CGEES computes for an N-by-N complex nonsymmetric matrix A, the


 eigenvalues, the Schur form T, and, optionally, the matrix of Schur


 vectors Z.  This gives the Schur factorization A = Z*T*(Z**H).


 Optionally, it also orders the eigenvalues on the diagonal of the


 Schur form so that selected eigenvalues are at the top left.


 The leading columns of Z then form an orthonormal basis for the


 invariant subspace corresponding to the selected eigenvalues.


 A complex matrix is in Schur form if it is upper triangular.

JOBVS

          JOBVS is CHARACTER*1


          = 'N': Schur vectors are not computed;


          = 'V': Schur vectors are computed.

SORT

          SORT is CHARACTER*1


          Specifies whether or not to order the eigenvalues on the


          diagonal of the Schur form.


          = 'N': Eigenvalues are not ordered:


          = 'S': Eigenvalues are ordered (see SELECT).

SELECT

          SELECT is a LOGICAL FUNCTION of one COMPLEX argument


          SELECT must be declared EXTERNAL in the calling subroutine.


          If SORT = 'S', SELECT is used to select eigenvalues to order


          to the top left of the Schur form.


          IF SORT = 'N', SELECT is not referenced.


          The eigenvalue W(j) is selected if SELECT(W(j)) is true.

N

          N is INTEGER


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

A

          A is COMPLEX array, dimension (LDA,N)


          On entry, the N-by-N matrix A.


          On exit, A has been overwritten by its Schur form T.

LDA

          LDA is INTEGER


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

SDIM

          SDIM is INTEGER


          If SORT = 'N', SDIM = 0.


          If SORT = 'S', SDIM = number of eigenvalues for which


                         SELECT is true.

W

          W is COMPLEX array, dimension (N)


          W contains the computed eigenvalues, in the same order that


          they appear on the diagonal of the output Schur form T.

VS

          VS is COMPLEX array, dimension (LDVS,N)


          If JOBVS = 'V', VS contains the unitary matrix Z of Schur


          vectors.


          If JOBVS = 'N', VS is not referenced.

LDVS

          LDVS is INTEGER


          The leading dimension of the array VS.  LDVS >= 1; if


          JOBVS = 'V', LDVS >= N.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.  LWORK >= max(1,2*N).


          For good performance, LWORK must generally be larger.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is REAL array, dimension (N)

BWORK

          BWORK is LOGICAL array, dimension (N)


          Not referenced if SORT = 'N'.

INFO

          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -i, the i-th argument had an illegal value.


          > 0: if INFO = i, and i is


               <= N:  the QR algorithm failed to compute all the


                      eigenvalues; elements 1:ILO-1 and i+1:N of W


                      contain those eigenvalues which have converged;


                      if JOBVS = 'V', VS contains the matrix which


                      reduces A to its partially converged Schur form.


               = N+1: the eigenvalues could not be reordered because


                      some eigenvalues were too close to separate (the


                      problem is very ill-conditioned);


               = N+2: after reordering, roundoff changed values of


                      some complex eigenvalues so that leading


                      eigenvalues in the Schur form no longer satisfy


                      SELECT = .TRUE..  This could also be caused by


                      underflow due to scaling.

Univ. of Tennessee

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NAG Ltd.

 CGEESX computes for an N-by-N complex nonsymmetric matrix A, the


 eigenvalues, the Schur form T, and, optionally, the matrix of Schur


 vectors Z.  This gives the Schur factorization A = Z*T*(Z**H).


 Optionally, it also orders the eigenvalues on the diagonal of the


 Schur form so that selected eigenvalues are at the top left;


 computes a reciprocal condition number for the average of the


 selected eigenvalues (RCONDE); and computes a reciprocal condition


 number for the right invariant subspace corresponding to the


 selected eigenvalues (RCONDV).  The leading columns of Z form an


 orthonormal basis for this invariant subspace.


 For further explanation of the reciprocal condition numbers RCONDE


 and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where


 these quantities are called s and sep respectively).


 A complex matrix is in Schur form if it is upper triangular.

JOBVS

          JOBVS is CHARACTER*1


          = 'N': Schur vectors are not computed;


          = 'V': Schur vectors are computed.

SORT

          SORT is CHARACTER*1


          Specifies whether or not to order the eigenvalues on the


          diagonal of the Schur form.


          = 'N': Eigenvalues are not ordered;


          = 'S': Eigenvalues are ordered (see SELECT).

SELECT

          SELECT is a LOGICAL FUNCTION of one COMPLEX argument


          SELECT must be declared EXTERNAL in the calling subroutine.


          If SORT = 'S', SELECT is used to select eigenvalues to order


          to the top left of the Schur form.


          If SORT = 'N', SELECT is not referenced.


          An eigenvalue W(j) is selected if SELECT(W(j)) is true.

SENSE

          SENSE is CHARACTER*1


          Determines which reciprocal condition numbers are computed.


          = 'N': None are computed;


          = 'E': Computed for average of selected eigenvalues only;


          = 'V': Computed for selected right invariant subspace only;


          = 'B': Computed for both.


          If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.

N

          N is INTEGER


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

A

          A is COMPLEX array, dimension (LDA, N)


          On entry, the N-by-N matrix A.


          On exit, A is overwritten by its Schur form T.

LDA

          LDA is INTEGER


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

SDIM

          SDIM is INTEGER


          If SORT = 'N', SDIM = 0.


          If SORT = 'S', SDIM = number of eigenvalues for which


                         SELECT is true.

W

          W is COMPLEX array, dimension (N)


          W contains the computed eigenvalues, in the same order


          that they appear on the diagonal of the output Schur form T.

