dtrsyl.f.html

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      SUBROUTINE <a name="DTRSYL.1"></a><a href="dtrsyl.f.html#DTRSYL.1">DTRSYL</a>( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
     $                   LDC, SCALE, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      CHARACTER          TRANA, TRANB
      INTEGER            INFO, ISGN, LDA, LDB, LDC, M, N
      DOUBLE PRECISION   SCALE
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="DTRSYL.20"></a><a href="dtrsyl.f.html#DTRSYL.1">DTRSYL</a> solves the real Sylvester matrix equation:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     op(A)*X + X*op(B) = scale*C or
</span><span class="comment">*</span><span class="comment">     op(A)*X - X*op(B) = scale*C,
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where op(A) = A or A**T, and  A and B are both upper quasi-
</span><span class="comment">*</span><span class="comment">  triangular. A is M-by-M and B is N-by-N; the right hand side C and
</span><span class="comment">*</span><span class="comment">  the solution X are M-by-N; and scale is an output scale factor, set
</span><span class="comment">*</span><span class="comment">  &lt;= 1 to avoid overflow in X.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A and B must be in Schur canonical form (as returned by <a name="DHSEQR.30"></a><a href="dhseqr.f.html#DHSEQR.1">DHSEQR</a>), that
</span><span class="comment">*</span><span class="comment">  is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks;
</span><span class="comment">*</span><span class="comment">  each 2-by-2 diagonal block has its diagonal elements equal and its
</span><span class="comment">*</span><span class="comment">  off-diagonal elements of opposite sign.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  TRANA   (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          Specifies the option op(A):
</span><span class="comment">*</span><span class="comment">          = 'N': op(A) = A    (No transpose)
</span><span class="comment">*</span><span class="comment">          = 'T': op(A) = A**T (Transpose)
</span><span class="comment">*</span><span class="comment">          = 'C': op(A) = A**H (Conjugate transpose = Transpose)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  TRANB   (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          Specifies the option op(B):
</span><span class="comment">*</span><span class="comment">          = 'N': op(B) = B    (No transpose)
</span><span class="comment">*</span><span class="comment">          = 'T': op(B) = B**T (Transpose)
</span><span class="comment">*</span><span class="comment">          = 'C': op(B) = B**H (Conjugate transpose = Transpose)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  ISGN    (input) INTEGER
</span><span class="comment">*</span><span class="comment">          Specifies the sign in the equation:
</span><span class="comment">*</span><span class="comment">          = +1: solve op(A)*X + X*op(B) = scale*C
</span><span class="comment">*</span><span class="comment">          = -1: solve op(A)*X - X*op(B) = scale*C
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  M       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The order of the matrix A, and the number of rows in the
</span><span class="comment">*</span><span class="comment">          matrices X and C. M &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The order of the matrix B, and the number of columns in the
</span><span class="comment">*</span><span class="comment">          matrices X and C. N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A       (input) DOUBLE PRECISION array, dimension (LDA,M)
</span><span class="comment">*</span><span class="comment">          The upper quasi-triangular matrix A, in Schur canonical form.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDA     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array A. LDA &gt;= max(1,M).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  B       (input) DOUBLE PRECISION array, dimension (LDB,N)
</span><span class="comment">*</span><span class="comment">          The upper quasi-triangular matrix B, in Schur canonical form.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDB     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array B. LDB &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
</span><span class="comment">*</span><span class="comment">          On entry, the M-by-N right hand side matrix C.
</span><span class="comment">*</span><span class="comment">          On exit, C is overwritten by the solution matrix X.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDC     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array C. LDC &gt;= max(1,M)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SCALE   (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">          The scale factor, scale, set &lt;= 1 to avoid overflow in X.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  INFO    (output) INTEGER
</span><span class="comment">*</span><span class="comment">          = 0: successful exit
</span><span class="comment">*</span><span class="comment">          &lt; 0: if INFO = -i, the i-th argument had an illegal value
</span><span class="comment">*</span><span class="comment">          = 1: A and B have common or very close eigenvalues; perturbed
</span><span class="comment">*</span><span class="comment">               values were used to solve the equation (but the matrices
</span><span class="comment">*</span><span class="comment">               A and B are unchanged).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      LOGICAL            NOTRNA, NOTRNB
      INTEGER            IERR, J, K, K1, K2, KNEXT, L, L1, L2, LNEXT
      DOUBLE PRECISION   A11, BIGNUM, DA11, DB, EPS, SCALOC, SGN, SMIN,
     $                   SMLNUM, SUML, SUMR, XNORM
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Arrays ..
</span>      DOUBLE PRECISION   DUM( 1 ), VEC( 2, 2 ), X( 2, 2 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Functions ..
