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



NAME
       DTGEVC

SYNOPSIS
       SUBROUTINE DTGEVC (SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL,
           VR, LDVR, MM, M, WORK, INFO)



PURPOSE
            DTGEVC computes some or all of the right and/or left eigenvectors of
            a pair of real matrices (S,P), where S is a quasi-triangular matrix
            and P is upper triangular.  Matrix pairs of this type are produced by
            the generalized Schur factorization of a matrix pair (A,B):

               A = Q*S*Z**T,  B = Q*P*Z**T

            as computed by DGGHRD + DHGEQZ.

            The right eigenvector x and the left eigenvector y of (S,P)
            corresponding to an eigenvalue w are defined by:

               S*x = w*P*x,  (y**H)*S = w*(y**H)*P,

            where y**H denotes the conjugate tranpose of y.
            The eigenvalues are not input to this routine, but are computed
            directly from the diagonal blocks of S and P.

            This routine returns the matrices X and/or Y of right and left
            eigenvectors of (S,P), or the products Z*X and/or Q*Y,
            where Z and Q are input matrices.
            If Q and Z are the orthogonal factors from the generalized Schur
            factorization of a matrix pair (A,B), then Z*X and Q*Y
            are the matrices of right and left eigenvectors of (A,B).




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 by the matrices in VR and/or VL;
                     = 'S': compute selected right and/or left eigenvectors,
                            specified by the logical array SELECT.

           SELECT    (input)
                     SELECT is LOGICAL array, dimension (N)
                     If HOWMNY='S', SELECT specifies the eigenvectors to be
                     computed.  If w(j) is a real eigenvalue, the corresponding
                     real eigenvector is computed if SELECT(j) is .TRUE..
                     If w(j) and w(j+1) are the real and imaginary parts of a
                     complex eigenvalue, the corresponding complex eigenvector
                     is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
                     and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
                     set to .FALSE..
                     Not referenced if HOWMNY = 'A' or 'B'.

           N         (input)
                     N is INTEGER
                     The order of the matrices S and P.  N >= 0.

           S         (input)
                     S is DOUBLE PRECISION array, dimension (LDS,N)
                     The upper quasi-triangular matrix S from a generalized Schur
                     factorization, as computed by DHGEQZ.

           LDS       (input)
                     LDS is INTEGER
                     The leading dimension of array S.  LDS >= max(1,N).

           P         (input)
                     P is DOUBLE PRECISION array, dimension (LDP,N)
                     The upper triangular matrix P from a generalized Schur
                     factorization, as computed by DHGEQZ.
                     2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
                     of S must be in positive diagonal form.

           LDP       (input)
                     LDP is INTEGER
                     The leading dimension of array P.  LDP >= max(1,N).

           VL        (input/output)
                     VL is DOUBLE PRECISION 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 orthogonal matrix Q
                     of left Schur vectors returned by DHGEQZ).
                     On exit, if SIDE = 'L' or 'B', VL contains:
                     if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
                     if HOWMNY = 'B', the matrix Q*Y;
                     if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
                                 SELECT, stored consecutively in the columns of
                                 VL, in the same order as their eigenvalues.

                     A complex eigenvector corresponding to a complex eigenvalue
                     is stored in two consecutive columns, the first holding the
                     real part, and the second the imaginary part.

                     Not referenced if SIDE = 'R'.

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

           VR        (input/output)
                     VR is DOUBLE PRECISION array, dimension (LDVR,MM)
                     On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
                     contain an N-by-N matrix Z (usually the orthogonal matrix Z
                     of right Schur vectors returned by DHGEQZ).

                     On exit, if SIDE = 'R' or 'B', VR contains:
                     if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
                     if HOWMNY = 'B' or 'b', the matrix Z*X;
                     if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
                                 specified by SELECT, stored consecutively in the
                                 columns of VR, in the same order as their
                                 eigenvalues.

                     A complex eigenvector corresponding to a complex eigenvalue
                     is stored in two consecutive columns, the first holding the
                     real part and the second the imaginary part.

                     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 real eigenvector occupies one
                     column and each selected complex eigenvector occupies two
                     columns.

           WORK      (output)
                     WORK is DOUBLE PRECISION array, dimension (6*N)

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.
                     > 0:  the 2-by-2 block (INFO:INFO+1) does not have a complex
                           eigenvalue.






FURTHER DETAILS
             Allocation of workspace:
             ---------- -- ---------

                WORK( j ) = 1-norm of j-th column of A, above the diagonal
                WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
                WORK( 2*N+1:3*N ) = real part of eigenvector
                WORK( 3*N+1:4*N ) = imaginary part of eigenvector
                WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
                WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector

             Rowwise vs. columnwise solution methods:
             ------- --  ---------- -------- -------

             Finding a generalized eigenvector consists basically of solving the
             singular triangular system

              (A - w B) x = 0     (for right) or:   (A - w B)**H y = 0  (for left)

             Consider finding the i-th right eigenvector (assume all eigenvalues
             are real). The equation to be solved is:
                  n                   i
             0 = sum  C(j,k) v(k)  = sum  C(j,k) v(k)     for j = i,. . .,1
                 k=j                 k=j

             where  C = (A - w B)  (The components v(i+1:n) are 0.)

             The "rowwise" method is:

             (1)  v(i) := 1
             for j = i-1,. . .,1:
                                     i
                 (2) compute  s = - sum C(j,k) v(k)   and
                                   k=j+1

                 (3) v(j) := s / C(j,j)

             Step 2 is sometimes called the "dot product" step, since it is an
             inner product between the j-th row and the portion of the eigenvector
             that has been computed so far.

             The "columnwise" method consists basically in doing the sums
             for all the rows in parallel.  As each v(j) is computed, the
             contribution of v(j) times the j-th column of C is added to the
             partial sums.  Since FORTRAN arrays are stored columnwise, this has
             the advantage that at each step, the elements of C that are accessed
             are adjacent to one another, whereas with the rowwise method, the
             elements accessed at a step are spaced LDS (and LDP) words apart.

             When finding left eigenvectors, the matrix in question is the
             transpose of the one in storage, so the rowwise method then
             actually accesses columns of A and B at each step, and so is the
             preferred method.



LAPACK routine                  31 October 2017                      DTGEVC(3)