ggevx(3)

NAME
ggevx - ggevx: eig, expert

SYNOPSIS

  Functions
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

Detailed Description

Function Documentation

  subroutine cggevx (character balanc, character jobvl, character jobvr, character sense, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, integer ilo, integer ihi, real, dimension( * ) lscale, real, dimension( * ) rscale, real abnrm, real bbnrm, real, dimension( * ) rconde, real, dimension( * ) rcondv, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, logical, dimension( * ) bwork, integer info)

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.

  subroutine dggevx (character balanc, character jobvl, character jobvr, character sense, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integer ilo, integer ihi, double precision, dimension( * ) lscale, double precision, dimension( * ) rscale, double precision abnrm, double precision bbnrm, double precision, dimension( * ) rconde, double precision, dimension( * ) rcondv, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, logical, dimension( * ) bwork, integer info)

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.

  subroutine sggevx (character balanc, character jobvl, character jobvr, character sense, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, integer ilo, integer ihi, real, dimension( * ) lscale, real, dimension( * ) rscale, real abnrm, real bbnrm, real, dimension( * ) rconde, real, dimension( * ) rcondv, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, logical, dimension( * ) bwork, integer info)

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.

  subroutine zggevx (character balanc, character jobvl, character jobvr, character sense, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, integer ilo, integer ihi, double precision, dimension( * ) lscale, double precision, dimension( * ) rscale, double precision abnrm, double precision bbnrm, double precision, dimension( * ) rconde, double precision, dimension( * ) rcondv, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, logical, dimension( * ) bwork, integer info)

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.

Author

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