tgevc(3)

NAME
tgevc - tgevc: eigvec of pair of matrices

SYNOPSIS

  Functions

subroutine ctgevc (side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, rwork, info)
CTGEVC
subroutine dtgevc (side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, info)
DTGEVC
subroutine stgevc (side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, info)
STGEVC
subroutine ztgevc (side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, rwork, info)
ZTGEVC

Detailed Description

Function Documentation

  subroutine ctgevc (character side, character howmny, logical, dimension( * ) select, integer n, complex, dimension( lds, * ) s, integer lds, complex, dimension( ldp, * ) p, integer ldp, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CTGEVC

Purpose:

    !>
    !> CTGEVC computes some or all of the right and/or left eigenvectors of
    !> a pair of complex matrices (S,P), where S and P are upper triangular.
    !> Matrix pairs of this type are produced by the generalized Schur
    !> factorization of a complex matrix pair (A,B):
    !>
    !>    A = Q*S*Z**H,  B = Q*P*Z**H
    !>
    !> as computed by CGGHRD + CHGEQZ.
    !>
    !> 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 transpose of y.
    !> The eigenvalues are not input to this routine, but are computed
    !> directly from the diagonal elements 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 unitary 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).
    !>

Parameters
    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.  The eigenvector corresponding to the j-th
    !>          eigenvalue is computed if SELECT(j) = .TRUE..
    !>          Not referenced if HOWMNY = 'A' or 'B'.
    !>

    N

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

    S

    !>          S is COMPLEX array, dimension (LDS,N)
    !>          The upper triangular matrix S from a generalized Schur
    !>          factorization, as computed by CHGEQZ.
    !>

    LDS

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

    P

    !>          P is COMPLEX array, dimension (LDP,N)
    !>          The upper triangular matrix P from a generalized Schur
    !>          factorization, as computed by CHGEQZ.  P must have real
    !>          diagonal elements.
    !>

    LDP

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

    VL

    !>          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 left Schur vectors returned by CHGEQZ).
    !>          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.
    !>          Not referenced if SIDE = 'R'.
    !>

    LDVL

    !>          LDVL is INTEGER
    !>          The leading dimension of array VL.  LDVL >= 1, and if
    !>          SIDE = 'L' or 'l' or 'B' or 'b', LDVL >= N.
    !>

    VR

    !>          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 Z (usually the unitary matrix Z
    !>          of right Schur vectors returned by CHGEQZ).
    !>          On exit, if SIDE = 'R' or 'B', VR contains:
    !>          if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
    !>          if HOWMNY = 'B', the matrix Z*X;
    !>          if HOWMNY = 'S', the right eigenvectors of (S,P) specified by
    !>                      SELECT, stored consecutively in the columns of
    !>                      VR, in the same order as their eigenvalues.
    !>          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 eigenvector occupies one column.
    !>

    WORK

    !>          WORK is COMPLEX array, dimension (2*N)
    !>

    RWORK

    !>          RWORK is REAL array, dimension (2*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine dtgevc (character side, character howmny, logical, dimension( * ) select, integer n, double precision, dimension( lds, * ) s, integer lds, double precision, dimension( ldp, * ) p, integer ldp, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, double precision, dimension( * ) work, integer info)

DTGEVC

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 transpose 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).
    !>
    !>

Parameters
    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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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  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  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  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.
    !>

  subroutine stgevc (character side, character howmny, logical, dimension( * ) select, integer n, real, dimension( lds, * ) s, integer lds, real, dimension( ldp, * ) p, integer ldp, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, real, dimension( * ) work, integer info)

STGEVC

Purpose:

    !>
    !> STGEVC 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 SGGHRD + SHGEQZ.
    !>
    !> 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 transpose 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).
    !>
    !>

Parameters
    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 REAL array, dimension (LDS,N)
    !>          The upper quasi-triangular matrix S from a generalized Schur
    !>          factorization, as computed by SHGEQZ.
    !>

    LDS

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

    P

    !>          P is REAL array, dimension (LDP,N)
    !>          The upper triangular matrix P from a generalized Schur
    !>          factorization, as computed by SHGEQZ.
    !>          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 REAL 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 SHGEQZ).
    !>          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 REAL 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 SHGEQZ).
    !>
    !>          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 REAL 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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  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  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  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.
    !>

  subroutine ztgevc (character side, character howmny, logical, dimension( * ) select, integer n, complex*16, dimension( lds, * ) s, integer lds, complex*16, dimension( ldp, * ) p, integer ldp, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZTGEVC

Purpose:

    !>
    !> ZTGEVC computes some or all of the right and/or left eigenvectors of
    !> a pair of complex matrices (S,P), where S and P are upper triangular.
    !> Matrix pairs of this type are produced by the generalized Schur
    !> factorization of a complex matrix pair (A,B):
    !>
    !>    A = Q*S*Z**H,  B = Q*P*Z**H
    !>
    !> as computed by ZGGHRD + ZHGEQZ.
    !>
    !> 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 transpose of y.
    !> The eigenvalues are not input to this routine, but are computed
    !> directly from the diagonal elements 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 unitary 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).
    !>

Parameters
    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.  The eigenvector corresponding to the j-th
    !>          eigenvalue is computed if SELECT(j) = .TRUE..
    !>          Not referenced if HOWMNY = 'A' or 'B'.
    !>

    N

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

    S

    !>          S is COMPLEX*16 array, dimension (LDS,N)
    !>          The upper triangular matrix S from a generalized Schur
    !>          factorization, as computed by ZHGEQZ.
    !>

    LDS

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

    P

    !>          P is COMPLEX*16 array, dimension (LDP,N)
    !>          The upper triangular matrix P from a generalized Schur
    !>          factorization, as computed by ZHGEQZ.  P must have real
    !>          diagonal elements.
    !>

    LDP

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

    VL

    !>          VL is COMPLEX*16 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 left Schur vectors returned by ZHGEQZ).
    !>          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.
    !>          Not referenced if SIDE = 'R'.
    !>

    LDVL

    !>          LDVL is INTEGER
    !>          The leading dimension of array VL.  LDVL >= 1, and if
    !>          SIDE = 'L' or 'l' or 'B' or 'b', LDVL >= N.
    !>

    VR

    !>          VR is COMPLEX*16 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 unitary matrix Z
    !>          of right Schur vectors returned by ZHGEQZ).
    !>          On exit, if SIDE = 'R' or 'B', VR contains:
    !>          if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
    !>          if HOWMNY = 'B', the matrix Z*X;
    !>          if HOWMNY = 'S', the right eigenvectors of (S,P) specified by
    !>                      SELECT, stored consecutively in the columns of
    !>                      VR, in the same order as their eigenvalues.
    !>          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 eigenvector occupies one column.
    !>

    WORK

    !>          WORK is COMPLEX*16 array, dimension (2*N)
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (2*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit.
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Author

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