table of contents
| ggesx(3) | LAPACK | ggesx(3) |
NAME¶
ggesx - ggesx: Schur form, expert
SYNOPSIS¶
Functions¶
subroutine cggesx (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)
CGGESX computes the eigenvalues, the Schur form, and, optionally, the
matrix of Schur vectors for GE matrices subroutine dggesx
(jobvsl, jobvsr, sort, selctg, sense, n, a, lda, b, ldb, sdim, alphar,
alphai, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, work, lwork, iwork,
liwork, bwork, info)
DGGESX computes the eigenvalues, the Schur form, and, optionally, the
matrix of Schur vectors for GE matrices subroutine sggesx
(jobvsl, jobvsr, sort, selctg, sense, n, a, lda, b, ldb, sdim, alphar,
alphai, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, work, lwork, iwork,
liwork, bwork, info)
SGGESX computes the eigenvalues, the Schur form, and, optionally, the
matrix of Schur vectors for GE matrices 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)
ZGGESX computes the eigenvalues, the Schur form, and, optionally, the
matrix of Schur vectors for GE matrices
Detailed Description¶
Function Documentation¶
subroutine cggesx (character jobvsl, character jobvsr, character sort, external selctg, character sense, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, integer sdim, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldvsl, * ) vsl, integer ldvsl, complex, dimension( ldvsr, * ) vsr, integer ldvsr, real, dimension( 2 ) rconde, real, dimension( 2 ) rcondv, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)¶
CGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
Purpose:
CGGESX 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.
Parameters
JOBVSL
JOBVSL is CHARACTER*1
= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.
JOBVSR
JOBVSR is CHARACTER*1
= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.
SORT
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
SELCTG is a LOGICAL FUNCTION of two COMPLEX 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
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
N is INTEGER
The order of the matrices A, B, VSL, and VSR. N >= 0.
A
A is COMPLEX 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
LDA is INTEGER
The leading dimension of A. LDA >= max(1,N).
B
B is COMPLEX 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
LDB is INTEGER
The leading dimension of B. LDB >= max(1,N).
SDIM
SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELCTG is true.
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. 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
VSL is COMPLEX array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors.
Not referenced if JOBVSL = 'N'.
LDVSL
LDVSL is INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and
if JOBVSL = 'V', LDVSL >= N.
VSR
VSR is COMPLEX array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors.
Not referenced if JOBVSR = 'N'.
LDVSR
LDVSR is INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and
if JOBVSR = 'V', LDVSR >= N.
RCONDE
RCONDE is REAL 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
RCONDV is REAL 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
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.
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
RWORK is REAL array, dimension ( 8*N )
Real workspace.
IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array WORK.
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
BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
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. (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 CHGEQZ
=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 CTGSEN.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dggesx (character jobvsl, character jobvsr, character sort, external selctg, character sense, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, integer sdim, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldvsl, * ) vsl, integer ldvsl, double precision, dimension( ldvsr, * ) vsr, integer ldvsr, double precision, dimension( 2 ) rconde, double precision, dimension( 2 ) rcondv, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)¶
DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
Purpose:
DGGESX computes for a pair of N-by-N real nonsymmetric matrices
(A,B), the generalized eigenvalues, the real 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)**T, (VSL) T (VSR)**T )
Optionally, it also orders the eigenvalues so that a selected cluster
of eigenvalues appears in the leading diagonal blocks of the upper
quasi-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 real Schur form if T is
upper triangular with non-negative diagonal and S is block upper
triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
to real generalized eigenvalues, while 2-by-2 blocks of S will be
'standardized' by making the corresponding elements of T have the
form:
[ a 0 ]
[ 0 b ]
and the pair of corresponding 2-by-2 blocks in S and T will have a
complex conjugate pair of generalized eigenvalues.
Parameters
JOBVSL
JOBVSL is CHARACTER*1
= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.
JOBVSR
JOBVSR is CHARACTER*1
= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.
SORT
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
SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION 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.
An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
one of a complex conjugate pair of eigenvalues is selected,
then both complex eigenvalues are selected.
Note that a selected complex eigenvalue may no longer satisfy
SELCTG(ALPHAR(j),ALPHAI(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.
SENSE
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
N is INTEGER
The order of the matrices A, B, VSL, and VSR. N >= 0.
A
A is DOUBLE PRECISION 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
LDA is INTEGER
The leading dimension of A. LDA >= max(1,N).
B
B is DOUBLE PRECISION 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
LDB is INTEGER
The leading dimension of B. LDB >= max(1,N).
SDIM
SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELCTG is true. (Complex conjugate pairs for which
SELCTG is true for either eigenvalue count as 2.)
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. ALPHAR(j) + ALPHAI(j)*i
and BETA(j),j=1,...,N are the diagonals of the complex Schur
form (S,T) that would result if the 2-by-2 diagonal blocks of
the real Schur form of (A,B) were further reduced to
triangular form using 2-by-2 complex unitary transformations.
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.
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).
VSL
VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors.
Not referenced if JOBVSL = 'N'.
LDVSL
LDVSL is INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and
if JOBVSL = 'V', LDVSL >= N.
VSR
VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors.
Not referenced if JOBVSR = 'N'.
LDVSR
LDVSR is INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and
if JOBVSR = 'V', LDVSR >= N.
RCONDE
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
RCONDV is DOUBLE PRECISION array, dimension ( 2 )
If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
reciprocal condition numbers for the selected deflating
subspaces.
Not referenced if SENSE = 'N' or 'E'.
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.
If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
LWORK >= max( 8*N, 6*N+16 ).
