLANGE( 'M', M, M, A, LDA, DUM ),
         $       EPS*ZLANGE( 'M', N, N, B, LDB, DUM ) )
          SCALE = ONE
          SGN = ISGN
    *
          IF( NOTRNA .AND. NOTRNB ) THEN
    *
    *        Solve    A*X + ISGN*X*B = scale*C.
    *
    *        The (K,L)th block of X is determined starting from
    *        bottom-left corner column by column by
    *
    *            A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
    *
    *        Where
    *                    M                        L-1
    *          R(K,L) = SUM [A(K,I)*X(I,L)] +ISGN*SUM [X(K,J)*B(J,L)].
    *                  I=K+1                      J=1
    *
             DO 30 L = 1, N
                DO 20 K = M, 1, -1
    *
                   SUML = ZDOTU( M-K, A( K, MIN( K+1, M ) ), LDA,
         $                C( MIN( K+1, M ), L ), 1 )
                   SUMR = ZDOTU( L-1, C( K, 1 ), LDC, B( 1, L ), 1 )
                   VEC = C( K, L ) - ( SUML+SGN*SUMR )
    *
                   SCALOC = ONE
                   A11 = A( K, K ) + SGN*B( L, L )
                   DA11 = ABS( DBLE( A11 ) ) + ABS( DIMAG( A11 ) )
                   IF( DA11.LE.SMIN ) THEN
                      A11 = SMIN
                      DA11 = SMIN
                      INFO = 1
                   END IF
                   DB = ABS( DBLE( VEC ) ) + ABS( DIMAG( VEC ) )
                   IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
                      IF( DB.GT.BIGNUM*DA11 )
         $               SCALOC = ONE / DB
                   END IF
                   X11 = ZLADIV( VEC*DCMPLX( SCALOC ), A11 )
    *
                   IF( SCALOC.NE.ONE ) THEN
                      DO 10 J = 1, N
                         CALL ZDSCAL( M, SCALOC, C( 1, J ), 1 )
       10             CONTINUE
                      SCALE = SCALE*SCALOC
                   END IF
                   C( K, L ) = X11
    *
       20       CONTINUE
       30    CONTINUE
    *
          ELSE IF( .NOT.NOTRNA .AND. NOTRNB ) THEN
    *
    *        Solve    A' *X + ISGN*X*B = scale*C.
    *
    *        The (K,L)th block of X is determined starting from
    *        upper-left corner column by column by
    *
    *            A'(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
    *
    *        Where
    *                   K-1                         L-1
    *          R(K,L) = SUM [A'(I,K)*X(I,L)] + ISGN*SUM [X(K,J)*B(J,L)]
    *                   I=1                         J=1
    *
             DO 60 L = 1, N
                DO 50 K = 1, M
    *
                   SUML = ZDOTC( K-1, A( 1, K ), 1, C( 1, L ), 1 )
                   SUMR = ZDOTU( L-1, C( K, 1 ), LDC, B( 1, L ), 1 )
                   VEC = C( K, L ) - ( SUML+SGN*SUMR )
    *
                   SCALOC = ONE
                   A11 = DCONJG( A( K, K ) ) + SGN*B( L, L )
                   DA11 = ABS( DBLE( A11 ) ) + ABS( DIMAG( A11 ) )
                   IF( DA11.LE.SMIN ) THEN
                      A11 = SMIN
                      DA11 = SMIN
                      INFO = 1
                   END IF
                   DB = ABS( DBLE( VEC ) ) + ABS( DIMAG( VEC ) )
                   IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
                      IF( DB.GT.BIGNUM*DA11 )
         $               SCALOC = ONE / DB
                   END IF
    *
                   X11 = ZLADIV( VEC*DCMPLX( SCALOC ), A11 )
    *
                   IF( SCALOC.NE.ONE ) THEN
                      DO 40 J = 1, N
                         CALL ZDSCAL( M, SCALOC, C( 1, J ), 1 )
       40             CONTINUE
                      SCALE = SCALE*SCALOC
                   END IF
                   C( K, L ) = X11
    *
       50       CONTINUE
       60    CONTINUE
    *
          ELSE IF( .NOT.NOTRNA .AND. .NOT.NOTRNB ) THEN
    *
    *        Solve    A'*X + ISGN*X*B' = C.
