!>
!> 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).
!>
!> 

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  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  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.
!> 

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!>
!>  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.
!> 

!>
!> DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
!> the generalized eigenvalues, and optionally, the left and/or right
!> generalized eigenvectors.
!>
!> Optionally also, it 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).
!>
!> 

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 DOUBLE PRECISION 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 real Schur
!>          form of the  versions of the input A and B.
!> 

LDA

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

B

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

LDB

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

ALPHAR

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

ALPHAI

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

BETA

!>          BETA is DOUBLE PRECISION array, dimension (N)
!>          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
!>          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
!>          the j-th eigenvalue is real; if positive, then the j-th and
!>          (j+1)-st eigenvalues are a complex conjugate pair, with
!>          ALPHAI(j+1) negative.
!>
!>          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
!>          may easily over- or underflow, and BETA(j) may even be zero.
!>          Thus, the user should avoid naively computing the ratio
!>          ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
!>          than and usually comparable with norm(A) in magnitude, and
!>          BETA always less than and usually comparable with norm(B).
!> 

VL

!>          VL is DOUBLE PRECISION array, dimension (LDVL,N)
!>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
!>          after another in the columns of VL, in the same order as
!>          their eigenvalues. If the j-th eigenvalue is real, then
!>          u(j) = VL(:,j), the j-th column of VL. If the j-th and
!>          (j+1)-th eigenvalues form a complex conjugate pair, then
!>          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
!>          Each eigenvector will be scaled so the largest component 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 DOUBLE PRECISION array, dimension (LDVR,N)
!>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
!>          after another in the columns of VR, in the same order as
!>          their eigenvalues. If the j-th eigenvalue is real, then
!>          v(j) = VR(:,j), the j-th column of VR. If the j-th and
!>          (j+1)-th eigenvalues form a complex conjugate pair, then
!>          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
!>          Each eigenvector will be scaled so the largest component 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.
!>          For a complex conjugate pair of eigenvalues two consecutive
!>          elements of RCONDE are set to the same value. Thus RCONDE(j),
!>          RCONDV(j), and the j-th columns of VL and VR all correspond
!>          to the j-th eigenpair.
!>          If SENSE = 'N or 'V', RCONDE is not referenced.
!> 

RCONDV

!>          RCONDV is DOUBLE PRECISION array, dimension (N)
!>          If SENSE = 'V' or 'B', the estimated reciprocal condition
!>          numbers of the eigenvectors, stored in consecutive elements
!>          of the array. For a complex eigenvector two consecutive
!>          elements of RCONDV are set to the same value. 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 DOUBLE PRECISION 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 BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
!>          LWORK >= max(1,6*N).
!>          If SENSE = 'E' or 'B', LWORK >= max(1,10*N).
!>          If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
!>
!>          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.
!> 

IWORK

!>          IWORK is INTEGER array, dimension (N+6)
!>          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 ALPHAR(j), ALPHAI(j), and BETA(j)
!>                should be correct for j=INFO+1,...,N.
!>          > N:  =N+1: other than QZ iteration failed in DHGEQZ.
!>                =N+2: error return from DTGEVC.
!> 

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!>
!>  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.
!> 

!>
!> SGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
!> the generalized eigenvalues, and optionally, the left and/or right
!> generalized eigenvectors.
!>
!> Optionally also, it 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).
!>
!> 

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 REAL 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 real Schur
!>          form of the  versions of the input A and B.
!> 

LDA

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

B

!>          B is REAL array, dimension (LDB, N)
!>          On entry, the matrix B 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 real Schur
!>          form of the  versions of the input A and B.
!> 

LDB

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

ALPHAR

!>          ALPHAR is REAL array, dimension (N)
!> 

ALPHAI

!>          ALPHAI is REAL array, dimension (N)
!> 

BETA

!>          BETA is REAL array, dimension (N)
!>          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
!>          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
!>          the j-th eigenvalue is real; if positive, then the j-th and
!>          (j+1)-st eigenvalues are a complex conjugate pair, with
!>          ALPHAI(j+1) negative.
!>
!>          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
!>          may easily over- or underflow, and BETA(j) may even be zero.
!>          Thus, the user should avoid naively computing the ratio
!>          ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
!>          than and usually comparable with norm(A) in magnitude, and
!>          BETA always less than and usually comparable with norm(B).
!> 

VL

!>          VL is REAL array, dimension (LDVL,N)
!>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
!>          after another in the columns of VL, in the same order as
!>          their eigenvalues. If the j-th eigenvalue is real, then
!>          u(j) = VL(:,j), the j-th column of VL. If the j-th and
!>          (j+1)-th eigenvalues form a complex conjugate pair, then
!>          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
!>          Each eigenvector will be scaled so the largest component 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 REAL array, dimension (LDVR,N)
!>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
!>          after another in the columns of VR, in the same order as
!>          their eigenvalues. If the j-th eigenvalue is real, then
!>          v(j) = VR(:,j), the j-th column of VR. If the j-th and
!>          (j+1)-th eigenvalues form a complex conjugate pair, then
!>          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
!>          Each eigenvector will be scaled so the largest component 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.
!>          For a complex conjugate pair of eigenvalues two consecutive
!>          elements of RCONDE are set to the same value. Thus RCONDE(j),
!>          RCONDV(j), and the j-th columns of VL and VR all correspond
!>          to the j-th eigenpair.
!>          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. For a complex eigenvector two consecutive
!>          elements of RCONDV are set to the same value. 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 REAL array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!> 

LWORK

!>          LWORK is INTEGER
!>          The dimension of the array WORK. LWORK >= max(1,2*N).
!>          If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
!>          LWORK >= max(1,6*N).
!>          If SENSE = 'E', LWORK >= max(1,10*N).
!>          If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
!>
!>          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.
!> 

IWORK

!>          IWORK is INTEGER array, dimension (N+6)
!>          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 ALPHAR(j), ALPHAI(j), and BETA(j)
!>                should be correct for j=INFO+1,...,N.
!>          > N:  =N+1: other than QZ iteration failed in SHGEQZ.
!>                =N+2: error return from STGEVC.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>  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.
!> 

!>
!> 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).
!>
!> 

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  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  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.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>  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.
!>