Note that 2*SDIM*(N-SDIM) <= N*N/2.
Note also that an error is only returned if
LWORK < max( 8*N, 6*N+16), 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.
IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK.
If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
LIWORK >= N+6.
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
BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
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. (A,B) are not in Schur
form, 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: 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 DTGSEN.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
An approximate (asymptotic) bound on the average absolute error of
the selected eigenvalues is
EPS * norm((A, B)) / RCONDE( 1 ).
An approximate (asymptotic) bound on the maximum angular error in
the computed deflating subspaces is
EPS * norm((A, B)) / RCONDV( 2 ).
See LAPACK User's Guide, section 4.11 for more information.
subroutine sggesx (character jobvsl, character jobvsr, character sort, external selctg, character sense, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, integer sdim, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( ldvsl, * ) vsl, integer ldvsl, real, dimension( ldvsr, * ) vsr, integer ldvsr, real, dimension( 2 ) rconde, real, dimension( 2 ) rcondv, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)¶
SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
Purpose:
SGGESX computes for a pair of N-by-N real nonsymmetric matrices
(A,B), the generalized eigenvalues, the real 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)**T, (VSL) T (VSR)**T )
Optionally, it also orders the eigenvalues so that a selected cluster
of eigenvalues appears in the leading diagonal blocks of the upper
quasi-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 real Schur form if T is
upper triangular with non-negative diagonal and S is block upper
triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
to real generalized eigenvalues, while 2-by-2 blocks of S will be
'standardized' by making the corresponding elements of T have the
form:
[ a 0 ]
[ 0 b ]
and the pair of corresponding 2-by-2 blocks in S and T will have a
complex conjugate pair of generalized eigenvalues.
Parameters
JOBVSL
JOBVSL is CHARACTER*1
= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.
JOBVSR
JOBVSR is CHARACTER*1
= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.
SORT
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
SELCTG is a LOGICAL FUNCTION of three REAL 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.
An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
one of a complex conjugate pair of eigenvalues is selected,
then both complex eigenvalues are selected.
Note that a selected complex eigenvalue may no longer satisfy
SELCTG(ALPHAR(j),ALPHAI(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.
SENSE
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
N is INTEGER
The order of the matrices A, B, VSL, and VSR. N >= 0.
A
A is REAL 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
LDA is INTEGER
The leading dimension of A. LDA >= max(1,N).
B
B is REAL 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
LDB is INTEGER
The leading dimension of B. LDB >= max(1,N).
SDIM
SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELCTG is true. (Complex conjugate pairs for which
SELCTG is true for either eigenvalue count as 2.)
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. ALPHAR(j) + ALPHAI(j)*i
and BETA(j),j=1,...,N are the diagonals of the complex Schur
form (S,T) that would result if the 2-by-2 diagonal blocks of
the real Schur form of (A,B) were further reduced to
triangular form using 2-by-2 complex unitary transformations.
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.
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).
VSL
VSL is REAL array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors.
Not referenced if JOBVSL = 'N'.
LDVSL
LDVSL is INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and
if JOBVSL = 'V', LDVSL >= N.
VSR
VSR is REAL array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors.
Not referenced if JOBVSR = 'N'.
LDVSR
LDVSR is INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and
if JOBVSR = 'V', LDVSR >= N.
RCONDE
RCONDE is REAL 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
RCONDV is REAL array, dimension ( 2 )
If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
reciprocal condition numbers for the selected deflating
subspaces.
Not referenced if SENSE = 'N' or 'E'.
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.
If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
LWORK >= max( 8*N, 6*N+16 ).
Note that 2*SDIM*(N-SDIM) <= N*N/2.
Note also that an error is only returned if
LWORK < max( 8*N, 6*N+16), 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.
IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK.
If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
LIWORK >= N+6.
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
BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
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. (A,B) are not in Schur
form, 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: 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 STGSEN.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
An approximate (asymptotic) bound on the average absolute error of
the selected eigenvalues is
EPS * norm((A, B)) / RCONDE( 1 ).
An approximate (asymptotic) bound on the maximum angular error in
the computed deflating subspaces is
EPS * norm((A, B)) / RCONDV( 2 ).
See LAPACK User's Guide, section 4.11 for more information.
subroutine zggesx (character jobvsl, character jobvsr, character sort, external selctg, character sense, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, integer sdim, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldvsl, * ) vsl, integer ldvsl, complex*16, dimension( ldvsr, * ) vsr, integer ldvsr, double precision, dimension( 2 ) rconde, double precision, dimension( 2 ) rcondv, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)¶
ZGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
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.
Parameters
JOBVSL
JOBVSL is CHARACTER*1
= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.
JOBVSR
JOBVSR is CHARACTER*1
= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.
SORT
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
SELCTG is a 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
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
N is INTEGER
The order of the matrices A, B, VSL, and VSR. N >= 0.
A
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
LDA is INTEGER
The leading dimension of A. LDA >= max(1,N).
B
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
LDB is INTEGER
The leading dimension of B. LDB >= max(1,N).
SDIM
SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELCTG is true.
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. 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
VSL is COMPLEX*16 array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors.
Not referenced if JOBVSL = 'N'.
LDVSL
LDVSL is INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and
if JOBVSL = 'V', LDVSL >= N.
VSR
VSR is COMPLEX*16 array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors.
Not referenced if JOBVSR = 'N'.
LDVSR
LDVSR is INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and
if JOBVSR = 'V', LDVSR >= N.
RCONDE
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
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
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.
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
RWORK is DOUBLE PRECISION array, dimension ( 8*N )
Real workspace.
IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
LIWORK
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
BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
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. (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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Author¶
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