LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ dggevx()

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

Download DGGEVX + dependencies [TGZ] [ZIP] [TXT]

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
[in]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.
!>
[in]JOBVL
!>          JOBVL is CHARACTER*1
!>          = 'N':  do not compute the left generalized eigenvectors;
!>          = 'V':  compute the left generalized eigenvectors.
!>
[in]JOBVR
!>          JOBVR is CHARACTER*1
!>          = 'N':  do not compute the right generalized eigenvectors;
!>          = 'V':  compute the right generalized eigenvectors.
!>
[in]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.
!>
[in]N
!>          N is INTEGER
!>          The order of the matrices A, B, VL, and VR.  N >= 0.
!>
[in,out]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  versions of the input A and B.
!>
[in]LDA
!>          LDA is INTEGER
!>          The leading dimension of A.  LDA >= max(1,N).
!>
[in,out]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  versions of the input A and B.
!>
[in]LDB
!>          LDB is INTEGER
!>          The leading dimension of B.  LDB >= max(1,N).
!>
[out]ALPHAR
!>          ALPHAR is DOUBLE PRECISION array, dimension (N)
!>
[out]ALPHAI
!>          ALPHAI is DOUBLE PRECISION array, dimension (N)
!>
[out]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).
!>
[out]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'.
!>
[in]LDVL
!>          LDVL is INTEGER
!>          The leading dimension of the matrix VL. LDVL >= 1, and
!>          if JOBVL = 'V', LDVL >= N.
!>
[out]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'.
!>
[in]LDVR
!>          LDVR is INTEGER
!>          The leading dimension of the matrix VR. LDVR >= 1, and
!>          if JOBVR = 'V', LDVR >= N.
!>
[out]ILO
!>          ILO is INTEGER
!>
[out]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.
!>
[out]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.
!>
[out]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.
!>
[out]ABNRM
!>          ABNRM is DOUBLE PRECISION
!>          The one-norm of the balanced matrix A.
!>
[out]BBNRM
!>          BBNRM is DOUBLE PRECISION
!>          The one-norm of the balanced matrix B.
!>
[out]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.
!>
[out]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.
!>
[out]WORK
!>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>
[in]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.
!>
[out]IWORK
!>          IWORK is INTEGER array, dimension (N+6)
!>          If SENSE = 'E', IWORK is not referenced.
!>
[out]BWORK
!>          BWORK is LOGICAL array, dimension (N)
!>          If SENSE = 'N', BWORK is not referenced.
!>
[out]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.
!>