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 "balanced" 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 "balanced" 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.

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April 2012

  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.