sgegv.f(3)

NAME
sgegv.f -

SYNOPSIS

  Functions/Subroutines

subroutine sgegv (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Function/Subroutine Documentation

  subroutine sgegv (characterJOBVL, characterJOBVR, integerN, real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B, integerLDB, real, dimension( * )ALPHAR, real, dimension( * )ALPHAI, real, dimension( * )BETA, real, dimension( ldvl, * )VL, integerLDVL, real, dimension( ldvr, * )VR, integerLDVR, real, dimension( * )WORK, integerLWORK, integerINFO)

SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

     This routine is deprecated and has been replaced by routine SGGEV.

     SGEGV computes the eigenvalues and, optionally, the left and/or right
     eigenvectors of a real matrix pair (A,B).
     Given two square matrices A and B,
     the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
     eigenvalues lambda and corresponding (non-zero) eigenvectors x such
     that

        A*x = lambda*B*x.

     An alternate form is to find the eigenvalues mu and corresponding
     eigenvectors y such that

        mu*A*y = B*y.

     These two forms are equivalent with mu = 1/lambda and x = y if
     neither lambda nor mu is zero.  In order to deal with the case that
     lambda or mu is zero or small, two values alpha and beta are returned
     for each eigenvalue, such that lambda = alpha/beta and
     mu = beta/alpha.

     The vectors x and y in the above equations are right eigenvectors of
     the matrix pair (A,B).  Vectors u and v satisfying

        u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B

     are left eigenvectors of (A,B).

     Note: this routine performs "full balancing" on A and B

Parameters:
    JOBVL

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

    JOBVR

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

    N

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

    A

              A is REAL array, dimension (LDA, N)
              On entry, the matrix A.
              If JOBVL = 'V' or JOBVR = 'V', then on exit A
              contains the real Schur form of A from the generalized Schur
              factorization of the pair (A,B) after balancing.
              If no eigenvectors were computed, then only the diagonal
              blocks from the Schur form will be correct.  See SGGHRD and
              SHGEQZ for details.

    LDA

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

    B

              B is REAL array, dimension (LDB, N)
              On entry, the matrix B.
              If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
              upper triangular matrix obtained from B in the generalized
              Schur factorization of the pair (A,B) after balancing.
              If no eigenvectors were computed, then only those elements of
              B corresponding to the diagonal blocks from the Schur form of
              A will be correct.  See SGGHRD and SHGEQZ for details.

    LDB

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

    ALPHAR

              ALPHAR is REAL array, dimension (N)
              The real parts of each scalar alpha defining an eigenvalue of
              GNEP.

    ALPHAI

              ALPHAI is REAL array, dimension (N)
              The imaginary parts of each scalar alpha defining an
              eigenvalue of GNEP.  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) = -ALPHAI(j).

    BETA

              BETA is REAL array, dimension (N)
              The scalars beta that define the eigenvalues of GNEP.

              Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
              beta = BETA(j) represent the j-th eigenvalue of the matrix
              pair (A,B), in one of the forms lambda = alpha/beta or
              mu = beta/alpha.  Since either lambda or mu may overflow,
              they should not, in general, be computed.

    VL

              VL is REAL array, dimension (LDVL,N)
              If JOBVL = 'V', the left eigenvectors u(j) are stored
              in the columns of VL, in the same order as their eigenvalues.
              If the j-th eigenvalue is real, then u(j) = VL(:,j).
              If the j-th and (j+1)-st eigenvalues form a complex conjugate
              pair, then
                 u(j) = VL(:,j) + i*VL(:,j+1)
              and
                u(j+1) = VL(:,j) - i*VL(:,j+1).

              Each eigenvector is scaled so that its largest component has
              abs(real part) + abs(imag. part) = 1, except for eigenvectors
              corresponding to an eigenvalue with alpha = beta = 0, which
              are set to zero.
              Not referenced if JOBVL = 'N'.

    LDVL

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

    VR

              VR is REAL array, dimension (LDVR,N)
              If JOBVR = 'V', the right eigenvectors x(j) are stored
              in the columns of VR, in the same order as their eigenvalues.
              If the j-th eigenvalue is real, then x(j) = VR(:,j).
              If the j-th and (j+1)-st eigenvalues form a complex conjugate
              pair, then
                x(j) = VR(:,j) + i*VR(:,j+1)
              and
                x(j+1) = VR(:,j) - i*VR(:,j+1).

              Each eigenvector is scaled so that its largest component has
              abs(real part) + abs(imag. part) = 1, except for eigenvalues
              corresponding to an eigenvalue with alpha = beta = 0, which
              are set to zero.
              Not referenced if JOBVR = 'N'.

    LDVR

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

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is 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) ).

              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is 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)

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    November 2011

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.

Definition at line 306 of file sgegv.f.

Author

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