ztrevc.f.html

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      SUBROUTINE <a name="ZTREVC.1"></a><a href="ztrevc.f.html#ZTREVC.1">ZTREVC</a>( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
     $                   LDVR, MM, M, WORK, RWORK, 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          HOWMNY, SIDE
      INTEGER            INFO, LDT, LDVL, LDVR, M, MM, N
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      LOGICAL            SELECT( * )
      DOUBLE PRECISION   RWORK( * )
      COMPLEX*16         T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
     $                   WORK( * )
<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="ZTREVC.22"></a><a href="ztrevc.f.html#ZTREVC.1">ZTREVC</a> computes some or all of the right and/or left eigenvectors of
</span><span class="comment">*</span><span class="comment">  a complex upper triangular matrix T.
</span><span class="comment">*</span><span class="comment">  Matrices of this type are produced by the Schur factorization of
</span><span class="comment">*</span><span class="comment">  a complex general matrix:  A = Q*T*Q**H, as computed by <a name="ZHSEQR.25"></a><a href="zhseqr.f.html#ZHSEQR.1">ZHSEQR</a>.
</span><span class="comment">*</span><span class="comment">  
</span><span class="comment">*</span><span class="comment">  The right eigenvector x and the left eigenvector y of T corresponding
</span><span class="comment">*</span><span class="comment">  to an eigenvalue w are defined by:
</span><span class="comment">*</span><span class="comment">  
</span><span class="comment">*</span><span class="comment">               T*x = w*x,     (y**H)*T = w*(y**H)
</span><span class="comment">*</span><span class="comment">  
</span><span class="comment">*</span><span class="comment">  where y**H denotes the conjugate transpose of the vector y.
</span><span class="comment">*</span><span class="comment">  The eigenvalues are not input to this routine, but are read directly
</span><span class="comment">*</span><span class="comment">  from the diagonal of T.
</span><span class="comment">*</span><span class="comment">  
</span><span class="comment">*</span><span class="comment">  This routine returns the matrices X and/or Y of right and left
</span><span class="comment">*</span><span class="comment">  eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
</span><span class="comment">*</span><span class="comment">  input matrix.  If Q is the unitary factor that reduces a matrix A to
</span><span class="comment">*</span><span class="comment">  Schur form T, then Q*X and Q*Y are the matrices of right and left
</span><span class="comment">*</span><span class="comment">  eigenvectors of A.
</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">  SIDE    (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          = 'R':  compute right eigenvectors only;
</span><span class="comment">*</span><span class="comment">          = 'L':  compute left eigenvectors only;
</span><span class="comment">*</span><span class="comment">          = 'B':  compute both right and left eigenvectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  HOWMNY  (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          = 'A':  compute all right and/or left eigenvectors;
</span><span class="comment">*</span><span class="comment">          = 'B':  compute all right and/or left eigenvectors,
</span><span class="comment">*</span><span class="comment">                  backtransformed using the matrices supplied in
</span><span class="comment">*</span><span class="comment">                  VR and/or VL;
</span><span class="comment">*</span><span class="comment">          = 'S':  compute selected right and/or left eigenvectors,
</span><span class="comment">*</span><span class="comment">                  as indicated by the logical array SELECT.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SELECT  (input) LOGICAL array, dimension (N)
</span><span class="comment">*</span><span class="comment">          If HOWMNY = 'S', SELECT specifies the eigenvectors to be
</span><span class="comment">*</span><span class="comment">          computed.
</span><span class="comment">*</span><span class="comment">          The eigenvector corresponding to the j-th eigenvalue is
</span><span class="comment">*</span><span class="comment">          computed if SELECT(j) = .TRUE..
</span><span class="comment">*</span><span class="comment">          Not referenced if HOWMNY = 'A' or 'B'.
</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 T. N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  T       (input/output) COMPLEX*16 array, dimension (LDT,N)
</span><span class="comment">*</span><span class="comment">          The upper triangular matrix T.  T is modified, but restored
</span><span class="comment">*</span><span class="comment">          on exit.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDT     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array T. LDT &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  VL      (input/output) COMPLEX*16 array, dimension (LDVL,MM)
</span><span class="comment">*</span><span class="comment">          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
</span><span class="comment">*</span><span class="comment">          contain an N-by-N matrix Q (usually the unitary matrix Q of
</span><span class="comment">*</span><span class="comment">          Schur vectors returned by <a name="ZHSEQR.78"></a><a href="zhseqr.f.html#ZHSEQR.1">ZHSEQR</a>).
</span><span class="comment">*</span><span class="comment">          On exit, if SIDE = 'L' or 'B', VL contains:
</span><span class="comment">*</span><span class="comment">          if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
</span><span class="comment">*</span><span class="comment">          if HOWMNY = 'B', the matrix Q*Y;
</span><span class="comment">*</span><span class="comment">          if HOWMNY = 'S', the left eigenvectors of T specified by
</span><span class="comment">*</span><span class="comment">                           SELECT, stored consecutively in the columns
</span><span class="comment">*</span><span class="comment">                           of VL, in the same order as their
</span><span class="comment">*</span><span class="comment">                           eigenvalues.
