complexGEeigen(3)

NAME
complexGEeigen - complex

SYNOPSIS

  Functions
subroutine cgees (jobvs, sort, select, n, a, lda, sdim, w, vs, ldvs, work, lwork, rwork, bwork, info)
CGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine cgeesx (jobvs, sort, select, sense, n, a, lda, sdim, w, vs, ldvs, rconde, rcondv, work, lwork, rwork, bwork, info)
CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine cgeev (jobvl, jobvr, n, a, lda, w, vl, ldvl, vr, ldvr, work, lwork, rwork, info)
CGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine cgeevx (balanc, jobvl, jobvr, sense, n, a, lda, w, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, rwork, info)
CGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine cgges (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork, bwork, info)
CGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine cgges3 (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork, bwork, info)
CGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)
subroutine cggesx (jobvsl, jobvsr, sort, selctg, sense, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, work, lwork, rwork, iwork, liwork, bwork, info)
CGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine cggev (jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)
CGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine cggev3 (jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)
CGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)
subroutine cggevx (balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, rwork, iwork, bwork, info)
CGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Detailed Description

This is the group of complex eigenvalue driver functions for GE matrices

Function Documentation

  subroutine cgees (character jobvs, character sort, external select, integer n, complex, dimension( lda, * ) a, integer lda, integer sdim, complex, dimension( * ) w, complex, dimension( ldvs, * ) vs, integer ldvs, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, logical, dimension( * ) bwork, integer info)

CGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

     CGEES computes for an N-by-N complex nonsymmetric matrix A, the
     eigenvalues, the Schur form T, and, optionally, the matrix of Schur
     vectors Z.  This gives the Schur factorization A = Z*T*(Z**H).
     Optionally, it also orders the eigenvalues on the diagonal of the
     Schur form so that selected eigenvalues are at the top left.
     The leading columns of Z then form an orthonormal basis for the
     invariant subspace corresponding to the selected eigenvalues.
     A complex matrix is in Schur form if it is upper triangular.

Parameters
    JOBVS

              JOBVS is CHARACTER*1
              = 'N': Schur vectors are not computed;
              = 'V': Schur vectors are computed.

    SORT

              SORT is CHARACTER*1
              Specifies whether or not to order the eigenvalues on the
              diagonal of the Schur form.
              = 'N': Eigenvalues are not ordered:
              = 'S': Eigenvalues are ordered (see SELECT).

    SELECT

              SELECT is a LOGICAL FUNCTION of one COMPLEX argument
              SELECT must be declared EXTERNAL in the calling subroutine.
              If SORT = 'S', SELECT is used to select eigenvalues to order
              to the top left of the Schur form.
              IF SORT = 'N', SELECT is not referenced.
              The eigenvalue W(j) is selected if SELECT(W(j)) is true.

    N

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

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, the N-by-N matrix A.
              On exit, A has been overwritten by its Schur form T.

    LDA

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

    SDIM

              SDIM is INTEGER
              If SORT = 'N', SDIM = 0.
              If SORT = 'S', SDIM = number of eigenvalues for which
                             SELECT is true.

    W

              W is COMPLEX array, dimension (N)
              W contains the computed eigenvalues, in the same order that
              they appear on the diagonal of the output Schur form T.

    VS

              VS is COMPLEX array, dimension (LDVS,N)
              If JOBVS = 'V', VS contains the unitary matrix Z of Schur
              vectors.
              If JOBVS = 'N', VS is not referenced.

    LDVS

              LDVS is INTEGER
              The leading dimension of the array VS.  LDVS >= 1; if
              JOBVS = 'V', LDVS >= N.

    WORK

              WORK is COMPLEX 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(1,2*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.

    RWORK

              RWORK is REAL array, dimension (N)

    BWORK

              BWORK is LOGICAL array, dimension (N)
              Not referenced if SORT = 'N'.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value.
              > 0: if INFO = i, and i is
                   <= N:  the QR algorithm failed to compute all the
                          eigenvalues; elements 1:ILO-1 and i+1:N of W
                          contain those eigenvalues which have converged;
                          if JOBVS = 'V', VS contains the matrix which
                          reduces A to its partially converged Schur form.
                   = N+1: the eigenvalues could not be reordered because
                          some eigenvalues were too close to separate (the
                          problem is very ill-conditioned);
                   = N+2: after reordering, roundoff changed values of
                          some complex eigenvalues so that leading
                          eigenvalues in the Schur form no longer satisfy
                          SELECT = .TRUE..  This could also be caused by
                          underflow due to scaling.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 195 of file cgees.f.

