ggevx - Man Page

subroutine cggevx (balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, rwork, iwork, bwork, info)
CGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine dggevx (balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, iwork, bwork, info)
DGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sggevx (balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, iwork, bwork, info)
SGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine zggevx (balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, rwork, iwork, bwork, info)
ZGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

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

Purpose:

 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).

Parameters

BALANC

          BALANC is CHARACTER*1
          Specifies the balance option to be performed:
          = 'N':  do not diagonally scale or permute;
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale.
          Computed reciprocal condition numbers will be for the
          matrices after permuting and/or balancing. Permuting does
          not change condition numbers (in exact arithmetic), but
          balancing does.

JOBVL

          JOBVL is CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors.

JOBVR

          JOBVR is CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors.

SENSE

          SENSE is CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': none are computed;
          = 'E': computed for eigenvalues only;
          = 'V': computed for eigenvectors only;
          = 'B': computed for eigenvalues and eigenvectors.

N

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

A

          A is COMPLEX 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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  Balancing a matrix pair (A,B) includes, first, permuting rows and
  columns to isolate eigenvalues, second, applying diagonal similarity
  transformation to the rows and columns to make the rows and columns
  as close in norm as possible. The computed reciprocal condition
  numbers correspond to the balanced matrix. Permuting rows and columns
  will not change the condition numbers (in exact arithmetic) but
  diagonal scaling will.  For further explanation of balancing, see
  section 4.11.1.2 of LAPACK Users' Guide.

  An approximate error bound on the chordal distance between the i-th
  computed generalized eigenvalue w and the corresponding exact
  eigenvalue lambda is

       chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

  An approximate error bound for the angle between the i-th computed
  eigenvector VL(i) or VR(i) is given by

       EPS * norm(ABNRM, BBNRM) / DIF(i).

  For further explanation of the reciprocal condition numbers RCONDE
  and RCONDV, see section 4.11 of LAPACK User's Guide.

Definition at line 370 of file cggevx.f.

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

Purpose:

 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).

Parameters

BALANC

          BALANC is CHARACTER*1
          Specifies the balance option to be performed.
          = 'N':  do not diagonally scale or permute;
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale.
          Computed reciprocal condition numbers will be for the
          matrices after permuting and/or balancing. Permuting does
          not change condition numbers (in exact arithmetic), but
          balancing does.

JOBVL

          JOBVL is CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors.

JOBVR

          JOBVR is CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors.

SENSE

          SENSE is CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': none are computed;
          = 'E': computed for eigenvalues only;
          = 'V': computed for eigenvectors only;
          = 'B': computed for eigenvalues and eigenvectors.

N

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

A

          A is 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 'balanced' 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 'balanced' 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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  Balancing a matrix pair (A,B) includes, first, permuting rows and
  columns to isolate eigenvalues, second, applying diagonal similarity
  transformation to the rows and columns to make the rows and columns
  as close in norm as possible. The computed reciprocal condition
  numbers correspond to the balanced matrix. Permuting rows and columns
  will not change the condition numbers (in exact arithmetic) but
  diagonal scaling will.  For further explanation of balancing, see
  section 4.11.1.2 of LAPACK Users' Guide.

  An approximate error bound on the chordal distance between the i-th
  computed generalized eigenvalue w and the corresponding exact
  eigenvalue lambda is

       chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

  An approximate error bound for the angle between the i-th computed
  eigenvector VL(i) or VR(i) is given by

       EPS * norm(ABNRM, BBNRM) / DIF(i).

  For further explanation of the reciprocal condition numbers RCONDE
  and RCONDV, see section 4.11 of LAPACK User's Guide.

Definition at line 387 of file dggevx.f.

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

Purpose:

 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).

Parameters

BALANC

          BALANC is CHARACTER*1
          Specifies the balance option to be performed.
          = 'N':  do not diagonally scale or permute;
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale.
          Computed reciprocal condition numbers will be for the
          matrices after permuting and/or balancing. Permuting does
          not change condition numbers (in exact arithmetic), but
          balancing does.

