SYNOPSIS
 SUBROUTINE DTGEVC(
 SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, INFO )
 CHARACTER HOWMNY, SIDE
 INTEGER INFO, LDP, LDS, LDVL, LDVR, M, MM, N
 LOGICAL SELECT( * )
 DOUBLE PRECISION P( LDP, * ), S( LDS, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
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 quasitriangular 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) CHARACTER*1

= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors.  HOWMNY (input) 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) 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) INTEGER
 The order of the matrices S and P. N >= 0.
 S (input) DOUBLE PRECISION array, dimension (LDS,N)
 The upper quasitriangular matrix S from a generalized Schur factorization, as computed by DHGEQZ.
 LDS (input) INTEGER
 The leading dimension of array S. LDS >= max(1,N).
 P (input) DOUBLE PRECISION array, dimension (LDP,N)
 The upper triangular matrix P from a generalized Schur factorization, as computed by DHGEQZ. 2by2 diagonal blocks of P corresponding to 2by2 blocks of S must be in positive diagonal form.
 LDP (input) INTEGER
 The leading dimension of array P. LDP >= max(1,N).
 VL (input/output) DOUBLE PRECISION array, dimension (LDVL,MM)
 On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must contain an NbyN 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) INTEGER
 The leading dimension of array VL. LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.
 VR (input/output) DOUBLE PRECISION array, dimension (LDVR,MM)
 On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must contain an NbyN 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) INTEGER
 The leading dimension of the array VR. LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.
 MM (input) INTEGER
 The number of columns in the arrays VL and/or VR. MM >= M.
 M (output) 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 (workspace) DOUBLE PRECISION array, dimension (6*N)
 INFO (output) INTEGER

= 0: successful exit.
< 0: if INFO = i, the ith argument had an illegal value.
> 0: the 2by2 block (INFO:INFO+1) does not have a complex eigenvalue.
FURTHER DETAILS
Allocation of workspace:  
WORK( j ) = 1norm of jth column of A, above the diagonal
WORK( N+j ) = 1norm of jth 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 backtransformed eigenvector
WORK( 5*N+1:6*N ) = imaginary part of backtransformed 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 ith 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 = i1,. . .,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 jth 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 jth 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.