  subroutine cgeesx (character jobvs, character sort, external select, character sense, integer n, complex, dimension( lda, * ) a, integer lda, integer sdim, complex, dimension( * ) w, complex, dimension( ldvs, * ) vs, integer ldvs, real rconde, real rcondv, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, logical, dimension( * ) bwork, integer info)

CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

     CGEESX computes for an N-by-N complex nonsymmetric matrix A, the
     eigenvalues, the Schur form T, and, optionally, the matrix of Schur
     vectors Z.  This gives the Schur factorization A = Z*T*(Z**H).
     Optionally, it also orders the eigenvalues on the diagonal of the
     Schur form so that selected eigenvalues are at the top left;
     computes a reciprocal condition number for the average of the
     selected eigenvalues (RCONDE); and computes a reciprocal condition
     number for the right invariant subspace corresponding to the
     selected eigenvalues (RCONDV).  The leading columns of Z form an
     orthonormal basis for this invariant subspace.
     For further explanation of the reciprocal condition numbers RCONDE
     and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
     these quantities are called s and sep respectively).
     A complex matrix is in Schur form if it is upper triangular.

Parameters
    JOBVS

              JOBVS is CHARACTER*1
              = 'N': Schur vectors are not computed;
              = 'V': Schur vectors are computed.

    SORT

              SORT is CHARACTER*1
              Specifies whether or not to order the eigenvalues on the
              diagonal of the Schur form.
              = 'N': Eigenvalues are not ordered;
              = 'S': Eigenvalues are ordered (see SELECT).

    SELECT

              SELECT is a LOGICAL FUNCTION of one COMPLEX argument
              SELECT must be declared EXTERNAL in the calling subroutine.
              If SORT = 'S', SELECT is used to select eigenvalues to order
              to the top left of the Schur form.
              If SORT = 'N', SELECT is not referenced.
              An eigenvalue W(j) is selected if SELECT(W(j)) is true.

    SENSE

              SENSE is CHARACTER*1
              Determines which reciprocal condition numbers are computed.
              = 'N': None are computed;
              = 'E': Computed for average of selected eigenvalues only;
              = 'V': Computed for selected right invariant subspace only;
              = 'B': Computed for both.
              If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.

    N

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

    A

              A is COMPLEX array, dimension (LDA, N)
              On entry, the N-by-N matrix A.
              On exit, A is overwritten by its Schur form T.

    LDA

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

    SDIM

              SDIM is INTEGER
              If SORT = 'N', SDIM = 0.
              If SORT = 'S', SDIM = number of eigenvalues for which
                             SELECT is true.

    W

              W is COMPLEX array, dimension (N)
              W contains the computed eigenvalues, in the same order
              that they appear on the diagonal of the output Schur form T.

    VS

              VS is COMPLEX array, dimension (LDVS,N)
              If JOBVS = 'V', VS contains the unitary matrix Z of Schur
              vectors.
              If JOBVS = 'N', VS is not referenced.

    LDVS

              LDVS is INTEGER
              The leading dimension of the array VS.  LDVS >= 1, and if
              JOBVS = 'V', LDVS >= N.

    RCONDE

              RCONDE is REAL
              If SENSE = 'E' or 'B', RCONDE contains the reciprocal
              condition number for the average of the selected eigenvalues.
              Not referenced if SENSE = 'N' or 'V'.

    RCONDV

              RCONDV is REAL
              If SENSE = 'V' or 'B', RCONDV contains the reciprocal
              condition number for the selected right invariant subspace.
              Not referenced if SENSE = 'N' or 'E'.

    WORK

              WORK is COMPLEX 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(1,2*N).
              Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
              where SDIM is the number of selected eigenvalues computed by
              this routine.  Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also
              that an error is only returned if LWORK < max(1,2*N), but if
              SENSE = 'E' or 'V' or 'B' this may not be large enough.
              For good performance, LWORK must generally be larger.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates upper bound on the optimal size of the
              array WORK, returns this value as the first entry of the WORK
              array, and no error message related to LWORK is issued by
              XERBLA.

    RWORK

              RWORK is REAL array, dimension (N)

    BWORK

              BWORK is LOGICAL array, dimension (N)
              Not referenced if SORT = 'N'.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value.
              > 0: if INFO = i, and i is
                 <= N: the QR algorithm failed to compute all the
                       eigenvalues; elements 1:ILO-1 and i+1:N of W
                       contain those eigenvalues which have converged; if
                       JOBVS = 'V', VS contains the transformation which
                       reduces A to its partially converged Schur form.
                 = N+1: the eigenvalues could not be reordered because some
                       eigenvalues were too close to separate (the problem
                       is very ill-conditioned);
                 = N+2: after reordering, roundoff changed values of some
                       complex eigenvalues so that leading eigenvalues in
                       the Schur form no longer satisfy SELECT=.TRUE.  This
                       could also be caused by underflow due to scaling.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 236 of file cgeesx.f.

  subroutine cgeev (character jobvl, character jobvr, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) w, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer info)

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

Purpose:

     CGEEV computes for an N-by-N complex nonsymmetric matrix A, the
     eigenvalues and, optionally, the left and/or right eigenvectors.
     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.

