ZGGESX(3) LAPACK routine of NEC Numeric Library Collection ZGGESX(3)
NAME
ZGGESX
SYNOPSIS
SUBROUTINE ZGGESX (JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B,
LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV,
WORK, LWORK, RWORK, IWORK, LIWORK, BWORK, INFO)
PURPOSE
ZGGESX computes for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, the complex Schur form (S,T),
and, optionally, the left and/or right matrices of Schur vectors (VSL
and VSR). This gives the generalized Schur factorization
(A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H )
where (VSR)**H is the conjugate-transpose of VSR.
Optionally, it also orders the eigenvalues so that a selected cluster
of eigenvalues appears in the leading diagonal blocks of the upper
triangular matrix S and the upper triangular matrix T; computes
a reciprocal condition number for the average of the selected
eigenvalues (RCONDE); and computes a reciprocal condition number for
the right and left deflating subspaces corresponding to the selected
eigenvalues (RCONDV). The leading columns of VSL and VSR then form
an orthonormal basis for the corresponding left and right eigenspaces
(deflating subspaces).
A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
or a ratio alpha/beta = w, such that A - w*B is singular. It is
usually represented as the pair (alpha,beta), as there is a
reasonable interpretation for beta=0 or for both being zero.
A pair of matrices (S,T) is in generalized complex Schur form if T is
upper triangular with non-negative diagonal and S is upper
triangular.
ARGUMENTS
JOBVSL (input)
JOBVSL is CHARACTER*1
= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.
JOBVSR (input)
JOBVSR is CHARACTER*1
= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.
SORT (input)
SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the generalized Schur form.
= 'N': Eigenvalues are not ordered;
= 'S': Eigenvalues are ordered (see SELCTG).
SELCTG (input)
SELCTG is procedure) LOGICAL FUNCTION of two COMPLEX*16 arguments
SELCTG must be declared EXTERNAL in the calling subroutine.
If SORT = 'N', SELCTG is not referenced.
If SORT = 'S', SELCTG is used to select eigenvalues to sort
to the top left of the Schur form.
Note that a selected complex eigenvalue may no longer satisfy
SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
ordering may change the value of complex eigenvalues
(especially if the eigenvalue is ill-conditioned), in this
case INFO is set to N+3 see INFO below).
SENSE (input)
SENSE is CHARACTER*1
Determines which reciprocal condition numbers are computed.
= 'N' : None are computed;
= 'E' : Computed for average of selected eigenvalues only;
= 'V' : Computed for selected deflating subspaces only;
= 'B' : Computed for both.
If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
N (input)
N is INTEGER
The order of the matrices A, B, VSL, and VSR. N >= 0.
A (input/output)
A is COMPLEX*16 array, dimension (LDA, N)
On entry, the first of the pair of matrices.
On exit, A has been overwritten by its generalized Schur
form S.
LDA (input)
LDA is INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output)
B is COMPLEX*16 array, dimension (LDB, N)
On entry, the second of the pair of matrices.
On exit, B has been overwritten by its generalized Schur
form T.
LDB (input)
LDB is INTEGER
The leading dimension of B. LDB >= max(1,N).
SDIM (output)
SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELCTG is true.
ALPHA (output)
ALPHA is COMPLEX*16 array, dimension (N)
BETA (output)
BETA is COMPLEX*16 array, dimension (N)
On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
generalized eigenvalues. ALPHA(j) and BETA(j),j=1,...,N are
the diagonals of the complex Schur form (S,T). BETA(j) will
be non-negative real.
Note: the quotients 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).
VSL (output)
VSL is COMPLEX*16 array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors.
Not referenced if JOBVSL = 'N'.
LDVSL (input)
LDVSL is INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and
if JOBVSL = 'V', LDVSL >= N.
VSR (output)
VSR is COMPLEX*16 array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors.
Not referenced if JOBVSR = 'N'.
LDVSR (input)
LDVSR is INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and
if JOBVSR = 'V', LDVSR >= N.
RCONDE (output)
RCONDE is DOUBLE PRECISION array, dimension ( 2 )
If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
reciprocal condition numbers for the average of the selected
eigenvalues.
Not referenced if SENSE = 'N' or 'V'.
RCONDV (output)
RCONDV is DOUBLE PRECISION array, dimension ( 2 )
If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
reciprocal condition number for the selected deflating
subspaces.
Not referenced if SENSE = 'N' or 'E'.
WORK (output)
WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
LWORK is INTEGER
The dimension of the array WORK.
If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else
LWORK >= MAX(1,2*N). Note that 2*SDIM*(N-SDIM) <= N*N/2.
Note also that an error is only returned if
LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may
not be large enough.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the bound on the optimal size of the WORK
array and the minimum size of the IWORK array, returns these
values as the first entries of the WORK and IWORK arrays, and
no error message related to LWORK or LIWORK is issued by
XERBLA.
RWORK (output)
RWORK is DOUBLE PRECISION array, dimension ( 8*N )
Real workspace.
IWORK (output)
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
LIWORK (input)
LIWORK is INTEGER
The dimension of the array IWORK.
If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
LIWORK >= N+2.
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the bound on the optimal size of the
WORK array and the minimum size of the IWORK array, returns
these values as the first entries of the WORK and IWORK
arrays, and no error message related to LWORK or LIWORK is
issued by XERBLA.
BWORK (output)
BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
INFO (output)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1,...,N:
The QZ iteration failed. (A,B) are not in Schur
form, 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: after reordering, roundoff changed values of
some complex eigenvalues so that leading
eigenvalues in the Generalized Schur form no
longer satisfy SELCTG=.TRUE. This could also
be caused due to scaling.
=N+3: reordering failed in ZTGSEN.
LAPACK routine 31 October 2017 ZGGESX(3)