</span><span class="comment">*</span><span class="comment">          Not referenced if SIDE = 'R'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDVL    (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array VL.  LDVL &gt;= 1, and if
</span><span class="comment">*</span><span class="comment">          SIDE = 'L' or 'B', LDVL &gt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  VR      (input/output) COMPLEX*16 array, dimension (LDVR,MM)
</span><span class="comment">*</span><span class="comment">          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
</span><span class="comment">*</span><span class="comment">          contain an N-by-N matrix Q (usually the unitary matrix Q of
</span><span class="comment">*</span><span class="comment">          Schur vectors returned by <a name="ZHSEQR.95"></a><a href="zhseqr.f.html#ZHSEQR.1">ZHSEQR</a>).
</span><span class="comment">*</span><span class="comment">          On exit, if SIDE = 'R' or 'B', VR contains:
</span><span class="comment">*</span><span class="comment">          if HOWMNY = 'A', the matrix X of right eigenvectors of T;
</span><span class="comment">*</span><span class="comment">          if HOWMNY = 'B', the matrix Q*X;
</span><span class="comment">*</span><span class="comment">          if HOWMNY = 'S', the right eigenvectors of T specified by
</span><span class="comment">*</span><span class="comment">                           SELECT, stored consecutively in the columns
</span><span class="comment">*</span><span class="comment">                           of VR, in the same order as their
</span><span class="comment">*</span><span class="comment">                           eigenvalues.
</span><span class="comment">*</span><span class="comment">          Not referenced if SIDE = 'L'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDVR    (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array VR.  LDVR &gt;= 1, and if
</span><span class="comment">*</span><span class="comment">          SIDE = 'R' or 'B'; LDVR &gt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  MM      (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of columns in the arrays VL and/or VR. MM &gt;= M.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  M       (output) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of columns in the arrays VL and/or VR actually
</span><span class="comment">*</span><span class="comment">          used to store the eigenvectors.  If HOWMNY = 'A' or 'B', M
</span><span class="comment">*</span><span class="comment">          is set to N.  Each selected eigenvector occupies one
</span><span class="comment">*</span><span class="comment">          column.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WORK    (workspace) COMPLEX*16 array, dimension (2*N)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
</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">
</span><span class="comment">*</span><span class="comment">  Further Details
</span><span class="comment">*</span><span class="comment">  ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The algorithm used in this program is basically backward (forward)
</span><span class="comment">*</span><span class="comment">  substitution, with scaling to make the the code robust against
</span><span class="comment">*</span><span class="comment">  possible overflow.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Each eigenvector is normalized so that the element of largest
</span><span class="comment">*</span><span class="comment">  magnitude has magnitude 1; here the magnitude of a complex number
</span><span class="comment">*</span><span class="comment">  (x,y) is taken to be |x| + |y|.
</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 )
      COMPLEX*16         CMZERO, CMONE
      PARAMETER          ( CMZERO = ( 0.0D+0, 0.0D+0 ),
     $                   CMONE = ( 1.0D+0, 0.0D+0 ) )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      LOGICAL            ALLV, BOTHV, LEFTV, OVER, RIGHTV, SOMEV
      INTEGER            I, II, IS, J, K, KI
      DOUBLE PRECISION   OVFL, REMAX, SCALE, SMIN, SMLNUM, ULP, UNFL
      COMPLEX*16         CDUM
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Functions ..
</span>      LOGICAL            <a name="LSAME.153"></a><a href="lsame.f.html#LSAME.1">LSAME</a>
      INTEGER            IZAMAX
      DOUBLE PRECISION   <a name="DLAMCH.155"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>, DZASUM
      EXTERNAL           <a name="LSAME.156"></a><a href="lsame.f.html#LSAME.1">LSAME</a>, IZAMAX, <a name="DLAMCH.156"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>, DZASUM
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Subroutines ..
</span>      EXTERNAL           <a name="XERBLA.159"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>, ZCOPY, ZDSCAL, ZGEMV, <a name="ZLATRS.159"></a><a href="zlatrs.f.html#ZLATRS.1">ZLATRS</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Statement Functions ..
</span>      DOUBLE PRECISION   CABS1
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Statement Function definitions ..
</span>      CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
<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 the input parameters
</span><span class="comment">*</span><span class="comment">
</span>      BOTHV = <a name="LSAME.174"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( SIDE, <span class="string">'B'</span> )
      RIGHTV = <a name="LSAME.175"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( SIDE, <span class="string">'R'</span> ) .OR. BOTHV
      LEFTV = <a name="LSAME.176"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( SIDE, <span class="string">'L'</span> ) .OR. BOTHV
<span class="comment">*</span><span class="comment">
</span>      ALLV = <a name="LSAME.178"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( HOWMNY, <span class="string">'A'</span> )
      OVER = <a name="LSAME.179"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( HOWMNY, <span class="string">'B'</span> )
      SOMEV = <a name="LSAME.180"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( HOWMNY, <span class="string">'S'</span> )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Set M to the number of columns required to store the selected
</span><span class="comment">*</span><span class="comment">     eigenvectors.
</span><span class="comment">*</span><span class="comment">
</span>      IF( SOMEV ) THEN
         M = 0
         DO 10 J = 1, N
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