Parameters
    JOBVL

              JOBVL is CHARACTER*1
              = 'N': left eigenvectors of A are not computed;
              = 'V': left eigenvectors of are computed.

    JOBVR

              JOBVR is CHARACTER*1
              = 'N': right eigenvectors of A are not computed;
              = 'V': right eigenvectors of A are computed.

    N

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

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, the N-by-N matrix A.
              On exit, A has been overwritten.

    LDA

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

    W

              W is COMPLEX array, dimension (N)
              W contains the computed eigenvalues.

    VL

              VL is COMPLEX 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.

    LDVL

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

    VR

              VR is COMPLEX 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.

    LDVR

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

    WORK

              WORK is COMPLEX 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(1,2*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.

    RWORK

              RWORK is REAL array, dimension (2*N)

    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 have been computed;
                    elements i+1:N of W contain eigenvalues which have
                    converged.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 178 of file cgeev.f.

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

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

Purpose:

     CGEEVX 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
    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.

    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'.

    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'.

    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').

    N

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

    A

              A is COMPLEX 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.

    LDA

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

    W

              W is COMPLEX array, dimension (N)
              W contains the computed eigenvalues.

    VL

              VL is COMPLEX 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.

    LDVL

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

    VR

              VR is COMPLEX 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.

    LDVR

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

    ILO

              ILO is INTEGER

    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.

    SCALE

              SCALE is REAL 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.

    ABNRM

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

    RCONDE

              RCONDE is REAL array, dimension (N)
              RCONDE(j) is the reciprocal condition number of the j-th
              eigenvalue.

    RCONDV

              RCONDV is REAL array, dimension (N)
              RCONDV(j) is the reciprocal condition number of the j-th
              right eigenvector.

    WORK

              WORK is COMPLEX 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.  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.

    RWORK

              RWORK is REAL array, dimension (2*N)

    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.

Definition at line 285 of file cgeevx.f.

  subroutine cgges (character jobvsl, character jobvsr, character sort, external selctg, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, integer sdim, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldvsl, * ) vsl, integer ldvsl, complex, dimension( ldvsr, * ) vsr, integer ldvsr, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, logical, dimension( * ) bwork, integer info)

CGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

     CGGES computes for a pair of N-by-N complex nonsymmetric matrices
     (A,B), the generalized eigenvalues, the generalized complex Schur
     form (S, T), and optionally left and/or right Schur vectors (VSL
     and VSR). This gives the generalized Schur factorization
             (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )
     where (VSR)**H is the conjugate-transpose of VSR.
     Optionally, it also orders the eigenvalues so that a selected cluster
     of eigenvalues appears in the leading diagonal blocks of the upper
     triangular matrix S and the upper triangular matrix T. The leading
     columns of VSL and VSR then form an unitary basis for the
     corresponding left and right eigenspaces (deflating subspaces).
     (If only the generalized eigenvalues are needed, use the driver
     CGGEV instead, which is faster.)
     A generalized eigenvalue for a pair of matrices (A,B) is 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 pair of matrices (S,T) is in generalized complex Schur form if S
     and T are upper triangular and, in addition, the diagonal elements
     of T are non-negative real numbers.

Parameters
    JOBVSL

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

    JOBVSR

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

    SORT

              SORT is CHARACTER*1
              Specifies whether or not to order the eigenvalues on the
              diagonal of the generalized Schur form.
              = 'N':  Eigenvalues are not ordered;
              = 'S':  Eigenvalues are ordered (see SELCTG).

    SELCTG

              SELCTG is a LOGICAL FUNCTION of two COMPLEX arguments
              SELCTG must be declared EXTERNAL in the calling subroutine.
              If SORT = 'N', SELCTG is not referenced.
              If SORT = 'S', SELCTG is used to select eigenvalues to sort
              to the top left of the Schur form.
              An eigenvalue ALPHA(j)/BETA(j) is selected if
              SELCTG(ALPHA(j),BETA(j)) is true.
              Note that a selected complex eigenvalue may no longer satisfy
              SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
              ordering may change the value of complex eigenvalues
              (especially if the eigenvalue is ill-conditioned), in this
              case INFO is set to N+2 (See INFO below).

