| complex16GEeigen(3) | LAPACK | complex16GEeigen(3) |
complex16GEeigen - complex16
subroutine zgees (JOBVS, SORT, SELECT, N, A, LDA, SDIM, W,
VS, LDVS, WORK, LWORK, RWORK, BWORK, INFO)
ZGEES computes the eigenvalues, the Schur form, and, optionally, the
matrix of Schur vectors for GE matrices subroutine zgeesx (JOBVS,
SORT, SELECT, SENSE, N, A, LDA, SDIM, W, VS, LDVS, RCONDE, RCONDV, WORK,
LWORK, RWORK, BWORK, INFO)
ZGEESX computes the eigenvalues, the Schur form, and, optionally, the
matrix of Schur vectors for GE matrices subroutine zgeev (JOBVL,
JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)
ZGEEV computes the eigenvalues and, optionally, the left and/or right
eigenvectors for GE matrices subroutine zgeevx (BALANC, JOBVL,
JOBVR, SENSE, N, A, LDA, W, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
RCONDE, RCONDV, WORK, LWORK, RWORK, INFO)
ZGEEVX computes the eigenvalues and, optionally, the left and/or right
eigenvectors for GE matrices subroutine zgges (JOBVSL, JOBVSR,
SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR,
WORK, LWORK, RWORK, BWORK, INFO)
ZGGES computes the eigenvalues, the Schur form, and, optionally, the
matrix of Schur vectors for GE matrices subroutine zgges3
(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL,
LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO)
ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the
matrix of Schur vectors for GE matrices (blocked algorithm) 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 subroutine zggev (JOBVL,
JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, LWORK,
RWORK, INFO)
ZGGEV computes the eigenvalues and, optionally, the left and/or right
eigenvectors for GE matrices subroutine zggev3 (JOBVL, JOBVR, N,
A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)
ZGGEV3 computes the eigenvalues and, optionally, the left and/or right
eigenvectors for GE matrices (blocked algorithm) 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
This is the group of complex16 eigenvalue driver functions for GE matrices
ZGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
Purpose:
ZGEES computes for an N-by-N complex nonsymmetric matrix A, the
eigenvalues, the Schur form T, and, optionally, the matrix of Schur
vectors Z. This gives the Schur factorization A = Z*T*(Z**H).
Optionally, it also orders the eigenvalues on the diagonal of the
Schur form so that selected eigenvalues are at the top left.
The leading columns of Z then form an orthonormal basis for the
invariant subspace corresponding to the selected eigenvalues.
A complex matrix is in Schur form if it is upper triangular.
Parameters
JOBVS is CHARACTER*1
= 'N': Schur vectors are not computed;
= 'V': Schur vectors are computed.
SORT
SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the Schur form.
= 'N': Eigenvalues are not ordered:
= 'S': Eigenvalues are ordered (see SELECT).
SELECT
SELECT is a LOGICAL FUNCTION of one COMPLEX*16 argument
SELECT must be declared EXTERNAL in the calling subroutine.
If SORT = 'S', SELECT is used to select eigenvalues to order
to the top left of the Schur form.
IF SORT = 'N', SELECT is not referenced.
The eigenvalue W(j) is selected if SELECT(W(j)) is true.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten by its Schur form T.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
SDIM
SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues for which
SELECT is true.
W
W is COMPLEX*16 array, dimension (N)
W contains the computed eigenvalues, in the same order that
they appear on the diagonal of the output Schur form T.
VS
VS is COMPLEX*16 array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the unitary matrix Z of Schur
vectors.
If JOBVS = 'N', VS is not referenced.
LDVS
LDVS is INTEGER
The leading dimension of the array VS. LDVS >= 1; if
JOBVS = 'V', LDVS >= N.
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).
For good performance, LWORK must generally be larger.
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 (N)
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.
> 0: if INFO = i, and i is
<= N: the QR algorithm failed to compute all the
eigenvalues; elements 1:ILO-1 and i+1:N of W
contain those eigenvalues which have converged;
if JOBVS = 'V', VS contains the matrix which
reduces A to its partially converged Schur form.
