CXML

SGEGV (3lapack)


SYNOPSIS

  SUBROUTINE SGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA,
                    VL, LDVL, VR, LDVR, WORK, LWORK, INFO )

      CHARACTER     JOBVL, JOBVR

      INTEGER       INFO, LDA, LDB, LDVL, LDVR, LWORK, N

      REAL          A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA(
                    * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

PURPOSE

  SGEGV computes for a pair of n-by-n real nonsymmetric matrices A and B, the
  generalized eigenvalues (alphar +/- alphai*i, beta), and optionally, the
  left and/or right generalized eigenvectors (VL and VR).

  A generalized eigenvalue for a pair of matrices (A,B) is, roughly speaking,
  a scalar w or a ratio  alpha/beta = w, such that  A - w*B is singular.  It
  is usually represented as the pair (alpha,beta), as there is a reasonable
  interpretation for beta=0, and even for both being zero.  A good beginning
  reference is the book, "Matrix Computations", by G. Golub & C. van Loan
  (Johns Hopkins U. Press)

  A right generalized eigenvector corresponding to a generalized eigenvalue
  w  for a pair of matrices (A,B) is a vector  r  such that  (A - w B) r = 0
  .  A left generalized eigenvector is a vector l such that l**H * (A - w B)
  = 0, where l**H is the
  conjugate-transpose of l.

  Note: this routine performs "full balancing" on A and B -- see "Further
  Details", below.

ARGUMENTS

  JOBVL   (input) CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors.

  JOBVR   (input) CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors.

  N       (input) INTEGER
          The order of the matrices A, B, VL, and VR.  N >= 0.

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the first of the pair of matrices whose generalized
          eigenvalues and (optionally) generalized eigenvectors are to be
          computed.  On exit, the contents will have been destroyed.  (For a
          description of the contents of A on exit, see "Further Details",
          below.)

  LDA     (input) INTEGER
          The leading dimension of A.  LDA >= max(1,N).

  B       (input/output) REAL array, dimension (LDB, N)
          On entry, the second of the pair of matrices whose generalized
          eigenvalues and (optionally) generalized eigenvectors are to be
          computed.  On exit, the contents will have been destroyed.  (For a
          description of the contents of B on exit, see "Further Details",
          below.)

  LDB     (input) INTEGER
          The leading dimension of B.  LDB >= max(1,N).

  ALPHAR  (output) REAL array, dimension (N)
          ALPHAI  (output) REAL array, dimension (N) BETA    (output) REAL
          array, dimension (N) On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j),
          j=1,...,N, will be the generalized eigenvalues.  If ALPHAI(j) is
          zero, then the j-th eigenvalue is real; if positive, then the j-th
          and (j+1)-st eigenvalues are a complex conjugate pair, with
          ALPHAI(j+1) negative.

          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may
          easily over- or underflow, and BETA(j) may even be zero.  Thus, the
          user should avoid naively computing the ratio alpha/beta.  However,
          ALPHAR and ALPHAI will be always less than and usually comparable
          with norm(A) in magnitude, and BETA always less than and usually
          comparable with norm(B).

  VL      (output) REAL array, dimension (LDVL,N)
          If JOBVL = 'V', the left generalized eigenvectors.  (See "Purpose",
          above.)  Real eigenvectors take one column, complex take two
          columns, the first for the real part and the second for the
          imaginary part.  Complex eigenvectors correspond to an eigenvalue
          with positive imaginary part.  Each eigenvector will be scaled so
          the largest component will have abs(real part) + abs(imag. part) =
          1, *except* that for eigenvalues with alpha=beta=0, a zero vector
          will be returned as the corresponding eigenvector.  Not referenced
          if JOBVL = 'N'.

  LDVL    (input) INTEGER
          The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL =
          'V', LDVL >= N.

  VR      (output) REAL array, dimension (LDVR,N)
          If JOBVL = 'V', the right generalized eigenvectors.  (See
          "Purpose", above.)  Real eigenvectors take one column, complex take
          two columns, the first for the real part and the second for the
          imaginary part.  Complex eigenvectors correspond to an eigenvalue
          with positive imaginary part.  Each eigenvector will be scaled so
          the largest component will have abs(real part) + abs(imag. part) =
          1, *except* that for eigenvalues with alpha=beta=0, a zero vector
          will be returned as the corresponding eigenvector.  Not referenced
          if JOBVR = 'N'.

  LDVR    (input) INTEGER
          The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR =
          'V', LDVR >= N.

  WORK    (workspace/output) REAL array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= max(1,8*N).  For good
          performance, LWORK must generally be larger.  To compute the
          optimal value of LWORK, call ILAENV to get blocksizes (for SGEQRF,
          SORMQR, and SORGQR.)  Then compute: NB  -- MAX of the blocksizes
          for SGEQRF, SORMQR, and SORGQR; The optimal LWORK is: 2*N + MAX(
          6*N, N*(NB+1) ).

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N: The QZ iteration failed.  No eigenvectors have been
          calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct
          for j=INFO+1,...,N.  > N:  errors that usually indicate LAPACK
          problems:
          =N+1: error return from SGGBAL
          =N+2: error return from SGEQRF
          =N+3: error return from SORMQR
          =N+4: error return from SORGQR
          =N+5: error return from SGGHRD
          =N+6: error return from SHGEQZ (other than failed iteration) =N+7:
          error return from STGEVC
          =N+8: error return from SGGBAK (computing VL)
          =N+9: error return from SGGBAK (computing VR)
          =N+10: error return from SLASCL (various calls)

FURTHER DETAILS

  Balancing
  ---------

  This driver calls SGGBAL to both permute and scale rows and columns of A
  and B.  The permutations PL and PR are chosen so that PL*A*PR and PL*B*R
  will be upper triangular except for the diagonal blocks A(i:j,i:j) and
  B(i:j,i:j), with i and j as close together as possible.  The diagonal
  scaling matrices DL and DR are chosen so that the pair  DL*PL*A*PR*DR,
  DL*PL*B*PR*DR have elements close to one (except for the elements that
  start out zero.)

  After the eigenvalues and eigenvectors of the balanced matrices have been
  computed, SGGBAK transforms the eigenvectors back to what they would have
  been (in perfect arithmetic) if they had not been balanced.

  Contents of A and B on Exit
  -------- -- - --- - -- ----

  If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or both),
  then on exit the arrays A and B will contain the real Schur form[*] of the
  "balanced" versions of A and B.  If no eigenvectors are computed, then only
  the diagonal blocks will be correct.

  [*] See SHGEQZ, SGEGS, or read the book "Matrix Computations",
      by Golub & van Loan, pub. by Johns Hopkins U. Press.

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