    N

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

    A

              A is COMPLEX array, dimension (LDA, N)
              On entry, the first of the pair of matrices.
              On exit, A has been overwritten by its generalized Schur
              form S.

    LDA

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

    B

              B is COMPLEX array, dimension (LDB, N)
              On entry, the second of the pair of matrices.
              On exit, B has been overwritten by its generalized Schur
              form T.

    LDB

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

    SDIM

              SDIM is INTEGER
              If SORT = 'N', SDIM = 0.
              If SORT = 'S', SDIM = number of eigenvalues (after sorting)
              for which SELCTG is true.

    ALPHA

              ALPHA is COMPLEX array, dimension (N)

    BETA

              BETA is COMPLEX array, dimension (N)
              On exit,  ALPHA(j)/BETA(j), j=1,...,N, will be the
              generalized eigenvalues.  ALPHA(j), j=1,...,N  and  BETA(j),
              j=1,...,N  are the diagonals of the complex Schur form (A,B)
              output by CGGES. The  BETA(j) will be non-negative real.
              Note: the quotients ALPHA(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, ALPHA 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

              VSL is COMPLEX array, dimension (LDVSL,N)
              If JOBVSL = 'V', VSL will contain the left Schur vectors.
              Not referenced if JOBVSL = 'N'.

    LDVSL

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

    VSR

              VSR is COMPLEX array, dimension (LDVSR,N)
              If JOBVSR = 'V', VSR will contain the right Schur vectors.
              Not referenced if JOBVSR = 'N'.

    LDVSR

              LDVSR is INTEGER
              The leading dimension of the matrix VSR. LDVSR >= 1, and
              if JOBVSR = 'V', LDVSR >= N.

    WORK

              WORK is COMPLEX 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(1,2*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.

    RWORK

              RWORK is REAL array, dimension (8*N)

    BWORK

              BWORK is LOGICAL array, dimension (N)
              Not referenced if SORT = 'N'.

    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.  (A,B) are not in Schur
                    form, but ALPHA(j) and BETA(j) should be correct for
                    j=INFO+1,...,N.
              > N:  =N+1: other than QZ iteration failed in CHGEQZ
                    =N+2: after reordering, roundoff changed values of
                          some complex eigenvalues so that leading
                          eigenvalues in the Generalized Schur form no
                          longer satisfy SELCTG=.TRUE.  This could also
                          be caused due to scaling.
                    =N+3: reordering failed in CTGSEN.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 267 of file cgges.f.

  subroutine cgges3 (character jobvsl, character jobvsr, character sort, external selctg, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, integer sdim, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldvsl, * ) vsl, integer ldvsl, complex, dimension( ldvsr, * ) vsr, integer ldvsr, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, logical, dimension( * ) bwork, integer info)

CGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Purpose:

     CGGES3 computes for a pair of N-by-N complex nonsymmetric matrices
     (A,B), the generalized eigenvalues, the generalized complex Schur
     form (S, T), and optionally left and/or right Schur vectors (VSL
     and VSR). This gives the generalized Schur factorization
             (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )
     where (VSR)**H is the conjugate-transpose of VSR.
     Optionally, it also orders the eigenvalues so that a selected cluster
     of eigenvalues appears in the leading diagonal blocks of the upper
     triangular matrix S and the upper triangular matrix T. The leading
     columns of VSL and VSR then form an unitary basis for the
     corresponding left and right eigenspaces (deflating subspaces).
     (If only the generalized eigenvalues are needed, use the driver
     CGGEV instead, which is faster.)
     A generalized eigenvalue for a pair of matrices (A,B) is 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 pair of matrices (S,T) is in generalized complex Schur form if S
     and T are upper triangular and, in addition, the diagonal elements
     of T are non-negative real numbers.

Parameters
    JOBVSL

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

    JOBVSR

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

    SORT

              SORT is CHARACTER*1
              Specifies whether or not to order the eigenvalues on the
              diagonal of the generalized Schur form.
              = 'N':  Eigenvalues are not ordered;
              = 'S':  Eigenvalues are ordered (see SELCTG).

    SELCTG

              SELCTG is a LOGICAL FUNCTION of two COMPLEX arguments
              SELCTG must be declared EXTERNAL in the calling subroutine.
              If SORT = 'N', SELCTG is not referenced.
              If SORT = 'S', SELCTG is used to select eigenvalues to sort
              to the top left of the Schur form.
              An eigenvalue ALPHA(j)/BETA(j) is selected if
              SELCTG(ALPHA(j),BETA(j)) is true.
              Note that a selected complex eigenvalue may no longer satisfy
              SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
              ordering may change the value of complex eigenvalues
              (especially if the eigenvalue is ill-conditioned), in this
              case INFO is set to N+2 (See INFO below).

