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## zgeevx()

subroutine zgeevx | ( | character | balanc,
|  | character | jobvl,
|  | character | jobvr,
|  | character | sense,
|  | integer | n,
|  | complex*16, dimension( lda, * ) | a,
|  | integer | lda,
|  | complex*16, dimension( * ) | w,
|  | complex*16, dimension( ldvl, * ) | vl,
|  | integer | ldvl,
|  | complex*16, dimension( ldvr, * ) | vr,
|  | integer | ldvr,
|  | integer | ilo,
|  | integer | ihi,
|  | double precision, dimension( * ) | scale,
|  | double precision | abnrm,
|  | double precision, dimension( * ) | rconde,
|  | double precision, dimension( * ) | rcondv,
|  | complex*16, dimension( * ) | work,
|  | integer | lwork,
|  | double precision, dimension( * ) | rwork,
|  | integer | info )

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

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

Purpose:

      !>
      !> ZGEEVX computes for an N-by-N complex nonsymmetric matrix A, the
      !> eigenvalues and, optionally, the left and/or right eigenvectors.
      !>
      !> Optionally also, it computes a balancing transformation to improve
      !> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
      !> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
      !> (RCONDE), and reciprocal condition numbers for the right
      !> eigenvectors (RCONDV).
      !>
      !> The right eigenvector v(j) of A satisfies
      !>                  A * v(j) = lambda(j) * v(j)
      !> where lambda(j) is its eigenvalue.
      !> The left eigenvector u(j) of A satisfies
      !>               u(j)**H * A = lambda(j) * u(j)**H
      !> where u(j)**H denotes the conjugate transpose of u(j).
      !>
      !> The computed eigenvectors are normalized to have Euclidean norm
      !> equal to 1 and largest component real.
      !>
      !> Balancing a matrix means permuting the rows and columns to make it
      !> more nearly upper triangular, and applying a diagonal similarity
      !> transformation D * A * D**(-1), where D is a diagonal matrix, to
      !> make its rows and columns closer in norm and the condition numbers
      !> of its eigenvalues and eigenvectors smaller.  The computed
      !> reciprocal condition numbers correspond to the balanced matrix.
      !> Permuting rows and columns will not change the condition numbers
      !> (in exact arithmetic) but diagonal scaling will.  For further
      !> explanation of balancing, see section 4.10.2 of the LAPACK
      !> Users' Guide.
      !>

Parameters

  [in]

  BALANC

      !>          BALANC is CHARACTER*1
      !>          Indicates how the input matrix should be diagonally scaled
      !>          and/or permuted to improve the conditioning of its
      !>          eigenvalues.
      !>          = 'N': Do not diagonally scale or permute;
      !>          = 'P': Perform permutations to make the matrix more nearly
      !>                 upper triangular. Do not diagonally scale;
      !>          = 'S': Diagonally scale the matrix, ie. replace A by
      !>                 D*A*D**(-1), where D is a diagonal matrix chosen
      !>                 to make the rows and columns of A more equal in
      !>                 norm. Do not permute;
      !>          = 'B': Both diagonally scale and permute A.
      !>
      !>          Computed reciprocal condition numbers will be for the matrix
      !>          after balancing and/or permuting. Permuting does not change
      !>          condition numbers (in exact arithmetic), but balancing does.
      !>

  [in]

  JOBVL

      !>          JOBVL is CHARACTER*1
      !>          = 'N': left eigenvectors of A are not computed;
      !>          = 'V': left eigenvectors of A are computed.
      !>          If SENSE = 'E' or 'B', JOBVL must = 'V'.
      !>

  [in]

  JOBVR

      !>          JOBVR is CHARACTER*1
      !>          = 'N': right eigenvectors of A are not computed;
      !>          = 'V': right eigenvectors of A are computed.
      !>          If SENSE = 'E' or 'B', JOBVR must = 'V'.
      !>

  [in]

  SENSE

      !>          SENSE is CHARACTER*1
      !>          Determines which reciprocal condition numbers are computed.
      !>          = 'N': None are computed;
      !>          = 'E': Computed for eigenvalues only;
      !>          = 'V': Computed for right eigenvectors only;
      !>          = 'B': Computed for eigenvalues and right eigenvectors.
      !>
      !>          If SENSE = 'E' or 'B', both left and right eigenvectors
      !>          must also be computed (JOBVL = 'V' and JOBVR = 'V').
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The order of the matrix A. N >= 0.
      !>

  [in,out]

  A

      !>          A is COMPLEX*16 array, dimension (LDA,N)
      !>          On entry, the N-by-N matrix A.
      !>          On exit, A has been overwritten.  If JOBVL = 'V' or
      !>          JOBVR = 'V', A contains the Schur form of the balanced
      !>          version of the matrix A.
      !>

  [in]

  LDA

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

  [out]

  W

      !>          W is COMPLEX*16 array, dimension (N)
      !>          W contains the computed eigenvalues.
      !>

  [out]

  VL

      !>          VL is COMPLEX*16 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 JOBVL = 'N', VL is not referenced.
      !>          u(j) = VL(:,j), the j-th column of VL.
      !>

  [in]

  LDVL

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

  [out]

  VR

      !>          VR is COMPLEX*16 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 JOBVR = 'N', VR is not referenced.
      !>          v(j) = VR(:,j), the j-th column of VR.
      !>

  [in]

  LDVR

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

  [out]

  ILO

      !>          ILO is INTEGER
      !>

  [out]

  IHI

      !>          IHI is INTEGER
      !>          ILO and IHI are integer values determined when A was
      !>          balanced.  The balanced A(i,j) = 0 if I > J and
      !>          J = 1,...,ILO-1 or I = IHI+1,...,N.
      !>

  [out]

  SCALE

      !>          SCALE is DOUBLE PRECISION array, dimension (N)
      !>          Details of the permutations and scaling factors applied
      !>          when balancing A.  If P(j) is the index of the row and column
      !>          interchanged with row and column j, and D(j) is the scaling
      !>          factor applied to row and column j, then
      !>          SCALE(J) = P(J),    for J = 1,...,ILO-1
      !>                   = D(J),    for J = ILO,...,IHI
      !>                   = P(J)     for J = IHI+1,...,N.
      !>          The order in which the interchanges are made is N to IHI+1,
      !>          then 1 to ILO-1.
      !>

  [out]

  ABNRM

      !>          ABNRM is DOUBLE PRECISION
      !>          The one-norm of the balanced matrix (the maximum
      !>          of the sum of absolute values of elements of any column).
      !>

  [out]

  RCONDE

      !>          RCONDE is DOUBLE PRECISION array, dimension (N)
      !>          RCONDE(j) is the reciprocal condition number of the j-th
      !>          eigenvalue.
      !>

  [out]

  RCONDV

      !>          RCONDV is DOUBLE PRECISION array, dimension (N)
      !>          RCONDV(j) is the reciprocal condition number of the j-th
      !>          right eigenvector.
      !>

  [out]

  WORK

      !>          WORK is COMPLEX*16 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.  If SENSE = 'N' or 'E',
      !>          LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',
      !>          LWORK >= N*N+2*N.
      !>          For good performance, LWORK must 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]

  RWORK

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

  [out]

  INFO

      !>          INFO is INTEGER
      !>          = 0:  successful exit
      !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
      !>          > 0:  if INFO = i, the QR algorithm failed to compute all the
      !>                eigenvalues, and no eigenvectors or condition numbers
      !>                have been computed; elements 1:ILO-1 and i+1:N of W
      !>                contain eigenvalues which have converged.
      !>

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