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).

SIDE

          SIDE is CHARACTER*1


          = 'R': compute right eigenvectors only;


          = 'L': compute left eigenvectors only;


          = 'B': compute both right and left eigenvectors.

HOWMNY

          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

          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

          N is INTEGER


          The order of the matrices S and P.  N >= 0.

S

          S is DOUBLE PRECISION array, dimension (LDS,N)


          The upper quasi-triangular matrix S from a generalized Schur


          factorization, as computed by DHGEQZ.

LDS

          LDS is INTEGER


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

P

          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

          LDP is INTEGER


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

VL

          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

          LDVL is INTEGER


          The leading dimension of array VL.  LDVL >= 1, and if


          SIDE = 'L' or 'B', LDVL >= N.

VR

          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

          LDVR is INTEGER


          The leading dimension of the array VR.  LDVR >= 1, and if


          SIDE = 'R' or 'B', LDVR >= N.

MM

          MM is INTEGER


          The number of columns in the arrays VL and/or VR. MM >= M.

M

          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

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

INFO

          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

  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.