    N

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

    A

              A is COMPLEX array, dimension (LDA, N)
              On entry, the first of the pair of matrices.
              On exit, A has been overwritten by its generalized Schur
              form S.

    LDA

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

    B

              B is COMPLEX array, dimension (LDB, N)
              On entry, the second of the pair of matrices.
              On exit, B has been overwritten by its generalized Schur
              form T.

    LDB

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

    SDIM

              SDIM is INTEGER
              If SORT = 'N', SDIM = 0.
              If SORT = 'S', SDIM = number of eigenvalues (after sorting)
              for which SELCTG is true.

    ALPHA

              ALPHA is COMPLEX array, dimension (N)

    BETA

              BETA is COMPLEX array, dimension (N)
              On exit,  ALPHA(j)/BETA(j), j=1,...,N, will be the
              generalized eigenvalues.  ALPHA(j), j=1,...,N  and  BETA(j),
              j=1,...,N  are the diagonals of the complex Schur form (A,B)
              output by CGGES3. The  BETA(j) will be non-negative real.
              Note: the quotients ALPHA(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, ALPHA 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

              VSL is COMPLEX array, dimension (LDVSL,N)
              If JOBVSL = 'V', VSL will contain the left Schur vectors.
              Not referenced if JOBVSL = 'N'.

    LDVSL

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

    VSR

              VSR is COMPLEX array, dimension (LDVSR,N)
              If JOBVSR = 'V', VSR will contain the right Schur vectors.
              Not referenced if JOBVSR = 'N'.

    LDVSR

              LDVSR is INTEGER
              The leading dimension of the matrix VSR. LDVSR >= 1, and
              if JOBVSR = 'V', LDVSR >= N.

    WORK

              WORK is COMPLEX 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.
              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.

    RWORK

              RWORK is REAL array, dimension (8*N)

    BWORK

              BWORK is LOGICAL array, dimension (N)
              Not referenced if SORT = 'N'.

    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.  (A,B) are not in Schur
                    form, but ALPHA(j) and BETA(j) should be correct for
                    j=INFO+1,...,N.
              > N:  =N+1: other than QZ iteration failed in CLAQZ0
                    =N+2: after reordering, roundoff changed values of
                          some complex eigenvalues so that leading
                          eigenvalues in the Generalized Schur form no
                          longer satisfy SELCTG=.TRUE.  This could also
                          be caused due to scaling.
                    =N+3: reordering failed in CTGSEN.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 266 of file cgges3.f.

  subroutine cggesx (character jobvsl, character jobvsr, character sort, external selctg, character sense, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, integer sdim, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldvsl, * ) vsl, integer ldvsl, complex, dimension( ldvsr, * ) vsr, integer ldvsr, real, dimension( 2 ) rconde, real, dimension( 2 ) rcondv, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)

CGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

     CGGESX computes for a pair of N-by-N complex nonsymmetric matrices
     (A,B), the generalized eigenvalues, the complex Schur form (S,T),
     and, optionally, the left and/or right matrices of Schur vectors (VSL
     and VSR).  This gives the generalized Schur factorization
          (A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H )
     where (VSR)**H is the conjugate-transpose of VSR.
     Optionally, it also orders the eigenvalues so that a selected cluster
     of eigenvalues appears in the leading diagonal blocks of the upper
     triangular matrix S and the upper triangular matrix T; computes
     a reciprocal condition number for the average of the selected
     eigenvalues (RCONDE); and computes a reciprocal condition number for
     the right and left deflating subspaces corresponding to the selected
     eigenvalues (RCONDV). The leading columns of VSL and VSR then form
     an orthonormal basis for the corresponding left and right eigenspaces
     (deflating subspaces).
     A generalized eigenvalue for a pair of matrices (A,B) is 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 or for both being zero.
     A pair of matrices (S,T) is in generalized complex Schur form if T is
     upper triangular with non-negative diagonal and S is upper
     triangular.

Parameters
    JOBVSL

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

    JOBVSR

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

    SORT

              SORT is CHARACTER*1
              Specifies whether or not to order the eigenvalues on the
              diagonal of the generalized Schur form.
              = 'N':  Eigenvalues are not ordered;
              = 'S':  Eigenvalues are ordered (see SELCTG).

    SELCTG

              SELCTG is a LOGICAL FUNCTION of two COMPLEX arguments
              SELCTG must be declared EXTERNAL in the calling subroutine.
              If SORT = 'N', SELCTG is not referenced.
              If SORT = 'S', SELCTG is used to select eigenvalues to sort
              to the top left of the Schur form.
              Note that a selected complex eigenvalue may no longer satisfy
              SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
              ordering may change the value of complex eigenvalues
              (especially if the eigenvalue is ill-conditioned), in this
              case INFO is set to N+3 see INFO below).