= N+1: the eigenvalues could not be reordered because
some eigenvalues were too close to separate (the
problem is very ill-conditioned);
= N+2: after reordering, roundoff changed values of
some complex eigenvalues so that leading
eigenvalues in the Schur form no longer satisfy
SELECT = .TRUE.. This could also be caused by
underflow due to scaling.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
ZGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
Purpose:
ZGEESX computes for an N-by-N complex nonsymmetric matrix A, the
eigenvalues, the Schur form T, and, optionally, the matrix of Schur
vectors Z. This gives the Schur factorization A = Z*T*(Z**H).
Optionally, it also orders the eigenvalues on the diagonal of the
Schur form so that selected eigenvalues are at the top left;
computes a reciprocal condition number for the average of the
selected eigenvalues (RCONDE); and computes a reciprocal condition
number for the right invariant subspace corresponding to the
selected eigenvalues (RCONDV). The leading columns of Z form an
orthonormal basis for this invariant subspace.
For further explanation of the reciprocal condition numbers RCONDE
and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
these quantities are called s and sep respectively).
A complex matrix is in Schur form if it is upper triangular.
Parameters
JOBVS is CHARACTER*1
= 'N': Schur vectors are not computed;
= 'V': Schur vectors are computed.
SORT
SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the Schur form.
= 'N': Eigenvalues are not ordered;
= 'S': Eigenvalues are ordered (see SELECT).
SELECT
SELECT is a LOGICAL FUNCTION of one COMPLEX*16 argument
SELECT must be declared EXTERNAL in the calling subroutine.
If SORT = 'S', SELECT is used to select eigenvalues to order
to the top left of the Schur form.
If SORT = 'N', SELECT is not referenced.
An eigenvalue W(j) is selected if SELECT(W(j)) is true.
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 right invariant subspace only;
= 'B': Computed for both.
If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA, N)
On entry, the N-by-N matrix A.
On exit, A is overwritten by its Schur form T.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
SDIM
SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues for which
SELECT is true.
W
W is COMPLEX*16 array, dimension (N)
W contains the computed eigenvalues, in the same order
that they appear on the diagonal of the output Schur form T.
VS
VS is COMPLEX*16 array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the unitary matrix Z of Schur
vectors.
If JOBVS = 'N', VS is not referenced.
LDVS
LDVS is INTEGER
The leading dimension of the array VS. LDVS >= 1, and if
JOBVS = 'V', LDVS >= N.
RCONDE
RCONDE is DOUBLE PRECISION
If SENSE = 'E' or 'B', RCONDE contains the reciprocal
condition number for the average of the selected eigenvalues.
Not referenced if SENSE = 'N' or 'V'.
RCONDV
RCONDV is DOUBLE PRECISION
If SENSE = 'V' or 'B', RCONDV contains the reciprocal
condition number for the selected right invariant subspace.
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. LWORK >= max(1,2*N).
Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
where SDIM is the number of selected eigenvalues computed by
this routine. 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.
For good performance, LWORK must generally be larger.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates upper bound on the optimal size of the
array WORK, 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 (N)
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.
> 0: if INFO = i, and i is
<= N: the QR algorithm failed to compute all the
eigenvalues; elements 1:ILO-1 and i+1:N of W
contain those eigenvalues which have converged; if
JOBVS = 'V', VS contains the transformation which
reduces A to its partially converged Schur form.
= N+1: the eigenvalues could not be reordered because some
eigenvalues were too close to separate (the problem
is very ill-conditioned);
= N+2: after reordering, roundoff changed values of some
complex eigenvalues so that leading eigenvalues in
the Schur form no longer satisfy SELECT=.TRUE. This
could also be caused by underflow due to scaling.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
ZGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
Purpose:
ZGEEV computes for an N-by-N complex nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.
The right eigenvector v(j) of A satisfies
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
u(j)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate transpose of u(j).
The computed eigenvectors are normalized to have Euclidean norm
equal to 1 and largest component real.