JOBVL

          JOBVL is CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors.

JOBVR

          JOBVR is CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors.

SENSE

          SENSE is CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': none are computed;
          = 'E': computed for eigenvalues only;
          = 'V': computed for eigenvectors only;
          = 'B': computed for eigenvalues and eigenvectors.

N

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

A

          A is 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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  Balancing a matrix pair (A,B) includes, first, permuting rows and
  columns to isolate eigenvalues, second, applying diagonal similarity
  transformation to the rows and columns to make the rows and columns
  as close in norm as possible. The computed reciprocal condition
  numbers correspond to the balanced matrix. Permuting rows and columns
  will not change the condition numbers (in exact arithmetic) but
  diagonal scaling will.  For further explanation of balancing, see
  section 4.11.1.2 of LAPACK Users' Guide.

  An approximate error bound on the chordal distance between the i-th
  computed generalized eigenvalue w and the corresponding exact
  eigenvalue lambda is

       chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

  An approximate error bound for the angle between the i-th computed
  eigenvector VL(i) or VR(i) is given by

       EPS * norm(ABNRM, BBNRM) / DIF(i).

  For further explanation of the reciprocal condition numbers RCONDE
  and RCONDV, see section 4.11 of LAPACK User's Guide.

Definition at line 387 of file sggevx.f.

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

Purpose:

 ZGGEVX computes for a pair of N-by-N complex nonsymmetric matrices
 (A,B) the generalized eigenvalues, and optionally, the left and/or
 right generalized eigenvectors.

 Optionally, it also computes a balancing transformation to improve
 the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
 LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
 the eigenvalues (RCONDE), and reciprocal condition numbers for the
 right eigenvectors (RCONDV).

 A generalized eigenvalue for a pair of matrices (A,B) is a scalar
 lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
 singular. It is usually represented as the pair (alpha,beta), as
 there is a reasonable interpretation for beta=0, and even for both
 being zero.

 The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
 of (A,B) satisfies
                  A * v(j) = lambda(j) * B * v(j) .
 The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
 of (A,B) satisfies
                  u(j)**H * A  = lambda(j) * u(j)**H * B.
 where u(j)**H is the conjugate-transpose of u(j).

Parameters

BALANC

          BALANC is CHARACTER*1
          Specifies the balance option to be performed:
          = 'N':  do not diagonally scale or permute;
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale.
          Computed reciprocal condition numbers will be for the
          matrices after permuting and/or balancing. Permuting does
          not change condition numbers (in exact arithmetic), but
          balancing does.

JOBVL

          JOBVL is CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors.

JOBVR

          JOBVR is CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors.

SENSE

          SENSE is CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': none are computed;
          = 'E': computed for eigenvalues only;
          = 'V': computed for eigenvectors only;
          = 'B': computed for eigenvalues and eigenvectors.

N

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

A

          A is COMPLEX*16 array, dimension (LDA, N)
          On entry, the matrix A in the pair (A,B).
          On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
          or both, then A contains the first part of the complex Schur
          form of the 'balanced' versions of the input A and B.

LDA

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

B

          B is COMPLEX*16 array, dimension (LDB, N)
          On entry, the matrix B in the pair (A,B).
          On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
          or both, then B contains the second part of the complex
          Schur form of the 'balanced' versions of the input A and B.

LDB

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

ALPHA

          ALPHA is COMPLEX*16 array, dimension (N)

BETA

          BETA is COMPLEX*16 array, dimension (N)
          On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized
          eigenvalues.

          Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or
          underflow, and BETA(j) may even be zero.  Thus, the user
          should avoid naively computing the ratio ALPHA/BETA.
          However, ALPHA will be always less than and usually
          comparable with norm(A) in magnitude, and BETA always less
          than and usually comparable with norm(B).

VL

          VL is COMPLEX*16 array, dimension (LDVL,N)
          If JOBVL = 'V', the left generalized eigenvectors u(j) are
          stored one after another in the columns of VL, in the same
          order as their eigenvalues.
          Each eigenvector will be scaled so the largest component
          will have abs(real part) + abs(imag. part) = 1.
          Not referenced if JOBVL = 'N'.