VS

          VS is COMPLEX array, dimension (LDVS,N)


          If JOBVS = 'V', VS contains the unitary matrix Z of Schur


          vectors.


          If JOBVS = 'N', VS is not referenced.

LDVS

          LDVS is INTEGER


          The leading dimension of the array VS.  LDVS >= 1, and if


          JOBVS = 'V', LDVS >= N.

RCONDE

          RCONDE is REAL


          If SENSE = 'E' or 'B', RCONDE contains the reciprocal


          condition number for the average of the selected eigenvalues.


          Not referenced if SENSE = 'N' or 'V'.

RCONDV

          RCONDV is REAL


          If SENSE = 'V' or 'B', RCONDV contains the reciprocal


          condition number for the selected right invariant subspace.


          Not referenced if SENSE = 'N' or 'E'.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.  LWORK >= max(1,2*N).


          Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),


          where SDIM is the number of selected eigenvalues computed by


          this routine.  Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also


          that an error is only returned if LWORK < max(1,2*N), but if


          SENSE = 'E' or 'V' or 'B' this may not be large enough.


          For good performance, LWORK must generally be larger.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates upper bound on the optimal size of the


          array WORK, returns this value as the first entry of the WORK


          array, and no error message related to LWORK is issued by


          XERBLA.

RWORK

          RWORK is REAL array, dimension (N)

BWORK

          BWORK is LOGICAL array, dimension (N)


          Not referenced if SORT = 'N'.

INFO

          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -i, the i-th argument had an illegal value.


          > 0: if INFO = i, and i is


             <= N: the QR algorithm failed to compute all the


                   eigenvalues; elements 1:ILO-1 and i+1:N of W


                   contain those eigenvalues which have converged; if


                   JOBVS = 'V', VS contains the transformation which


                   reduces A to its partially converged Schur form.


             = N+1: the eigenvalues could not be reordered because some


                   eigenvalues were too close to separate (the problem


                   is very ill-conditioned);


             = N+2: after reordering, roundoff changed values of some


                   complex eigenvalues so that leading eigenvalues in


                   the Schur form no longer satisfy SELECT=.TRUE.  This


                   could also be caused by underflow due to scaling.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CGEEV computes for an N-by-N complex nonsymmetric matrix A, the


 eigenvalues and, optionally, the left and/or right eigenvectors.


 The right eigenvector v(j) of A satisfies


                  A * v(j) = lambda(j) * v(j)


 where lambda(j) is its eigenvalue.


 The left eigenvector u(j) of A satisfies


               u(j)**H * A = lambda(j) * u(j)**H


 where u(j)**H denotes the conjugate transpose of u(j).


 The computed eigenvectors are normalized to have Euclidean norm


 equal to 1 and largest component real.

JOBVL

          JOBVL is CHARACTER*1


          = 'N': left eigenvectors of A are not computed;


          = 'V': left eigenvectors of are computed.

JOBVR

          JOBVR is CHARACTER*1


          = 'N': right eigenvectors of A are not computed;


          = 'V': right eigenvectors of A are computed.

N

          N is INTEGER


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

A

          A is COMPLEX array, dimension (LDA,N)


          On entry, the N-by-N matrix A.


          On exit, A has been overwritten.

LDA

          LDA is INTEGER


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

W

          W is COMPLEX array, dimension (N)


          W contains the computed eigenvalues.

VL

          VL is COMPLEX array, dimension (LDVL,N)


          If JOBVL = 'V', the left eigenvectors u(j) are stored one


          after another in the columns of VL, in the same order


          as their eigenvalues.


          If JOBVL = 'N', VL is not referenced.


          u(j) = VL(:,j), the j-th column of VL.

LDVL

          LDVL is INTEGER


          The leading dimension of the array VL.  LDVL >= 1; if


          JOBVL = 'V', LDVL >= N.

VR

          VR is COMPLEX array, dimension (LDVR,N)


          If JOBVR = 'V', the right eigenvectors v(j) are stored one


          after another in the columns of VR, in the same order


          as their eigenvalues.


          If JOBVR = 'N', VR is not referenced.


          v(j) = VR(:,j), the j-th column of VR.

LDVR

          LDVR is INTEGER


          The leading dimension of the array VR.  LDVR >= 1; if


          JOBVR = 'V', LDVR >= N.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.  LWORK >= max(1,2*N).


          For good performance, LWORK must generally be larger.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is REAL array, dimension (2*N)

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          > 0:  if INFO = i, the QR algorithm failed to compute all the


                eigenvalues, and no eigenvectors have been computed;


                elements i+1:N of W contain eigenvalues which have


                converged.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the


 eigenvalues and, optionally, the left and/or right eigenvectors.


 Optionally also, it computes a balancing transformation to improve


 the conditioning of the eigenvalues and eigenvectors (ILO, IHI,


 SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues


 (RCONDE), and reciprocal condition numbers for the right


 eigenvectors (RCONDV).


 The right eigenvector v(j) of A satisfies


                  A * v(j) = lambda(j) * v(j)


 where lambda(j) is its eigenvalue.


 The left eigenvector u(j) of A satisfies


               u(j)**H * A = lambda(j) * u(j)**H


 where u(j)**H denotes the conjugate transpose of u(j).


 The computed eigenvectors are normalized to have Euclidean norm


 equal to 1 and largest component real.


 Balancing a matrix means permuting the rows and columns to make it


 more nearly upper triangular, and applying a diagonal similarity


 transformation D * A * D**(-1), where D is a diagonal matrix, to


 make its rows and columns closer in norm and the condition numbers


 of its eigenvalues and eigenvectors smaller.  The computed


 reciprocal condition numbers correspond to the balanced matrix.