    SENSE

              SENSE is CHARACTER*1
              Determines which reciprocal condition numbers are computed.
              = 'N':  None are computed;
              = 'E':  Computed for average of selected eigenvalues only;
              = 'V':  Computed for selected deflating subspaces only;
              = 'B':  Computed for both.
              If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

    N

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

    A

              A is COMPLEX array, dimension (LDA, N)
              On entry, the first of the pair of matrices.
              On exit, A has been overwritten by its generalized Schur
              form S.

    LDA

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

    B

              B is COMPLEX array, dimension (LDB, N)
              On entry, the second of the pair of matrices.
              On exit, B has been overwritten by its generalized Schur
              form T.

    LDB

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

    SDIM

              SDIM is INTEGER
              If SORT = 'N', SDIM = 0.
              If SORT = 'S', SDIM = number of eigenvalues (after sorting)
              for which SELCTG is true.

    ALPHA

              ALPHA is COMPLEX array, dimension (N)

    BETA

              BETA is COMPLEX array, dimension (N)
              On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
              generalized eigenvalues.  ALPHA(j) and BETA(j),j=1,...,N  are
              the diagonals of the complex Schur form (S,T).  BETA(j) will
              be non-negative real.
              Note: the quotients ALPHA(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, ALPHA 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

              VSL is COMPLEX array, dimension (LDVSL,N)
              If JOBVSL = 'V', VSL will contain the left Schur vectors.
              Not referenced if JOBVSL = 'N'.

    LDVSL

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

    VSR

              VSR is COMPLEX array, dimension (LDVSR,N)
              If JOBVSR = 'V', VSR will contain the right Schur vectors.
              Not referenced if JOBVSR = 'N'.

    LDVSR

              LDVSR is INTEGER
              The leading dimension of the matrix VSR. LDVSR >= 1, and
              if JOBVSR = 'V', LDVSR >= N.

    RCONDE

              RCONDE is REAL array, dimension ( 2 )
              If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
              reciprocal condition numbers for the average of the selected
              eigenvalues.
              Not referenced if SENSE = 'N' or 'V'.

    RCONDV

              RCONDV is REAL array, dimension ( 2 )
              If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
              reciprocal condition number for the selected deflating
              subspaces.
              Not referenced if SENSE = 'N' or 'E'.

    WORK

              WORK is COMPLEX 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.
              If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
              LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else
              LWORK >= MAX(1,2*N).  Note that 2*SDIM*(N-SDIM) <= N*N/2.
              Note also that an error is only returned if
              LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may
              not be large enough.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the bound on the optimal size of the WORK
              array and the minimum size of the IWORK array, returns these
              values as the first entries of the WORK and IWORK arrays, and
              no error message related to LWORK or LIWORK is issued by
              XERBLA.

    RWORK

              RWORK is REAL array, dimension ( 8*N )
              Real workspace.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of the array WORK.
              If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
              LIWORK >= N+2.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the bound on the optimal size of the
              WORK array and the minimum size of the IWORK array, returns
              these values as the first entries of the WORK and IWORK
              arrays, and no error message related to LWORK or LIWORK is
              issued by XERBLA.

    BWORK

              BWORK is LOGICAL array, dimension (N)
              Not referenced if SORT = 'N'.

    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.  (A,B) are not in Schur
                    form, but ALPHA(j) and BETA(j) should be correct for
                    j=INFO+1,...,N.
              > N:  =N+1: other than QZ iteration failed in CHGEQZ
                    =N+2: after reordering, roundoff changed values of
                          some complex eigenvalues so that leading
                          eigenvalues in the Generalized Schur form no
                          longer satisfy SELCTG=.TRUE.  This could also
                          be caused due to scaling.
                    =N+3: reordering failed in CTGSEN.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 326 of file cggesx.f.

  subroutine cggev (character jobvl, character jobvr, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer info)

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

Purpose:

     CGGEV computes for a pair of N-by-N complex 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 generalized eigenvector v(j) corresponding to the
     generalized eigenvalue lambda(j) of (A,B) satisfies
                  A * v(j) = lambda(j) * B * v(j).
     The left generalized eigenvector u(j) corresponding to the
     generalized eigenvalues 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
    JOBVL

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

    JOBVR

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

    N

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

    A

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

    LDA

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

    B

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

    LDB

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

    ALPHA

              ALPHA is COMPLEX array, dimension (N)

    BETA

              BETA is COMPLEX array, dimension (N)
              On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
              generalized eigenvalues.
              Note: the quotients ALPHA(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, ALPHA 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