LDVL

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

VR

          VR is COMPLEX*16 array, dimension (LDVR,N)
          If JOBVR = 'V', the right generalized eigenvectors v(j) are
          stored one after another in the columns of VR, in the same
          order as their eigenvalues.
          Each eigenvector will be scaled so the largest component
          will have abs(real part) + abs(imag. part) = 1.
          Not referenced if JOBVR = 'N'.

LDVR

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

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
          ILO and IHI are integer values such that on exit
          A(i,j) = 0 and B(i,j) = 0 if i > j and
          j = 1,...,ILO-1 or i = IHI+1,...,N.
          If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

LSCALE

          LSCALE is DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          to the left side of A and B.  If PL(j) is the index of the
          row interchanged with row j, and DL(j) is the scaling
          factor applied to row j, then
            LSCALE(j) = PL(j)  for j = 1,...,ILO-1
                      = DL(j)  for j = ILO,...,IHI
                      = PL(j)  for j = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

RSCALE

          RSCALE is DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          to the right side of A and B.  If PR(j) is the index of the
          column interchanged with column j, and DR(j) is the scaling
          factor applied to column j, then
            RSCALE(j) = PR(j)  for j = 1,...,ILO-1
                      = DR(j)  for j = ILO,...,IHI
                      = PR(j)  for j = IHI+1,...,N
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

ABNRM

          ABNRM is DOUBLE PRECISION
          The one-norm of the balanced matrix A.

BBNRM

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

RCONDE

          RCONDE is DOUBLE PRECISION array, dimension (N)
          If SENSE = 'E' or 'B', the reciprocal condition numbers of
          the eigenvalues, stored in consecutive elements of the array.
          If SENSE = 'N' or 'V', RCONDE is not referenced.

RCONDV

          RCONDV is DOUBLE PRECISION array, dimension (N)
          If JOB = 'V' or 'B', the estimated reciprocal condition
          numbers of the eigenvectors, stored in consecutive elements
          of the array. If the eigenvalues cannot be reordered to
          compute RCONDV(j), RCONDV(j) is set to 0; this can only occur
          when the true value would be very small anyway.
          If SENSE = 'N' or 'E', RCONDV is not referenced.

WORK

          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,2*N).
          If SENSE = 'E', LWORK >= max(1,4*N).
          If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is DOUBLE PRECISION array, dimension (lrwork)
          lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B',
          and at least max(1,2*N) otherwise.
          Real workspace.

IWORK

          IWORK is INTEGER array, dimension (N+2)
          If SENSE = 'E', IWORK is not referenced.

BWORK

          BWORK is LOGICAL array, dimension (N)
          If SENSE = 'N', BWORK is not referenced.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  No eigenvectors have been
                calculated, but ALPHA(j) and BETA(j) should be correct
                for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in ZHGEQZ.
                =N+2: error return from ZTGEVC.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  Balancing a matrix pair (A,B) includes, first, permuting rows and
  columns to isolate eigenvalues, second, applying diagonal similarity
  transformation to the rows and columns to make the rows and columns
  as close in norm as possible. The computed reciprocal condition
  numbers correspond to the balanced matrix. Permuting rows and columns
  will not change the condition numbers (in exact arithmetic) but
  diagonal scaling will.  For further explanation of balancing, see
  section 4.11.1.2 of LAPACK Users' Guide.

  An approximate error bound on the chordal distance between the i-th
  computed generalized eigenvalue w and the corresponding exact
  eigenvalue lambda is

       chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

  An approximate error bound for the angle between the i-th computed
  eigenvector VL(i) or VR(i) is given by

       EPS * norm(ABNRM, BBNRM) / DIF(i).

  For further explanation of the reciprocal condition numbers RCONDE
  and RCONDV, see section 4.11 of LAPACK User's Guide.

Definition at line 370 of file zggevx.f.