 Permuting rows and columns will not change the condition numbers


 (in exact arithmetic) but diagonal scaling will.  For further


 explanation of balancing, see section 4.10.2 of the LAPACK


 Users' Guide.

BALANC

          BALANC is CHARACTER*1


          Indicates how the input matrix should be diagonally scaled


          and/or permuted to improve the conditioning of its


          eigenvalues.


          = 'N': Do not diagonally scale or permute;


          = 'P': Perform permutations to make the matrix more nearly


                 upper triangular. Do not diagonally scale;


          = 'S': Diagonally scale the matrix, ie. replace A by


                 D*A*D**(-1), where D is a diagonal matrix chosen


                 to make the rows and columns of A more equal in


                 norm. Do not permute;


          = 'B': Both diagonally scale and permute A.


          Computed reciprocal condition numbers will be for the matrix


          after balancing and/or permuting. Permuting does not change


          condition numbers (in exact arithmetic), but balancing does.

JOBVL

          JOBVL is CHARACTER*1


          = 'N': left eigenvectors of A are not computed;


          = 'V': left eigenvectors of A are computed.


          If SENSE = 'E' or 'B', JOBVL must = 'V'.

JOBVR

          JOBVR is CHARACTER*1


          = 'N': right eigenvectors of A are not computed;


          = 'V': right eigenvectors of A are computed.


          If SENSE = 'E' or 'B', JOBVR must = 'V'.

SENSE

          SENSE is CHARACTER*1


          Determines which reciprocal condition numbers are computed.


          = 'N': None are computed;


          = 'E': Computed for eigenvalues only;


          = 'V': Computed for right eigenvectors only;


          = 'B': Computed for eigenvalues and right eigenvectors.


          If SENSE = 'E' or 'B', both left and right eigenvectors


          must also be computed (JOBVL = 'V' and JOBVR = 'V').

N

          N is INTEGER


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

A

          A is COMPLEX array, dimension (LDA,N)


          On entry, the N-by-N matrix A.


          On exit, A has been overwritten.  If JOBVL = 'V' or


          JOBVR = 'V', A contains the Schur form of the balanced


          version of the matrix A.

LDA

          LDA is INTEGER


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

W

          W is COMPLEX array, dimension (N)


          W contains the computed eigenvalues.

VL

          VL is COMPLEX array, dimension (LDVL,N)


          If JOBVL = 'V', the left eigenvectors u(j) are stored one


          after another in the columns of VL, in the same order


          as their eigenvalues.


          If JOBVL = 'N', VL is not referenced.


          u(j) = VL(:,j), the j-th column of VL.

LDVL

          LDVL is INTEGER


          The leading dimension of the array VL.  LDVL >= 1; if


          JOBVL = 'V', LDVL >= N.

VR

          VR is COMPLEX array, dimension (LDVR,N)


          If JOBVR = 'V', the right eigenvectors v(j) are stored one


          after another in the columns of VR, in the same order


          as their eigenvalues.


          If JOBVR = 'N', VR is not referenced.


          v(j) = VR(:,j), the j-th column of VR.

LDVR

          LDVR is INTEGER


          The leading dimension of the array VR.  LDVR >= 1; if


          JOBVR = 'V', LDVR >= N.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER


          ILO and IHI are integer values determined when A was


          balanced.  The balanced A(i,j) = 0 if I > J and


          J = 1,...,ILO-1 or I = IHI+1,...,N.

SCALE

          SCALE is REAL array, dimension (N)


          Details of the permutations and scaling factors applied


          when balancing A.  If P(j) is the index of the row and column


          interchanged with row and column j, and D(j) is the scaling


          factor applied to row and column j, then


          SCALE(J) = P(J),    for J = 1,...,ILO-1


                   = D(J),    for J = ILO,...,IHI


                   = P(J)     for J = IHI+1,...,N.


          The order in which the interchanges are made is N to IHI+1,


          then 1 to ILO-1.

ABNRM

          ABNRM is REAL


          The one-norm of the balanced matrix (the maximum


          of the sum of absolute values of elements of any column).

RCONDE

          RCONDE is REAL array, dimension (N)


          RCONDE(j) is the reciprocal condition number of the j-th


          eigenvalue.

RCONDV

          RCONDV is REAL array, dimension (N)


          RCONDV(j) is the reciprocal condition number of the j-th


          right eigenvector.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.  If SENSE = 'N' or 'E',


          LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',


          LWORK >= N*N+2*N.


          For good performance, LWORK must generally be larger.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is REAL array, dimension (2*N)

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          > 0:  if INFO = i, the QR algorithm failed to compute all the


                eigenvalues, and no eigenvectors or condition numbers


                have been computed; elements 1:ILO-1 and i+1:N of W


                contain eigenvalues which have converged.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CGGES computes for a pair of N-by-N complex nonsymmetric matrices


 (A,B), the generalized eigenvalues, the generalized complex Schur


 form (S, T), and optionally left and/or right Schur vectors (VSL


 and VSR). This gives the generalized Schur factorization


         (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )


 where (VSR)**H is the conjugate-transpose of VSR.


 Optionally, it also orders the eigenvalues so that a selected cluster


 of eigenvalues appears in the leading diagonal blocks of the upper


 triangular matrix S and the upper triangular matrix T. The leading


 columns of VSL and VSR then form an unitary basis for the


 corresponding left and right eigenspaces (deflating subspaces).


 (If only the generalized eigenvalues are needed, use the driver


 CGGEV instead, which is faster.)


 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w


 or a ratio alpha/beta = w, such that  A - w*B is singular.  It is


 usually represented as the pair (alpha,beta), as there is a


 reasonable interpretation for beta=0, and even for both being zero.


