Functions

subroutine zgegs (JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, INFO)
ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine zgegv (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)
ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine zgees (JOBVS, SORT, SELECT, N, A, LDA, SDIM, W, VS, LDVS, WORK, LWORK, RWORK, BWORK, INFO)
ZGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine zgeesx (JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK, BWORK, INFO)
ZGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine zgeev (JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)
ZGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine zgeevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, INFO)
ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine zgges (JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO)
ZGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine zgges3 (JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO)
ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)
subroutine zggesx (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)
ZGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine zggev (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)
ZGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine zggev3 (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)
ZGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)
subroutine zggevx (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)
ZGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Detailed Description

This is the group of complex16 eigenvalue driver functions for GE matrices

Function Documentation

subroutine zgees (character JOBVS, character SORT, external SELECT, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer SDIM, complex*16, dimension( * ) W, complex*16, dimension( ldvs, * ) VS, integer LDVS, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)

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

Purpose:

 ZGEES 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*16 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*16 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*16 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*16 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*16 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 DOUBLE PRECISION 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.

Date:

November 2011

subroutine zgeesx (character JOBVS, character SORT, external SELECT, character SENSE, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer SDIM, complex*16, dimension( * ) W, complex*16, dimension( ldvs, * ) VS, integer LDVS, double precision RCONDE, double precision RCONDV, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)

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

Purpose:

 ZGEESX 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*16 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*16 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*16 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*16 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 DOUBLE PRECISION
          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 DOUBLE PRECISION
          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*16 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 DOUBLE PRECISION 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.

Date:

June 2016

subroutine zgeev (character JOBVL, character JOBVR, 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, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer INFO)

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

Purpose:

 ZGEEV 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*16 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*16 array, dimension (N)
          W contains the computed eigenvalues.

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.

LDVL

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

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.

LDVR

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

WORK

          WORK is COMPLEX*16 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 DOUBLE PRECISION 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 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.

Date:

June 2016

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

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:

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*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.

LDA

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

W

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

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.

LDVL

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

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.

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 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.

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).

RCONDE

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

RCONDV

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

WORK

          WORK is COMPLEX*16 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 DOUBLE PRECISION 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.

Date:

June 2016

subroutine zgegs (character JOBVSL, character JOBVSR, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldvsl, * ) VSL, integer LDVSL, complex*16, dimension( ldvsr, * ) VSR, integer LDVSR, 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

Purpose:

 This routine is deprecated and has been replaced by routine ZGGES.
 ZGEGS computes the eigenvalues, Schur form, and, optionally, the
 left and or/right Schur vectors of a complex matrix pair (A,B).
 Given two square matrices A and B, the generalized Schur
 factorization has the form
 
    A = Q*S*Z**H,  B = Q*T*Z**H
 
 where Q and Z are unitary matrices and S and T are upper triangular.
 The columns of Q are the left Schur vectors
 and the columns of Z are the right Schur vectors.
 
 If only the eigenvalues of (A,B) are needed, the driver routine
 ZGEGV should be used instead.  See ZGEGV for a description of the
 eigenvalues of the generalized nonsymmetric eigenvalue problem
 (GNEP).

 

Parameters:

JOBVSL

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

JOBVSR

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

N

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

A

          A is COMPLEX*16 array, dimension (LDA, N)
          On entry, the matrix A.
          On exit, the upper triangular matrix S from the generalized
          Schur factorization.

LDA

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

B

          B is COMPLEX*16 array, dimension (LDB, N)
          On entry, the matrix B.
          On exit, the upper triangular matrix T from the generalized
          Schur factorization.

LDB

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

ALPHA

          ALPHA is COMPLEX*16 array, dimension (N)
          The complex scalars alpha that define the eigenvalues of
          GNEP.  ALPHA(j) = S(j,j), the diagonal element of the Schur
          form of A.

BETA

          BETA is COMPLEX*16 array, dimension (N)
          The non-negative real scalars beta that define the
          eigenvalues of GNEP.  BETA(j) = T(j,j), the diagonal element
          of the triangular factor T.
          Together, the quantities alpha = ALPHA(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.

VSL

          VSL is COMPLEX*16 array, dimension (LDVSL,N)
          If JOBVSL = 'V', the matrix of left Schur vectors Q.
          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*16 array, dimension (LDVSR,N)
          If JOBVSR = 'V', the matrix of right Schur vectors Z.
          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*16 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.
          To compute the optimal value of LWORK, call ILAENV to get
          blocksizes (for ZGEQRF, ZUNMQR, and CUNGQR.)  Then compute:
          NB  -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and CUNGQR;
          the optimal LWORK is 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.

RWORK

          RWORK is DOUBLE PRECISION array, dimension (3*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:  errors that usually indicate LAPACK problems:
                =N+1: error return from ZGGBAL
                =N+2: error return from ZGEQRF
                =N+3: error return from ZUNMQR
                =N+4: error return from ZUNGQR
                =N+5: error return from ZGGHRD
                =N+6: error return from ZHGEQZ (other than failed
                                               iteration)
                =N+7: error return from ZGGBAK (computing VSL)
                =N+8: error return from ZGGBAK (computing VSR)
                =N+9: error return from ZLASCL (various places)

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine zgegv (character JOBVL, character JOBVR, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, 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

Purpose:

 This routine is deprecated and has been replaced by routine ZGGEV.
 ZGEGV computes the eigenvalues and, optionally, the left and/or right
 eigenvectors of a complex 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 COMPLEX*16 array, dimension (LDA, N)
          On entry, the matrix A.
          If JOBVL = 'V' or JOBVR = 'V', then on exit A
          contains the 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 elements
          of the Schur form will be correct.  See ZGGHRD and ZHGEQZ
          for details.