              VL is COMPLEX array, dimension (LDVL,N)
              If JOBVL = 'V', the left generalized eigenvectors u(j) are
              stored one after another in the columns of VL, in the same
              order as their eigenvalues.
              Each eigenvector is scaled so the largest component has
              abs(real part) + abs(imag. part) = 1.
              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 COMPLEX array, dimension (LDVR,N)
              If JOBVR = 'V', the right generalized eigenvectors v(j) are
              stored one after another in the columns of VR, in the same
              order as their eigenvalues.
              Each eigenvector is scaled so the largest component has
              abs(real part) + abs(imag. part) = 1.
              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 COMPLEX 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(1,2*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.

    RWORK

              RWORK is REAL array, dimension (8*N)

    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 ALPHA(j) and BETA(j) should be
                    correct for j=INFO+1,...,N.
              > N:  =N+1: other then QZ iteration failed in CHGEQZ,
                    =N+2: error return from CTGEVC.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 215 of file cggev.f.

  subroutine cggev3 (character jobvl, character jobvr, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer info)

CGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

Purpose:

     CGGEV3 computes for a pair of N-by-N complex 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 generalized eigenvector v(j) corresponding to the
     generalized eigenvalue lambda(j) of (A,B) satisfies
                  A * v(j) = lambda(j) * B * v(j).
     The left generalized eigenvector u(j) corresponding to the
     generalized eigenvalues 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
    JOBVL

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

    JOBVR

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

    N

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

    A

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

    LDA

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

    B

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

    LDB

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

    ALPHA

              ALPHA is COMPLEX array, dimension (N)

    BETA

              BETA is COMPLEX array, dimension (N)
              On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
              generalized eigenvalues.
              Note: the quotients ALPHA(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, ALPHA 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

              VL is COMPLEX array, dimension (LDVL,N)
              If JOBVL = 'V', the left generalized eigenvectors u(j) are
              stored one after another in the columns of VL, in the same
              order as their eigenvalues.
              Each eigenvector is scaled so the largest component has
              abs(real part) + abs(imag. part) = 1.
              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 COMPLEX array, dimension (LDVR,N)
              If JOBVR = 'V', the right generalized eigenvectors v(j) are
              stored one after another in the columns of VR, in the same
              order as their eigenvalues.
              Each eigenvector is scaled so the largest component has
              abs(real part) + abs(imag. part) = 1.
              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 COMPLEX 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.
              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.

    RWORK

              RWORK is REAL array, dimension (8*N)

    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 ALPHA(j) and BETA(j) should be
                    correct for j=INFO+1,...,N.
              > N:  =N+1: other then QZ iteration failed in CHGEQZ,
                    =N+2: error return from CTGEVC.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 214 of file cggev3.f.

  subroutine cggevx (character balanc, character jobvl, character jobvr, character sense, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, integer ilo, integer ihi, real, dimension( * ) lscale, real, dimension( * ) rscale, real abnrm, real bbnrm, real, dimension( * ) rconde, real, dimension( * ) rcondv, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, logical, dimension( * ) bwork, integer info)

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

Purpose:

     CGGEVX computes for a pair of N-by-N complex nonsymmetric matrices
     (A,B) the generalized eigenvalues, and optionally, the left and/or
     right generalized eigenvectors.
     Optionally, it also computes a balancing transformation to improve
     the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
     LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
     the eigenvalues (RCONDE), and reciprocal condition numbers for the
     right eigenvectors (RCONDV).
     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
    BALANC

              BALANC is CHARACTER*1
              Specifies the balance option to be performed:
              = 'N':  do not diagonally scale or permute;
              = 'P':  permute only;
              = 'S':  scale only;
              = 'B':  both permute and scale.
              Computed reciprocal condition numbers will be for the
              matrices after permuting and/or balancing. Permuting does
              not change condition numbers (in exact arithmetic), but
              balancing does.

    JOBVL

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

    JOBVR

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

    SENSE

              SENSE is CHARACTER*1
              Determines which reciprocal condition numbers are computed.
              = 'N': none are computed;
              = 'E': computed for eigenvalues only;
              = 'V': computed for eigenvectors only;
              = 'B': computed for eigenvalues and eigenvectors.

    N

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

    A

              A is COMPLEX array, dimension (LDA, N)
              On entry, the matrix A in the pair (A,B).
              On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
              or both, then A contains the first part of the complex Schur
              form of the 'balanced' versions of the input A and B.

    LDA

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

    B

              B is COMPLEX array, dimension (LDB, N)
              On entry, the matrix B in the pair (A,B).
              On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
              or both, then B contains the second part of the complex
              Schur form of the 'balanced' versions of the input A and B.