 A pair of matrices (S,T) is in generalized complex Schur form if S


 and T are upper triangular and, in addition, the diagonal elements


 of T are non-negative real numbers.

JOBVSL

          JOBVSL is CHARACTER*1


          = 'N':  do not compute the left Schur vectors;


          = 'V':  compute the left Schur vectors.

JOBVSR

          JOBVSR is CHARACTER*1


          = 'N':  do not compute the right Schur vectors;


          = 'V':  compute the right Schur vectors.

SORT

          SORT is CHARACTER*1


          Specifies whether or not to order the eigenvalues on the


          diagonal of the generalized Schur form.


          = 'N':  Eigenvalues are not ordered;


          = 'S':  Eigenvalues are ordered (see SELCTG).

SELCTG

          SELCTG is a LOGICAL FUNCTION of two COMPLEX arguments


          SELCTG must be declared EXTERNAL in the calling subroutine.


          If SORT = 'N', SELCTG is not referenced.


          If SORT = 'S', SELCTG is used to select eigenvalues to sort


          to the top left of the Schur form.


          An eigenvalue ALPHA(j)/BETA(j) is selected if


          SELCTG(ALPHA(j),BETA(j)) is true.


          Note that a selected complex eigenvalue may no longer satisfy


          SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since


          ordering may change the value of complex eigenvalues


          (especially if the eigenvalue is ill-conditioned), in this


          case INFO is set to N+2 (See INFO below).

N

          N is INTEGER


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

A

          A is COMPLEX array, dimension (LDA, N)


          On entry, the first of the pair of matrices.


          On exit, A has been overwritten by its generalized Schur


          form S.

LDA

          LDA is INTEGER


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

B

          B is COMPLEX array, dimension (LDB, N)


          On entry, the second of the pair of matrices.


          On exit, B has been overwritten by its generalized Schur


          form T.

LDB

          LDB is INTEGER


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

SDIM

          SDIM is INTEGER


          If SORT = 'N', SDIM = 0.


          If SORT = 'S', SDIM = number of eigenvalues (after sorting)


          for which SELCTG is true.

ALPHA

          ALPHA is COMPLEX array, dimension (N)

BETA

          BETA is COMPLEX array, dimension (N)


          On exit,  ALPHA(j)/BETA(j), j=1,...,N, will be the


          generalized eigenvalues.  ALPHA(j), j=1,...,N  and  BETA(j),


          j=1,...,N  are the diagonals of the complex Schur form (A,B)


          output by CGGES. The  BETA(j) will be non-negative real.


          Note: the quotients ALPHA(j)/BETA(j) may easily over- or


          underflow, and BETA(j) may even be zero.  Thus, the user


          should avoid naively computing the ratio alpha/beta.


          However, ALPHA will be always less than and usually


          comparable with norm(A) in magnitude, and BETA always less


          than and usually comparable with norm(B).

VSL

          VSL is COMPLEX array, dimension (LDVSL,N)


          If JOBVSL = 'V', VSL will contain the left Schur vectors.


          Not referenced if JOBVSL = 'N'.

LDVSL

          LDVSL is INTEGER


          The leading dimension of the matrix VSL. LDVSL >= 1, and


          if JOBVSL = 'V', LDVSL >= N.

VSR

          VSR is COMPLEX array, dimension (LDVSR,N)


          If JOBVSR = 'V', VSR will contain the right Schur vectors.


          Not referenced if JOBVSR = 'N'.

LDVSR

          LDVSR is INTEGER


          The leading dimension of the matrix VSR. LDVSR >= 1, and


          if JOBVSR = 'V', LDVSR >= N.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.  LWORK >= max(1,2*N).


          For good performance, LWORK must generally be larger.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is REAL array, dimension (8*N)

BWORK

          BWORK is LOGICAL array, dimension (N)


          Not referenced if SORT = 'N'.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          =1,...,N:


                The QZ iteration failed.  (A,B) are not in Schur


                form, but ALPHA(j) and BETA(j) should be correct for


                j=INFO+1,...,N.


          > N:  =N+1: other than QZ iteration failed in CHGEQZ


                =N+2: after reordering, roundoff changed values of


                      some complex eigenvalues so that leading


                      eigenvalues in the Generalized Schur form no


                      longer satisfy SELCTG=.TRUE.  This could also


                      be caused due to scaling.


                =N+3: reordering failed in CTGSEN.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CGGES3 computes for a pair of N-by-N complex nonsymmetric matrices


 (A,B), the generalized eigenvalues, the generalized complex Schur


 form (S, T), and optionally left and/or right Schur vectors (VSL


 and VSR). This gives the generalized Schur factorization


         (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )


 where (VSR)**H is the conjugate-transpose of VSR.


 Optionally, it also orders the eigenvalues so that a selected cluster


 of eigenvalues appears in the leading diagonal blocks of the upper


 triangular matrix S and the upper triangular matrix T. The leading


 columns of VSL and VSR then form an unitary basis for the


 corresponding left and right eigenspaces (deflating subspaces).


 (If only the generalized eigenvalues are needed, use the driver


 CGGEV instead, which is faster.)


 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w


 or a ratio alpha/beta = w, such that  A - w*B is singular.  It is


 usually represented as the pair (alpha,beta), as there is a


 reasonable interpretation for beta=0, and even for both being zero.


 A pair of matrices (S,T) is in generalized complex Schur form if S


 and T are upper triangular and, in addition, the diagonal elements


 of T are non-negative real numbers.

JOBVSL

          JOBVSL is CHARACTER*1


          = 'N':  do not compute the left Schur vectors;


          = 'V':  compute the left Schur vectors.