LDA

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

B

          B is COMPLEX*16 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 the diagonal
          elements of B will be correct.  See ZGGHRD and ZHGEQZ for
          details.

LDB

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

ALPHA

          ALPHA is COMPLEX*16 array, dimension (N)
          The complex scalars alpha that define the eigenvalues of
          GNEP.

BETA

          BETA is COMPLEX*16 array, dimension (N)
          The complex scalars beta that define the eigenvalues of GNEP.
          
          Together, the quantities alpha = ALPHA(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 COMPLEX*16 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.
          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 COMPLEX*16 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.
          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 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*16 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.
          To compute the optimal value of LWORK, call ILAENV to get
          blocksizes (for ZGEQRF, ZUNMQR, and ZUNGQR.)  Then compute:
          NB  -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and ZUNGQR;
          The optimal LWORK is  MAX( 2*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.

RWORK

          RWORK is DOUBLE PRECISION 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:  errors that usually indicate LAPACK problems:
                =N+1: error return from ZGGBAL
                =N+2: error return from ZGEQRF
                =N+3: error return from ZUNMQR
                =N+4: error return from ZUNGQR
                =N+5: error return from ZGGHRD
                =N+6: error return from ZHGEQZ (other than failed
                                               iteration)
                =N+7: error return from ZTGEVC
                =N+8: error return from ZGGBAK (computing VL)
                =N+9: error return from ZGGBAK (computing VR)
                =N+10: error return from ZLASCL (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 ZGGBAL 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, ZGGBAK 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 complex Schur
  form[*] of the "balanced" versions of A and B.  If no eigenvectors
  are computed, then only the diagonal blocks will be correct.
  [*] In other words, upper triangular form.

 

subroutine zgges (character JOBVSL, character JOBVSR, character SORT, external SELCTG, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, integer SDIM, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldvsl, * ) VSL, integer LDVSL, complex*16, dimension( ldvsr, * ) VSR, integer LDVSR, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)

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

Purpose:

 ZGGES 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
 ZGGEV 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*16 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*16 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*16 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*16 array, dimension (N)

BETA

          BETA is COMPLEX*16 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 ZGGES. 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*16 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*16 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*16 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 DOUBLE PRECISION 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 ZHGEQZ
                =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 ZTGSEN.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2015

subroutine zgges3 (character JOBVSL, character JOBVSR, character SORT, external SELCTG, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, integer SDIM, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldvsl, * ) VSL, integer LDVSL, complex*16, dimension( ldvsr, * ) VSR, integer LDVSR, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)

ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Purpose:

 ZGGES3 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
 ZGGEV 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*16 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*16 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*16 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*16 array, dimension (N)

BETA

          BETA is COMPLEX*16 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 ZGGES3. 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*16 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*16 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*16 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 DOUBLE PRECISION 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 ZHGEQZ
                =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 ZTGSEN.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

January 2015

subroutine zggesx (character JOBVSL, character JOBVSR, character SORT, external SELCTG, character SENSE, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, integer SDIM, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldvsl, * ) VSL, integer LDVSL, complex*16, dimension( ldvsr, * ) VSR, integer LDVSR, double precision, dimension( 2 ) RCONDE, double precision, dimension( 2 ) RCONDV, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer LIWORK, logical, dimension( * ) BWORK, integer INFO)

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

Purpose:

 ZGGESX 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 procedure) LOGICAL FUNCTION of two COMPLEX*16 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*16 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*16 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*16 array, dimension (N)

BETA

          BETA is COMPLEX*16 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*16 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*16 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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*16 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 DOUBLE PRECISION 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 IWORK.
          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 ZHGEQZ
                =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 ZTGSEN.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine zggev (character JOBVL, character JOBVR, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer INFO)

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

Purpose:

 ZGGEV 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*16 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*16 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*16 array, dimension (N)

BETA

          BETA is COMPLEX*16 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*16 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*16 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*16 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 DOUBLE PRECISION 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 DHGEQZ,
                =N+2: error return from DTGEVC.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

April 2012

subroutine zggev3 (character JOBVL, character JOBVR, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer INFO)

ZGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

Purpose:

 ZGGEV3 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*16 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*16 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*16 array, dimension (N)

BETA

          BETA is COMPLEX*16 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*16 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*16 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*16 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 DOUBLE PRECISION 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 DHGEQZ,
                =N+2: error return from DTGEVC.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

January 2015

subroutine zggevx (character BALANC, character JOBVL, character JOBVR, character SENSE, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldvl, * ) VL, integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, integer ILO, integer IHI, double precision, dimension( * ) LSCALE, double precision, dimension( * ) RSCALE, double precision ABNRM, double precision BBNRM, double precision, dimension( * ) RCONDE, double precision, dimension( * ) RCONDV, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, logical, dimension( * ) BWORK, integer INFO)

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

Purpose:

 ZGGEVX 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*16 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*16 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*16 array, dimension (N)

BETA

          BETA is COMPLEX*16 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*16 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*16 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
          The one-norm of the balanced matrix A.

BBNRM

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

RCONDE

          RCONDE is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)
          If JOB = '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*16 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 DOUBLE PRECISION 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 ZHGEQZ.
                =N+2: error return from ZTGEVC.

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

April 2012

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.

 

Author

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