SGEGS (3lapack)

compute for a pair of N-by-N real nonsymmetric matrices A, B

SYNOPSIS

  SUBROUTINE SGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA,
                    VSL, LDVSL, VSR, LDVSR, WORK, LWORK, INFO )

      CHARACTER     JOBVSL, JOBVSR

      INTEGER       INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N

      REAL          A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA(
                    * ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * )

PURPOSE

  SGEGS computes for a pair of N-by-N real nonsymmetric matrices A, B: the
  generalized eigenvalues (alphar +/- alphai*i, beta), the real Schur form
  (A, B), and optionally left and/or right Schur vectors (VSL and VSR).

  (If only the generalized eigenvalues are needed, use the driver SGEGV
  instead.)

  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)

  The (generalized) Schur form of a pair of matrices is the result of
  multiplying both matrices on the left by one orthogonal matrix and both on
  the right by another orthogonal matrix, these two orthogonal matrices being
  chosen so as to bring the pair of matrices into (real) Schur form.

  A pair of matrices A, B is in generalized real Schur form if B is upper
  triangular with non-negative diagonal and A is block upper triangular with
  1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond to real generalized
  eigenvalues, while 2-by-2 blocks of A will be "standardized" by making the
  corresponding elements of B have the form:
          [  a  0  ]
          [  0  b  ]

  and the pair of corresponding 2-by-2 blocks in A and B will have a complex
  conjugate pair of generalized eigenvalues.

  The left and right Schur vectors are the columns of VSL and VSR,
  respectively, where VSL and VSR are the orthogonal matrices which reduce A
  and B to Schur form:

  Schur form of (A,B) = ( (VSL)**T A (VSR), (VSL)**T B (VSR) )

ARGUMENTS

  JOBVSL  (input) CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors.

  JOBVSR  (input) CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors.

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

  A       (input/output) REAL array, dimension (LDA, N)
          On entry, the first of the pair of matrices whose generalized
          eigenvalues and (optionally) Schur vectors are to be computed.  On
          exit, the generalized Schur form of A.  Note: to avoid overflow,
          the Frobenius norm of the matrix A should be less than the overflow
          threshold.

  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) Schur vectors are to be computed.  On
          exit, the generalized Schur form of B.  Note: to avoid overflow,
          the Frobenius norm of the matrix B should be less than the overflow
          threshold.

  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.  ALPHAR(j) +
          ALPHAI(j)*i, j=1,...,N  and  BETA(j),j=1,...,N  are the diagonals
          of the complex Schur form (A,B) that would result if the 2-by-2
          diagonal blocks of the real Schur form of (A,B) were further
          reduced to triangular form using 2-by-2 complex unitary
          transformations.  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).

  VSL     (output) REAL array, dimension (LDVSL,N)
          If JOBVSL = 'V', VSL will contain the left Schur vectors.  (See
          "Purpose", above.) Not referenced if JOBVSL = 'N'.

  LDVSL   (input) INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL =
          'V', LDVSL >= N.

  VSR     (output) REAL array, dimension (LDVSR,N)
          If JOBVSR = 'V', VSR will contain the right Schur vectors.  (See
          "Purpose", above.) Not referenced if JOBVSR = 'N'.

  LDVSR   (input) INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR
          = 'V', LDVSR >= 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(1,4*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 +
          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.  (A,B) are not in Schur form,
          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 SGGBAK (computing VSL)
          =N+8: error return from SGGBAK (computing VSR)
          =N+9: error return from SLASCL (various places)