JOBVSR

          JOBVSR is CHARACTER*1


          = 'N':  do not compute the right Schur vectors;


          = 'V':  compute the right Schur vectors.

SORT

          SORT is CHARACTER*1


          Specifies whether or not to order the eigenvalues on the


          diagonal of the generalized Schur form.


          = 'N':  Eigenvalues are not ordered;


          = 'S':  Eigenvalues are ordered (see SELCTG).

SELCTG

          SELCTG is a LOGICAL FUNCTION of two COMPLEX arguments


          SELCTG must be declared EXTERNAL in the calling subroutine.


          If SORT = 'N', SELCTG is not referenced.


          If SORT = 'S', SELCTG is used to select eigenvalues to sort


          to the top left of the Schur form.


          An eigenvalue ALPHA(j)/BETA(j) is selected if


          SELCTG(ALPHA(j),BETA(j)) is true.


          Note that a selected complex eigenvalue may no longer satisfy


          SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since


          ordering may change the value of complex eigenvalues


          (especially if the eigenvalue is ill-conditioned), in this


          case INFO is set to N+2 (See INFO below).

N

          N is INTEGER


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

A

          A is COMPLEX array, dimension (LDA, N)


          On entry, the first of the pair of matrices.


          On exit, A has been overwritten by its generalized Schur


          form S.

LDA

          LDA is INTEGER


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

B

          B is COMPLEX array, dimension (LDB, N)


          On entry, the second of the pair of matrices.


          On exit, B has been overwritten by its generalized Schur


          form T.

LDB

          LDB is INTEGER


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

SDIM

          SDIM is INTEGER


          If SORT = 'N', SDIM = 0.


          If SORT = 'S', SDIM = number of eigenvalues (after sorting)


          for which SELCTG is true.

ALPHA

          ALPHA is COMPLEX array, dimension (N)

BETA

          BETA is COMPLEX array, dimension (N)


          On exit,  ALPHA(j)/BETA(j), j=1,...,N, will be the


          generalized eigenvalues.  ALPHA(j), j=1,...,N  and  BETA(j),


          j=1,...,N  are the diagonals of the complex Schur form (A,B)


          output by CGGES3. The  BETA(j) will be non-negative real.


          Note: the quotients ALPHA(j)/BETA(j) may easily over- or


          underflow, and BETA(j) may even be zero.  Thus, the user


          should avoid naively computing the ratio alpha/beta.


          However, ALPHA will be always less than and usually


          comparable with norm(A) in magnitude, and BETA always less


          than and usually comparable with norm(B).

VSL

          VSL is COMPLEX array, dimension (LDVSL,N)


          If JOBVSL = 'V', VSL will contain the left Schur vectors.


          Not referenced if JOBVSL = 'N'.

LDVSL

          LDVSL is INTEGER


          The leading dimension of the matrix VSL. LDVSL >= 1, and


          if JOBVSL = 'V', LDVSL >= N.

VSR

          VSR is COMPLEX array, dimension (LDVSR,N)


          If JOBVSR = 'V', VSR will contain the right Schur vectors.


          Not referenced if JOBVSR = 'N'.

LDVSR

          LDVSR is INTEGER


          The leading dimension of the matrix VSR. LDVSR >= 1, and


          if JOBVSR = 'V', LDVSR >= N.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is REAL array, dimension (8*N)

BWORK

          BWORK is LOGICAL array, dimension (N)


          Not referenced if SORT = 'N'.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          =1,...,N:


                The QZ iteration failed.  (A,B) are not in Schur


                form, but ALPHA(j) and BETA(j) should be correct for


                j=INFO+1,...,N.


          > N:  =N+1: other than QZ iteration failed in CLAQZ0


                =N+2: after reordering, roundoff changed values of


                      some complex eigenvalues so that leading


                      eigenvalues in the Generalized Schur form no


                      longer satisfy SELCTG=.TRUE.  This could also


                      be caused due to scaling.


                =N+3: reordering failed in CTGSEN.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CGGESX computes for a pair of N-by-N complex nonsymmetric matrices


 (A,B), the generalized eigenvalues, the complex Schur form (S,T),


 and, optionally, the left and/or right matrices of Schur vectors (VSL


 and VSR).  This gives the generalized Schur factorization


      (A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H )


 where (VSR)**H is the conjugate-transpose of VSR.


 Optionally, it also orders the eigenvalues so that a selected cluster


 of eigenvalues appears in the leading diagonal blocks of the upper


 triangular matrix S and the upper triangular matrix T; computes


 a reciprocal condition number for the average of the selected


 eigenvalues (RCONDE); and computes a reciprocal condition number for


 the right and left deflating subspaces corresponding to the selected


 eigenvalues (RCONDV). The leading columns of VSL and VSR then form


 an orthonormal basis for the corresponding left and right eigenspaces


 (deflating subspaces).


 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w


 or a ratio alpha/beta = w, such that  A - w*B is singular.  It is


 usually represented as the pair (alpha,beta), as there is a


 reasonable interpretation for beta=0 or for both being zero.


 A pair of matrices (S,T) is in generalized complex Schur form if T is


 upper triangular with non-negative diagonal and S is upper


 triangular.

JOBVSL

          JOBVSL is CHARACTER*1


          = 'N':  do not compute the left Schur vectors;


          = 'V':  compute the left Schur vectors.

JOBVSR

          JOBVSR is CHARACTER*1


          = 'N':  do not compute the right Schur vectors;


          = 'V':  compute the right Schur vectors.

SORT

          SORT is CHARACTER*1


          Specifies whether or not to order the eigenvalues on the


          diagonal of the generalized Schur form.