    *
    *        The (K,L)th block of X is determined starting from
    *        upper-right corner column by column by
    *
    *            A'(K,K)*X(K,L) + ISGN*X(K,L)*B'(L,L) = C(K,L) - R(K,L)
    *
    *        Where
    *                    K-1
    *           R(K,L) = SUM [A'(I,K)*X(I,L)] +
    *                    I=1
    *                           N
    *                     ISGN*SUM [X(K,J)*B'(L,J)].
    *                          J=L+1
    *
             DO 90 L = N, 1, -1
                DO 80 K = 1, M
    *
                   SUML = ZDOTC( K-1, A( 1, K ), 1, C( 1, L ), 1 )
                   SUMR = ZDOTC( N-L, C( K, MIN( L+1, N ) ), LDC,
         $                B( L, MIN( L+1, N ) ), LDB )
                   VEC = C( K, L ) - ( SUML+SGN*DCONJG( SUMR ) )
    *
                   SCALOC = ONE
                   A11 = DCONJG( A( K, K )+SGN*B( L, L ) )
                   DA11 = ABS( DBLE( A11 ) ) + ABS( DIMAG( A11 ) )
                   IF( DA11.LE.SMIN ) THEN
                      A11 = SMIN
                      DA11 = SMIN
                      INFO = 1
                   END IF
                   DB = ABS( DBLE( VEC ) ) + ABS( DIMAG( VEC ) )
                   IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
                      IF( DB.GT.BIGNUM*DA11 )
         $               SCALOC = ONE / DB
                   END IF
    *
                   X11 = ZLADIV( VEC*DCMPLX( SCALOC ), A11 )
    *
                   IF( SCALOC.NE.ONE ) THEN
                      DO 70 J = 1, N
                         CALL ZDSCAL( M, SCALOC, C( 1, J ), 1 )
       70             CONTINUE
                      SCALE = SCALE*SCALOC
                   END IF
                   C( K, L ) = X11
    *
       80       CONTINUE
       90    CONTINUE
    *
          ELSE IF( NOTRNA .AND. .NOT.NOTRNB ) THEN
    *
    *        Solve    A*X + ISGN*X*B' = C.
    *
    *        The (K,L)th block of X is determined starting from
    *        bottom-left corner column by column by
    *
    *           A(K,K)*X(K,L) + ISGN*X(K,L)*B'(L,L) = C(K,L) - R(K,L)
    *
    *        Where
    *                    M                          N
    *          R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B'(L,J)]
    *                  I=K+1                      J=L+1
    *
             DO 120 L = N, 1, -1
                DO 110 K = M, 1, -1
    *
                   SUML = ZDOTU( M-K, A( K, MIN( K+1, M ) ), LDA,
         $                C( MIN( K+1, M ), L ), 1 )
                   SUMR = ZDOTC( N-L, C( K, MIN( L+1, N ) ), LDC,
         $                B( L, MIN( L+1, N ) ), LDB )
                   VEC = C( K, L ) - ( SUML+SGN*DCONJG( SUMR ) )
    *
                   SCALOC = ONE
                   A11 = A( K, K ) + SGN*DCONJG( B( L, L ) )
                   DA11 = ABS( DBLE( A11 ) ) + ABS( DIMAG( A11 ) )
                   IF( DA11.LE.SMIN ) THEN
                      A11 = SMIN
                      DA11 = SMIN
                      INFO = 1
                   END IF
                   DB = ABS( DBLE( VEC ) ) + ABS( DIMAG( VEC ) )
                   IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
                      IF( DB.GT.BIGNUM*DA11 )
         $               SCALOC = ONE / DB
                   END IF
    *
                   X11 = ZLADIV( VEC*DCMPLX( SCALOC ), A11 )
    *
                   IF( SCALOC.NE.ONE ) THEN
                      DO 100 J = 1, N
                         CALL ZDSCAL( M, SCALOC, C( 1, J ), 1 )
      100             CONTINUE
                      SCALE = SCALE*SCALOC
                   END IF
                   C( K, L ) = X11
    *
      110       CONTINUE
      120    CONTINUE
    *
          END IF
    *
          RETURN
    *
    *     End of ZTRSYL
    *
          END