</span>      LOGICAL            <a name="LSAME.108"></a><a href="lsame.f.html#LSAME.1">LSAME</a>
      DOUBLE PRECISION   DDOT, <a name="DLAMCH.109"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>, <a name="DLANGE.109"></a><a href="dlange.f.html#DLANGE.1">DLANGE</a>
      EXTERNAL           <a name="LSAME.110"></a><a href="lsame.f.html#LSAME.1">LSAME</a>, DDOT, <a name="DLAMCH.110"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>, <a name="DLANGE.110"></a><a href="dlange.f.html#DLANGE.1">DLANGE</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Subroutines ..
</span>      EXTERNAL           <a name="DLABAD.113"></a><a href="dlabad.f.html#DLABAD.1">DLABAD</a>, <a name="DLALN2.113"></a><a href="dlaln2.f.html#DLALN2.1">DLALN2</a>, <a name="DLASY2.113"></a><a href="dlasy2.f.html#DLASY2.1">DLASY2</a>, DSCAL, <a name="XERBLA.113"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          ABS, DBLE, MAX, MIN
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Decode and Test input parameters
</span><span class="comment">*</span><span class="comment">
</span>      NOTRNA = <a name="LSAME.122"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( TRANA, <span class="string">'N'</span> )
      NOTRNB = <a name="LSAME.123"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( TRANB, <span class="string">'N'</span> )
<span class="comment">*</span><span class="comment">
</span>      INFO = 0
      IF( .NOT.NOTRNA .AND. .NOT.<a name="LSAME.126"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( TRANA, <span class="string">'T'</span> ) .AND. .NOT.
     $    <a name="LSAME.127"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( TRANA, <span class="string">'C'</span> ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRNB .AND. .NOT.<a name="LSAME.129"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( TRANB, <span class="string">'T'</span> ) .AND. .NOT.
     $         <a name="LSAME.130"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( TRANB, <span class="string">'C'</span> ) ) 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 <a name="XERBLA.146"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>( <span class="string">'<a name="DTRSYL.146"></a><a href="dtrsyl.f.html#DTRSYL.1">DTRSYL</a>'</span>, -INFO )
         RETURN
      END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Quick return if possible
</span><span class="comment">*</span><span class="comment">
</span>      IF( M.EQ.0 .OR. N.EQ.0 )
     $   RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Set constants to control overflow
</span><span class="comment">*</span><span class="comment">
</span>      EPS = <a name="DLAMCH.157"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>( <span class="string">'P'</span> )
      SMLNUM = <a name="DLAMCH.158"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>( <span class="string">'S'</span> )
      BIGNUM = ONE / SMLNUM
      CALL <a name="DLABAD.160"></a><a href="dlabad.f.html#DLABAD.1">DLABAD</a>( SMLNUM, BIGNUM )
      SMLNUM = SMLNUM*DBLE( M*N ) / EPS
      BIGNUM = ONE / SMLNUM
<span class="comment">*</span><span class="comment">
</span>      SMIN = MAX( SMLNUM, EPS*<a name="DLANGE.164"></a><a href="dlange.f.html#DLANGE.1">DLANGE</a>( <span class="string">'M'</span>, M, M, A, LDA, DUM ),
     $       EPS*<a name="DLANGE.165"></a><a href="dlange.f.html#DLANGE.1">DLANGE</a>( <span class="string">'M'</span>, N, N, B, LDB, DUM ) )
<span class="comment">*</span><span class="comment">
</span>      SCALE = ONE
      SGN = ISGN
<span class="comment">*</span><span class="comment">
</span>      IF( NOTRNA .AND. NOTRNB ) THEN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Solve    A*X + ISGN*X*B = scale*C.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        The (K,L)th block of X is determined starting from
</span><span class="comment">*</span><span class="comment">        bottom-left corner column by column by
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">         A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Where
</span><span class="comment">*</span><span class="comment">                  M                         L-1
</span><span class="comment">*</span><span class="comment">        R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B(J,L)].
</span><span class="comment">*</span><span class="comment">                I=K+1                       J=1
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Start column loop (index = L)
</span><span class="comment">*</span><span class="comment">        L1 (L2) : column index of the first (first) row of X(K,L).
</span><span class="comment">*</span><span class="comment">
</span>         LNEXT = 1
         DO 60 L = 1, N
            IF( L.LT.LNEXT )
     $         GO TO 60
            IF( L.EQ.N ) THEN
               L1 = L
               L2 = L
            ELSE
               IF( B( L+1, L ).NE.ZERO ) THEN
                  L1 = L
                  L2 = L + 1
                  LNEXT = L + 2
               ELSE
                  L1 = L
                  L2 = L
                  LNEXT = L + 1
               END IF
            END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Start row loop (index = K)
</span><span class="comment">*</span><span class="comment">           K1 (K2): row index of the first (last) row of X(K,L).
</span><span class="comment">*</span><span class="comment">
</span>            KNEXT = M
            DO 50 K = M, 1, -1
               IF( K.GT.KNEXT )
     $            GO TO 50
               IF( K.EQ.1 ) THEN
                  K1 = K
                  K2 = K
               ELSE
1234

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