          = 'N':  Eigenvalues are not ordered;


          = 'S':  Eigenvalues are ordered (see SELCTG).

SELCTG

          SELCTG is a LOGICAL FUNCTION of two COMPLEX arguments


          SELCTG must be declared EXTERNAL in the calling subroutine.


          If SORT = 'N', SELCTG is not referenced.


          If SORT = 'S', SELCTG is used to select eigenvalues to sort


          to the top left of the Schur form.


          Note that a selected complex eigenvalue may no longer satisfy


          SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since


          ordering may change the value of complex eigenvalues


          (especially if the eigenvalue is ill-conditioned), in this


          case INFO is set to N+3 see INFO below).

SENSE

          SENSE is CHARACTER*1


          Determines which reciprocal condition numbers are computed.


          = 'N':  None are computed;


          = 'E':  Computed for average of selected eigenvalues only;


          = 'V':  Computed for selected deflating subspaces only;


          = 'B':  Computed for both.


          If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

N

          N is INTEGER


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

A

          A is COMPLEX array, dimension (LDA, N)


          On entry, the first of the pair of matrices.


          On exit, A has been overwritten by its generalized Schur


          form S.

LDA

          LDA is INTEGER


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

B

          B is COMPLEX array, dimension (LDB, N)


          On entry, the second of the pair of matrices.


          On exit, B has been overwritten by its generalized Schur


          form T.

LDB

          LDB is INTEGER


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

SDIM

          SDIM is INTEGER


          If SORT = 'N', SDIM = 0.


          If SORT = 'S', SDIM = number of eigenvalues (after sorting)


          for which SELCTG is true.

ALPHA

          ALPHA is COMPLEX array, dimension (N)

BETA

          BETA is COMPLEX array, dimension (N)


          On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the


          generalized eigenvalues.  ALPHA(j) and BETA(j),j=1,...,N  are


          the diagonals of the complex Schur form (S,T).  BETA(j) will


          be non-negative real.


          Note: the quotients ALPHA(j)/BETA(j) may easily over- or


          underflow, and BETA(j) may even be zero.  Thus, the user


          should avoid naively computing the ratio alpha/beta.


          However, ALPHA will be always less than and usually


          comparable with norm(A) in magnitude, and BETA always less


          than and usually comparable with norm(B).

VSL

          VSL is COMPLEX array, dimension (LDVSL,N)


          If JOBVSL = 'V', VSL will contain the left Schur vectors.


          Not referenced if JOBVSL = 'N'.

LDVSL

          LDVSL is INTEGER


          The leading dimension of the matrix VSL. LDVSL >=1, and


          if JOBVSL = 'V', LDVSL >= N.

VSR

          VSR is COMPLEX array, dimension (LDVSR,N)


          If JOBVSR = 'V', VSR will contain the right Schur vectors.


          Not referenced if JOBVSR = 'N'.

LDVSR

          LDVSR is INTEGER


          The leading dimension of the matrix VSR. LDVSR >= 1, and


          if JOBVSR = 'V', LDVSR >= N.

RCONDE

          RCONDE is REAL array, dimension ( 2 )


          If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the


          reciprocal condition numbers for the average of the selected


          eigenvalues.


          Not referenced if SENSE = 'N' or 'V'.

RCONDV

          RCONDV is REAL array, dimension ( 2 )


          If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the


          reciprocal condition number for the selected deflating


          subspaces.


          Not referenced if SENSE = 'N' or 'E'.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',


          LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else


          LWORK >= MAX(1,2*N).  Note that 2*SDIM*(N-SDIM) <= N*N/2.


          Note also that an error is only returned if


          LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may


          not be large enough.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the bound on the optimal size of the WORK


          array and the minimum size of the IWORK array, returns these


          values as the first entries of the WORK and IWORK arrays, and


          no error message related to LWORK or LIWORK is issued by


          XERBLA.

RWORK

          RWORK is REAL array, dimension ( 8*N )


          Real workspace.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))


          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

LIWORK

          LIWORK is INTEGER


          The dimension of the array WORK.


          If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise


          LIWORK >= N+2.


          If LIWORK = -1, then a workspace query is assumed; the


          routine only calculates the bound on the optimal size of the


          WORK array and the minimum size of the IWORK array, returns


          these values as the first entries of the WORK and IWORK


          arrays, and no error message related to LWORK or LIWORK is


          issued by XERBLA.

BWORK

          BWORK is LOGICAL array, dimension (N)


          Not referenced if SORT = 'N'.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          = 1,...,N:


                The QZ iteration failed.  (A,B) are not in Schur


                form, but ALPHA(j) and BETA(j) should be correct for


                j=INFO+1,...,N.


          > N:  =N+1: other than QZ iteration failed in CHGEQZ


                =N+2: after reordering, roundoff changed values of


                      some complex eigenvalues so that leading


                      eigenvalues in the Generalized Schur form no


                      longer satisfy SELCTG=.TRUE.  This could also


                      be caused due to scaling.


                =N+3: reordering failed in CTGSEN.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CGGEV computes for a pair of N-by-N complex nonsymmetric matrices


 (A,B), the generalized eigenvalues, and optionally, the left and/or


 right generalized eigenvectors.


 A generalized eigenvalue for a pair of matrices (A,B) is a scalar


 lambda or a ratio alpha/beta = lambda, such that A - lambda*B is


 singular. It is usually represented as the pair (alpha,beta), as


 there is a reasonable interpretation for beta=0, and even for both


 being zero.


 The right generalized eigenvector v(j) corresponding to the


 generalized eigenvalue lambda(j) of (A,B) satisfies


              A * v(j) = lambda(j) * B * v(j).