Parameters
JOBVL is CHARACTER*1
= 'N': left eigenvectors of A are not computed;
= 'V': left eigenvectors of are computed.
JOBVR
JOBVR is CHARACTER*1
= 'N': right eigenvectors of A are not computed;
= 'V': right eigenvectors of A are computed.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
W
W is COMPLEX*16 array, dimension (N)
W contains the computed eigenvalues.
VL
VL is COMPLEX*16 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 JOBVL = 'N', VL is not referenced.
u(j) = VL(:,j), the j-th column of VL.
LDVL
LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1; if
JOBVL = 'V', LDVL >= N.
VR
VR is COMPLEX*16 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 JOBVR = 'N', VR is not referenced.
v(j) = VR(:,j), the j-th column of VR.
LDVR
LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1; if
JOBVR = 'V', LDVR >= N.
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).
For good performance, LWORK must generally be larger.
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 (2*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors have been computed;
elements i+1:N of W contain eigenvalues which have
converged.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
Purpose:
ZGEEVX computes for an N-by-N complex nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.
Optionally also, it computes a balancing transformation to improve
the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
(RCONDE), and reciprocal condition numbers for the right
eigenvectors (RCONDV).
The right eigenvector v(j) of A satisfies
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
u(j)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate transpose of u(j).
The computed eigenvectors are normalized to have Euclidean norm
equal to 1 and largest component real.
Balancing a matrix means permuting the rows and columns to make it
more nearly upper triangular, and applying a diagonal similarity
transformation D * A * D**(-1), where D is a diagonal matrix, to
make its rows and columns closer in norm and the condition numbers
of its eigenvalues and eigenvectors smaller. 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 4.10.2 of the LAPACK
Users' Guide.
Parameters
BALANC is CHARACTER*1
Indicates how the input matrix should be diagonally scaled
and/or permuted to improve the conditioning of its
eigenvalues.
= 'N': Do not diagonally scale or permute;
= 'P': Perform permutations to make the matrix more nearly
upper triangular. Do not diagonally scale;
= 'S': Diagonally scale the matrix, ie. replace A by
D*A*D**(-1), where D is a diagonal matrix chosen
to make the rows and columns of A more equal in
norm. Do not permute;
= 'B': Both diagonally scale and permute A.
Computed reciprocal condition numbers will be for the matrix
after balancing and/or permuting. Permuting does not change
condition numbers (in exact arithmetic), but balancing does.
JOBVL
JOBVL is CHARACTER*1
= 'N': left eigenvectors of A are not computed;
= 'V': left eigenvectors of A are computed.
If SENSE = 'E' or 'B', JOBVL must = 'V'.
JOBVR
JOBVR is CHARACTER*1
= 'N': right eigenvectors of A are not computed;
= 'V': right eigenvectors of A are computed.
If SENSE = 'E' or 'B', JOBVR must = 'V'.
SENSE
SENSE is CHARACTER*1
Determines which reciprocal condition numbers are computed.
= 'N': None are computed;
= 'E': Computed for eigenvalues only;
= 'V': Computed for right eigenvectors only;
= 'B': Computed for eigenvalues and right eigenvectors.
If SENSE = 'E' or 'B', both left and right eigenvectors
must also be computed (JOBVL = 'V' and JOBVR = 'V').
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten. If JOBVL = 'V' or
JOBVR = 'V', A contains the Schur form of the balanced
version of the matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
W
W is COMPLEX*16 array, dimension (N)
W contains the computed eigenvalues.
VL
VL is COMPLEX*16 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 JOBVL = 'N', VL is not referenced.
u(j) = VL(:,j), the j-th column of VL.
LDVL
LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1; if
JOBVL = 'V', LDVL >= N.
VR
VR is COMPLEX*16 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 JOBVR = 'N', VR is not referenced.
v(j) = VR(:,j), the j-th column of VR.
LDVR
LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1; if
JOBVR = 'V', LDVR >= N.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
ILO and IHI are integer values determined when A was
balanced. The balanced A(i,j) = 0 if I > J and
J = 1,...,ILO-1 or I = IHI+1,...,N.