    LDB

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

    ALPHA

              ALPHA is COMPLEX array, dimension (N)

    BETA

              BETA is COMPLEX array, dimension (N)
              On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized
              eigenvalues.
              Note: the quotient ALPHA(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, ALPHA 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

              VL is COMPLEX array, dimension (LDVL,N)
              If JOBVL = 'V', the left generalized eigenvectors u(j) are
              stored one after another in the columns of VL, in the same
              order as their eigenvalues.
              Each eigenvector will be scaled so the largest component
              will have abs(real part) + abs(imag. part) = 1.
              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 COMPLEX array, dimension (LDVR,N)
              If JOBVR = 'V', the right generalized eigenvectors v(j) are
              stored one after another in the columns of VR, in the same
              order as their eigenvalues.
              Each eigenvector will be scaled so the largest component
              will have abs(real part) + abs(imag. part) = 1.
              Not referenced if JOBVR = 'N'.

    LDVR

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

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              ILO and IHI are integer values such that on exit
              A(i,j) = 0 and B(i,j) = 0 if i > j and
              j = 1,...,ILO-1 or i = IHI+1,...,N.
              If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

    LSCALE

              LSCALE is REAL array, dimension (N)
              Details of the permutations and scaling factors applied
              to the left side of A and B.  If PL(j) is the index of the
              row interchanged with row j, and DL(j) is the scaling
              factor applied to row j, then
                LSCALE(j) = PL(j)  for j = 1,...,ILO-1
                          = DL(j)  for j = ILO,...,IHI
                          = PL(j)  for j = IHI+1,...,N.
              The order in which the interchanges are made is N to IHI+1,
              then 1 to ILO-1.

    RSCALE

              RSCALE is REAL array, dimension (N)
              Details of the permutations and scaling factors applied
              to the right side of A and B.  If PR(j) is the index of the
              column interchanged with column j, and DR(j) is the scaling
              factor applied to column j, then
                RSCALE(j) = PR(j)  for j = 1,...,ILO-1
                          = DR(j)  for j = ILO,...,IHI
                          = PR(j)  for j = IHI+1,...,N
              The order in which the interchanges are made is N to IHI+1,
              then 1 to ILO-1.

    ABNRM

              ABNRM is REAL
              The one-norm of the balanced matrix A.

    BBNRM

              BBNRM is REAL
              The one-norm of the balanced matrix B.

    RCONDE

              RCONDE is REAL array, dimension (N)
              If SENSE = 'E' or 'B', the reciprocal condition numbers of
              the eigenvalues, stored in consecutive elements of the array.
              If SENSE = 'N' or 'V', RCONDE is not referenced.

    RCONDV

              RCONDV is REAL array, dimension (N)
              If SENSE = 'V' or 'B', the estimated reciprocal condition
              numbers of the eigenvectors, stored in consecutive elements
              of the array. If the eigenvalues cannot be reordered to
              compute RCONDV(j), RCONDV(j) is set to 0; this can only occur
              when the true value would be very small anyway.
              If SENSE = 'N' or 'E', RCONDV is not referenced.

    WORK

              WORK is COMPLEX 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(1,2*N).
              If SENSE = 'E', LWORK >= max(1,4*N).
              If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N).
              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.

    RWORK

              RWORK is REAL array, dimension (lrwork)
              lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B',
              and at least max(1,2*N) otherwise.
              Real workspace.

    IWORK

              IWORK is INTEGER array, dimension (N+2)
              If SENSE = 'E', IWORK is not referenced.

    BWORK

              BWORK is LOGICAL array, dimension (N)
              If SENSE = 'N', BWORK is not referenced.

    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 ALPHA(j) and BETA(j) should be correct
                    for j=INFO+1,...,N.
              > N:  =N+1: other than QZ iteration failed in CHGEQZ.
                    =N+2: error return from CTGEVC.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      Balancing a matrix pair (A,B) includes, first, permuting rows and
      columns to isolate eigenvalues, second, applying diagonal similarity
      transformation to the rows and columns to make the rows and columns
      as close in norm as possible. 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 4.11.1.2 of LAPACK Users' Guide.
      An approximate error bound on the chordal distance between the i-th
      computed generalized eigenvalue w and the corresponding exact
      eigenvalue lambda is
           chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
      An approximate error bound for the angle between the i-th computed
      eigenvector VL(i) or VR(i) is given by
           EPS * norm(ABNRM, BBNRM) / DIF(i).
      For further explanation of the reciprocal condition numbers RCONDE
      and RCONDV, see section 4.11 of LAPACK User's Guide.

Definition at line 370 of file cggevx.f.

Author

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