 The left generalized eigenvector u(j) corresponding to the


 generalized eigenvalues lambda(j) of (A,B) satisfies


              u(j)**H * A = lambda(j) * u(j)**H * B


 where u(j)**H is the conjugate-transpose of u(j).

JOBVL

          JOBVL is CHARACTER*1


          = 'N':  do not compute the left generalized eigenvectors;


          = 'V':  compute the left generalized eigenvectors.

JOBVR

          JOBVR is CHARACTER*1


          = 'N':  do not compute the right generalized eigenvectors;


          = 'V':  compute the right generalized eigenvectors.

N

          N is INTEGER


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

A

          A is COMPLEX array, dimension (LDA, N)


          On entry, the matrix A in the pair (A,B).


          On exit, A has been overwritten.

LDA

          LDA is INTEGER


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

B

          B is COMPLEX array, dimension (LDB, N)


          On entry, the matrix B in the pair (A,B).


          On exit, B has been overwritten.

LDB

          LDB is INTEGER


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

ALPHA

          ALPHA is COMPLEX array, dimension (N)

BETA

          BETA is COMPLEX array, dimension (N)


          On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the


          generalized eigenvalues.


          Note: the quotients ALPHA(j)/BETA(j) may easily over- or


          underflow, and BETA(j) may even be zero.  Thus, the user


          should avoid naively computing the ratio alpha/beta.


          However, ALPHA will be always less than and usually


          comparable with norm(A) in magnitude, and BETA always less


          than and usually comparable with norm(B).

VL

          VL is COMPLEX array, dimension (LDVL,N)


          If JOBVL = 'V', the left generalized eigenvectors u(j) are


          stored one after another in the columns of VL, in the same


          order as their eigenvalues.


          Each eigenvector is scaled so the largest component has


          abs(real part) + abs(imag. part) = 1.


          Not referenced if JOBVL = 'N'.

LDVL

          LDVL is INTEGER


          The leading dimension of the matrix VL. LDVL >= 1, and


          if JOBVL = 'V', LDVL >= N.

VR

          VR is COMPLEX array, dimension (LDVR,N)


          If JOBVR = 'V', the right generalized eigenvectors v(j) are


          stored one after another in the columns of VR, in the same


          order as their eigenvalues.


          Each eigenvector is scaled so the largest component has


          abs(real part) + abs(imag. part) = 1.


          Not referenced if JOBVR = 'N'.

LDVR

          LDVR is INTEGER


          The leading dimension of the matrix VR. LDVR >= 1, and


          if JOBVR = 'V', LDVR >= N.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.  LWORK >= max(1,2*N).


          For good performance, LWORK must generally be larger.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is REAL array, dimension (8*N)

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          =1,...,N:


                The QZ iteration failed.  No eigenvectors have been


                calculated, but ALPHA(j) and BETA(j) should be


                correct for j=INFO+1,...,N.


          > N:  =N+1: other then QZ iteration failed in CHGEQZ,


                =N+2: error return from CTGEVC.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 CGGEV3 computes for a pair of N-by-N complex nonsymmetric matrices


 (A,B), the generalized eigenvalues, and optionally, the left and/or


 right generalized eigenvectors.


 A generalized eigenvalue for a pair of matrices (A,B) is a scalar


 lambda or a ratio alpha/beta = lambda, such that A - lambda*B is


 singular. It is usually represented as the pair (alpha,beta), as


 there is a reasonable interpretation for beta=0, and even for both


 being zero.


 The right generalized eigenvector v(j) corresponding to the


 generalized eigenvalue lambda(j) of (A,B) satisfies


              A * v(j) = lambda(j) * B * v(j).


 The left generalized eigenvector u(j) corresponding to the


 generalized eigenvalues lambda(j) of (A,B) satisfies


              u(j)**H * A = lambda(j) * u(j)**H * B


 where u(j)**H is the conjugate-transpose of u(j).

JOBVL

          JOBVL is CHARACTER*1


          = 'N':  do not compute the left generalized eigenvectors;


          = 'V':  compute the left generalized eigenvectors.

JOBVR

          JOBVR is CHARACTER*1


          = 'N':  do not compute the right generalized eigenvectors;


          = 'V':  compute the right generalized eigenvectors.

N

          N is INTEGER


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

A

          A is COMPLEX array, dimension (LDA, N)


          On entry, the matrix A in the pair (A,B).


          On exit, A has been overwritten.

LDA

          LDA is INTEGER


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

B

          B is COMPLEX array, dimension (LDB, N)


          On entry, the matrix B in the pair (A,B).


          On exit, B has been overwritten.

LDB

          LDB is INTEGER


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

ALPHA

          ALPHA is COMPLEX array, dimension (N)

BETA

          BETA is COMPLEX array, dimension (N)


          On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the


          generalized eigenvalues.


          Note: the quotients ALPHA(j)/BETA(j) may easily over- or


          underflow, and BETA(j) may even be zero.  Thus, the user


          should avoid naively computing the ratio alpha/beta.


          However, ALPHA will be always less than and usually


          comparable with norm(A) in magnitude, and BETA always less


          than and usually comparable with norm(B).

VL

          VL is COMPLEX array, dimension (LDVL,N)


          If JOBVL = 'V', the left generalized eigenvectors u(j) are


          stored one after another in the columns of VL, in the same


          order as their eigenvalues.


          Each eigenvector is scaled so the largest component has


          abs(real part) + abs(imag. part) = 1.


          Not referenced if JOBVL = 'N'.

LDVL

          LDVL is INTEGER


          The leading dimension of the matrix VL. LDVL >= 1, and


          if JOBVL = 'V', LDVL >= N.

