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## dggev3()

subroutine dggev3 | ( | character | jobvl,
|  | character | jobvr,
|  | integer | n,
|  | double precision, dimension( lda, * ) | a,
|  | integer | lda,
|  | double precision, dimension( ldb, * ) | b,
|  | integer | ldb,
|  | double precision, dimension( * ) | alphar,
|  | double precision, dimension( * ) | alphai,
|  | double precision, dimension( * ) | beta,
|  | double precision, dimension( ldvl, * ) | vl,
|  | integer | ldvl,
|  | double precision, dimension( ldvr, * ) | vr,
|  | integer | ldvr,
|  | double precision, dimension( * ) | work,
|  | integer | lwork,
|  | integer | info )

DGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

Download DGGEV3 + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> DGGEV3 computes for a pair of N-by-N real nonsymmetric matrices (A,B)
      !> the generalized eigenvalues, and optionally, the left and/or right
      !> generalized eigenvectors.
      !>
      !> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
      !> lambda or a ratio alpha/beta = lambda, such that A - lambda*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.
      !>
      !> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
      !> of (A,B) satisfies
      !>
      !>                  A * v(j) = lambda(j) * B * v(j).
      !>
      !> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
      !> of (A,B) satisfies
      !>
      !>                  u(j)**H * A  = lambda(j) * u(j)**H * B .
      !>
      !> where u(j)**H is the conjugate-transpose of u(j).
      !>
      !>

Parameters

  [in]

  JOBVL

      !>          JOBVL is CHARACTER*1
      !>          = 'N':  do not compute the left generalized eigenvectors;
      !>          = 'V':  compute the left generalized eigenvectors.
      !>

  [in]

  JOBVR

      !>          JOBVR is CHARACTER*1
      !>          = 'N':  do not compute the right generalized eigenvectors;
      !>          = 'V':  compute the right generalized eigenvectors.
      !>

  [in]

  N

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

  [in,out]

  A

      !>          A is DOUBLE PRECISION array, dimension (LDA, N)
      !>          On entry, the matrix A in the pair (A,B).
      !>          On exit, A has been overwritten.
      !>

  [in]

  LDA

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

  [in,out]

  B

      !>          B is DOUBLE PRECISION array, dimension (LDB, N)
      !>          On entry, the matrix B in the pair (A,B).
      !>          On exit, B has been overwritten.
      !>

  [in]

  LDB

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

  [out]

  ALPHAR

      !>          ALPHAR is DOUBLE PRECISION array, dimension (N)
      !>

  [out]

  ALPHAI

      !>          ALPHAI is DOUBLE PRECISION array, dimension (N)
      !>

  [out]

  BETA

      !>          BETA is DOUBLE PRECISION 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).
      !>

  [out]

  VL

      !>          VL is DOUBLE PRECISION array, dimension (LDVL,N)
      !>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
      !>          after another in the columns of VL, in the same order as
      !>          their eigenvalues. If the j-th eigenvalue is real, then
      !>          u(j) = VL(:,j), the j-th column of VL. If the j-th and
      !>          (j+1)-th 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 the largest component has
      !>          abs(real part)+abs(imag. part)=1.
      !>          Not referenced if JOBVL = 'N'.
      !>

  [in]

  LDVL

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

  [out]

  VR

      !>          VR is DOUBLE PRECISION array, dimension (LDVR,N)
      !>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
      !>          after another in the columns of VR, in the same order as
      !>          their eigenvalues. If the j-th eigenvalue is real, then
      !>          v(j) = VR(:,j), the j-th column of VR. If the j-th and
      !>          (j+1)-th eigenvalues form a complex conjugate pair, then
      !>          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
      !>          Each eigenvector is scaled so the largest component has
      !>          abs(real part)+abs(imag. part)=1.
      !>          Not referenced if JOBVR = 'N'.
      !>

  [in]

  LDVR

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

  [out]

  WORK

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

  [in]

  LWORK

      !>          LWORK is INTEGER.
      !>          The dimension of the array WORK. LWORK >= MAX(1,8*N).
      !>          For good performance, LWORK should generally be larger.
      !>
      !>          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.
      !>

  [out]

  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:  =N+1: other than QZ iteration failed in DLAQZ0.
      !>                =N+2: error return from DTGEVC.
      !>

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.