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


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


 Optionally also, it computes a balancing transformation to improve


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


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


 (RCONDE), and reciprocal condition numbers for the right


 eigenvectors (RCONDV).


 The right eigenvector v(j) of A satisfies


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


 where lambda(j) is its eigenvalue.


 The left eigenvector u(j) of A satisfies


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


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


 The computed eigenvectors are normalized to have Euclidean norm


 equal to 1 and largest component real.


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


 more nearly upper triangular, and applying a diagonal similarity


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


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


 of its eigenvalues and eigenvectors smaller.  The computed


 reciprocal condition numbers correspond to the balanced matrix.


 Permuting rows and columns will not change the condition numbers


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


 explanation of balancing, see section 4.10.2 of the LAPACK


 Users' Guide.

BALANC

          BALANC is CHARACTER*1


          Indicates how the input matrix should be diagonally scaled


          and/or permuted to improve the conditioning of its


          eigenvalues.


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


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


                 upper triangular. Do not diagonally scale;


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


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


                 to make the rows and columns of A more equal in


                 norm. Do not permute;


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


          Computed reciprocal condition numbers will be for the matrix


          after balancing and/or permuting. Permuting does not change


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

JOBVL

          JOBVL is CHARACTER*1


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


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


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

JOBVR

          JOBVR is CHARACTER*1


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


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


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

SENSE

          SENSE is CHARACTER*1


          Determines which reciprocal condition numbers are computed.


          = 'N': None are computed;


          = 'E': Computed for eigenvalues only;


          = 'V': Computed for right eigenvectors only;


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


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


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

N

          N is INTEGER


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

A

          A is COMPLEX*16 array, dimension (LDA,N)


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


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


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


          version of the matrix A.

LDA

          LDA is INTEGER


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

W

          W is COMPLEX*16 array, dimension (N)


          W contains the computed eigenvalues.

VL

          VL is COMPLEX*16 array, dimension (LDVL,N)


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


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


          as their eigenvalues.


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


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

LDVL

          LDVL is INTEGER


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


          JOBVL = 'V', LDVL >= N.

VR

          VR is COMPLEX*16 array, dimension (LDVR,N)


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


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


          as their eigenvalues.


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


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

LDVR

          LDVR is INTEGER


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


          JOBVR = 'V', LDVR >= N.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER


          ILO and IHI are integer values determined when A was


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


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

SCALE

          SCALE is DOUBLE PRECISION array, dimension (N)


          Details of the permutations and scaling factors applied


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


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


          factor applied to row and column j, then


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


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


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


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


          then 1 to ILO-1.

ABNRM

          ABNRM is DOUBLE PRECISION


          The one-norm of the balanced matrix (the maximum


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

RCONDE

          RCONDE is DOUBLE PRECISION array, dimension (N)


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


          eigenvalue.

RCONDV

          RCONDV is DOUBLE PRECISION array, dimension (N)


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


          right eigenvector.

WORK

          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))


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

LWORK

          LWORK is INTEGER


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


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


          LWORK >= N*N+2*N.


          For good performance, LWORK must generally be larger.


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


          only calculates the optimal size of the WORK array, returns


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


          message related to LWORK is issued by XERBLA.

RWORK

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

INFO

          INFO is INTEGER


          = 0:  successful exit


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


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


                eigenvalues, and no eigenvectors or condition numbers


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


                contain eigenvalues which have converged.

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June 2016