VR

          VR is COMPLEX array, dimension (LDVR,N)


          If JOBVR = 'V', the right generalized eigenvectors v(j) are


          stored one after another in the columns of VR, in the same


          order as their eigenvalues.


          Each eigenvector is scaled so the largest component has


          abs(real part) + abs(imag. part) = 1.


          Not referenced if JOBVR = 'N'.

LDVR

          LDVR is INTEGER


          The leading dimension of the matrix VR. LDVR >= 1, and


          if JOBVR = 'V', LDVR >= N.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is REAL array, dimension (8*N)

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          =1,...,N:


                The QZ iteration failed.  No eigenvectors have been


                calculated, but ALPHA(j) and BETA(j) should be


                correct for j=INFO+1,...,N.


          > N:  =N+1: other then QZ iteration failed in CHGEQZ,


                =N+2: error return from CTGEVC.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 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 'balanced' versions of the input A and B.

LDA

          LDA is INTEGER


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

B

          B is COMPLEX array, dimension (LDB, N)


          On entry, the matrix B in the pair (A,B).


          On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'


          or both, then B contains the second part of the complex


          Schur form of the 'balanced' versions of the input A and B.

LDB

          LDB is INTEGER


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

ALPHA

          ALPHA is COMPLEX array, dimension (N)

BETA

          BETA is COMPLEX array, dimension (N)


          On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized


          eigenvalues.


          Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or


          underflow, and BETA(j) may even be zero.  Thus, the user


          should avoid naively computing the ratio ALPHA/BETA.


          However, ALPHA will be always less than and usually


          comparable with norm(A) in magnitude, and BETA always less


          than and usually comparable with norm(B).

VL

          VL is COMPLEX array, dimension (LDVL,N)


          If JOBVL = 'V', the left generalized eigenvectors u(j) are


          stored one after another in the columns of VL, in the same


          order as their eigenvalues.


          Each eigenvector will be scaled so the largest component


          will have abs(real part) + abs(imag. part) = 1.


          Not referenced if JOBVL = 'N'.

LDVL

          LDVL is INTEGER


          The leading dimension of the matrix VL. LDVL >= 1, and


          if JOBVL = 'V', LDVL >= N.

VR

          VR is COMPLEX array, dimension (LDVR,N)


          If JOBVR = 'V', the right generalized eigenvectors v(j) are


          stored one after another in the columns of VR, in the same


          order as their eigenvalues.


          Each eigenvector will be scaled so the largest component


          will have abs(real part) + abs(imag. part) = 1.


          Not referenced if JOBVR = 'N'.

LDVR

          LDVR is INTEGER


          The leading dimension of the matrix VR. LDVR >= 1, and


          if JOBVR = 'V', LDVR >= N.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER


          ILO and IHI are integer values such that on exit


          A(i,j) = 0 and B(i,j) = 0 if i > j and


          j = 1,...,ILO-1 or i = IHI+1,...,N.


          If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

LSCALE

          LSCALE is REAL array, dimension (N)


          Details of the permutations and scaling factors applied


          to the left side of A and B.  If PL(j) is the index of the


          row interchanged with row j, and DL(j) is the scaling


          factor applied to row j, then


            LSCALE(j) = PL(j)  for j = 1,...,ILO-1


                      = DL(j)  for j = ILO,...,IHI


                      = PL(j)  for j = IHI+1,...,N.


          The order in which the interchanges are made is N to IHI+1,


          then 1 to ILO-1.

RSCALE

          RSCALE is REAL array, dimension (N)


          Details of the permutations and scaling factors applied


          to the right side of A and B.  If PR(j) is the index of the


          column interchanged with column j, and DR(j) is the scaling


          factor applied to column j, then


            RSCALE(j) = PR(j)  for j = 1,...,ILO-1


                      = DR(j)  for j = ILO,...,IHI


                      = PR(j)  for j = IHI+1,...,N


          The order in which the interchanges are made is N to IHI+1,


          then 1 to ILO-1.

ABNRM

          ABNRM is REAL


          The one-norm of the balanced matrix A.

BBNRM

          BBNRM is REAL


          The one-norm of the balanced matrix B.

RCONDE

          RCONDE is REAL array, dimension (N)


          If SENSE = 'E' or 'B', the reciprocal condition numbers of


          the eigenvalues, stored in consecutive elements of the array.


          If SENSE = 'N' or 'V', RCONDE is not referenced.

RCONDV

          RCONDV is REAL array, dimension (N)


          If SENSE = 'V' or 'B', the estimated reciprocal condition


          numbers of the eigenvectors, stored in consecutive elements


          of the array. If the eigenvalues cannot be reordered to


          compute RCONDV(j), RCONDV(j) is set to 0; this can only occur


          when the true value would be very small anyway.


          If SENSE = 'N' or 'E', RCONDV is not referenced.

WORK

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK. LWORK >= max(1,2*N).


          If SENSE = 'E', LWORK >= max(1,4*N).


          If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N).


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is REAL array, dimension (lrwork)


          lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B',


          and at least max(1,2*N) otherwise.


          Real workspace.

IWORK

          IWORK is INTEGER array, dimension (N+2)


          If SENSE = 'E', IWORK is not referenced.

BWORK

          BWORK is LOGICAL array, dimension (N)


          If SENSE = 'N', BWORK is not referenced.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          = 1,...,N:


                The QZ iteration failed.  No eigenvectors have been


                calculated, but ALPHA(j) and BETA(j) should be correct


                for j=INFO+1,...,N.


          > N:  =N+1: other than QZ iteration failed in CHGEQZ.


                =N+2: error return from CTGEVC.

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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.