CTREVC(3)      LAPACK routine of NEC Numeric Library Collection      CTREVC(3)



NAME
       CTREVC

SYNOPSIS
       SUBROUTINE CTREVC (SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR,
           MM, M, WORK, RWORK, INFO)



PURPOSE
            CTREVC computes some or all of the right and/or left eigenvectors of
            a complex upper triangular matrix T.
            Matrices of this type are produced by the Schur factorization of
            a complex general matrix:  A = Q*T*Q**H, as computed by CHSEQR.

            The right eigenvector x and the left eigenvector y of T corresponding
            to an eigenvalue w are defined by:

                         T*x = w*x,     (y**H)*T = w*(y**H)

            where y**H denotes the conjugate transpose of the vector y.
            The eigenvalues are not input to this routine, but are read directly
            from the diagonal of T.

            This routine returns the matrices X and/or Y of right and left
            eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
            input matrix.  If Q is the unitary factor that reduces a matrix A to
            Schur form T, then Q*X and Q*Y are the matrices of right and left
            eigenvectors of A.




ARGUMENTS
           SIDE      (input)
                     SIDE is CHARACTER*1
                     = 'R':  compute right eigenvectors only;
                     = 'L':  compute left eigenvectors only;
                     = 'B':  compute both right and left eigenvectors.

           HOWMNY    (input)
                     HOWMNY is CHARACTER*1
                     = 'A':  compute all right and/or left eigenvectors;
                     = 'B':  compute all right and/or left eigenvectors,
                             backtransformed using the matrices supplied in
                             VR and/or VL;
                     = 'S':  compute selected right and/or left eigenvectors,
                             as indicated by the logical array SELECT.

           SELECT    (input)
                     SELECT is LOGICAL array, dimension (N)
                     If HOWMNY = 'S', SELECT specifies the eigenvectors to be
                     computed.
                     The eigenvector corresponding to the j-th eigenvalue is
                     computed if SELECT(j) = .TRUE..
                     Not referenced if HOWMNY = 'A' or 'B'.

           N         (input)
                     N is INTEGER
                     The order of the matrix T. N >= 0.

           T         (input/output)
                     T is COMPLEX array, dimension (LDT,N)
                     The upper triangular matrix T.  T is modified, but restored
                     on exit.

           LDT       (input)
                     LDT is INTEGER
                     The leading dimension of the array T. LDT >= max(1,N).

           VL        (input/output)
                     VL is COMPLEX array, dimension (LDVL,MM)
                     On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
                     contain an N-by-N matrix Q (usually the unitary matrix Q of
                     Schur vectors returned by CHSEQR).
                     On exit, if SIDE = 'L' or 'B', VL contains:
                     if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
                     if HOWMNY = 'B', the matrix Q*Y;
                     if HOWMNY = 'S', the left eigenvectors of T specified by
                                      SELECT, stored consecutively in the columns
                                      of VL, in the same order as their
                                      eigenvalues.
                     Not referenced if SIDE = 'R'.

           LDVL      (input)
                     LDVL is INTEGER
                     The leading dimension of the array VL.  LDVL >= 1, and if
                     SIDE = 'L' or 'B', LDVL >= N.

           VR        (input/output)
                     VR is COMPLEX array, dimension (LDVR,MM)
                     On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
                     contain an N-by-N matrix Q (usually the unitary matrix Q of
                     Schur vectors returned by CHSEQR).
                     On exit, if SIDE = 'R' or 'B', VR contains:
                     if HOWMNY = 'A', the matrix X of right eigenvectors of T;
                     if HOWMNY = 'B', the matrix Q*X;
                     if HOWMNY = 'S', the right eigenvectors of T specified by
                                      SELECT, stored consecutively in the columns
                                      of VR, in the same order as their
                                      eigenvalues.
                     Not referenced if SIDE = 'L'.

           LDVR      (input)
                     LDVR is INTEGER
                     The leading dimension of the array VR.  LDVR >= 1, and if
                     SIDE = 'R' or 'B'; LDVR >= N.

           MM        (input)
                     MM is INTEGER
                     The number of columns in the arrays VL and/or VR. MM >= M.

           M         (output)
                     M is INTEGER
                     The number of columns in the arrays VL and/or VR actually
                     used to store the eigenvectors.  If HOWMNY = 'A' or 'B', M
                     is set to N.  Each selected eigenvector occupies one
                     column.

           WORK      (output)
                     WORK is COMPLEX array, dimension (2*N)

           RWORK     (output)
                     RWORK is REAL array, dimension (N)

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value






FURTHER DETAILS
             The algorithm used in this program is basically backward (forward)
             substitution, with scaling to make the the code robust against
             possible overflow.

             Each eigenvector is normalized so that the element of largest
             magnitude has magnitude 1; here the magnitude of a complex number
             (x,y) is taken to be |x| + |y|.



LAPACK routine                  31 October 2017                      CTREVC(3)