SCALE
SCALE is DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied
when balancing A. If P(j) is the index of the row and column
interchanged with row and column j, and D(j) is the scaling
factor applied to row and column j, then
SCALE(J) = P(J), for J = 1,...,ILO-1
= D(J), for J = ILO,...,IHI
= P(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 (the maximum
of the sum of absolute values of elements of any column).
RCONDE
RCONDE is DOUBLE PRECISION array, dimension (N)
RCONDE(j) is the reciprocal condition number of the j-th
eigenvalue.
RCONDV
RCONDV is DOUBLE PRECISION array, dimension (N)
RCONDV(j) is the reciprocal condition number of the j-th
right eigenvector.
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 SENSE = 'N' or 'E',
LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',
LWORK >= N*N+2*N.
For good performance, LWORK must generally be larger.
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 (2*N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors or condition numbers
have been computed; elements 1:ILO-1 and i+1:N of W
contain eigenvalues which have converged.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
ZGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
Purpose:
ZGGES computes for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, the generalized complex Schur
form (S, T), and optionally left and/or right 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. The leading
columns of VSL and VSR then form an unitary basis for the
corresponding left and right eigenspaces (deflating subspaces).
(If only the generalized eigenvalues are needed, use the driver
ZGGEV instead, which is faster.)
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, and even for both being zero.
A pair of matrices (S,T) is in generalized complex Schur form if S
and T are upper triangular and, in addition, the diagonal elements
of T are non-negative real numbers.
Parameters
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.
An eigenvalue ALPHA(j)/BETA(j) is selected if
SELCTG(ALPHA(j),BETA(j)) is true.
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+2 (See INFO below).
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), j=1,...,N and BETA(j),
j=1,...,N are the diagonals of the complex Schur form (A,B)
output by ZGGES. The 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.
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).
For good performance, LWORK must generally be larger.
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 (8*N)
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 California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)
Purpose:
ZGGES3 computes for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, the generalized complex Schur
form (S, T), and optionally left and/or right 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. The leading
columns of VSL and VSR then form an unitary basis for the
corresponding left and right eigenspaces (deflating subspaces).
(If only the generalized eigenvalues are needed, use the driver
ZGGEV instead, which is faster.)
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, and even for both being zero.
A pair of matrices (S,T) is in generalized complex Schur form if S
and T are upper triangular and, in addition, the diagonal elements
of T are non-negative real numbers.
Parameters
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.
An eigenvalue ALPHA(j)/BETA(j) is selected if
SELCTG(ALPHA(j),BETA(j)) is true.
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+2 (See INFO below).
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), j=1,...,N and BETA(j),
j=1,...,N are the diagonals of the complex Schur form (A,B)
output by ZGGES3. The 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.
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 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 (8*N)
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 California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
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 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 California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
ZGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
Purpose:
ZGGEV 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.
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 generalized eigenvector v(j) corresponding to the
generalized eigenvalue lambda(j) of (A,B) satisfies
A * v(j) = lambda(j) * B * v(j).
The left generalized eigenvector u(j) corresponding to the
generalized eigenvalues 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
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.
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.
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.
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 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).
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 is scaled so the largest component has
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 is scaled so the largest component has
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.
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).
For good performance, LWORK must generally be larger.
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 (8*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. No eigenvectors have been
calculated, but ALPHA(j) and BETA(j) should be
correct for j=INFO+1,...,N.
> N: =N+1: other then QZ iteration failed in DHGEQZ,
=N+2: error return from DTGEVC.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
ZGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)
Purpose:
ZGGEV3 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.
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 generalized eigenvector v(j) corresponding to the
generalized eigenvalue lambda(j) of (A,B) satisfies
A * v(j) = lambda(j) * B * v(j).
The left generalized eigenvector u(j) corresponding to the
generalized eigenvalues 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
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.
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.
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.
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 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).
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 is scaled so the largest component has
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 is scaled so the largest component has
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.
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 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 (8*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. No eigenvectors have been
calculated, but ALPHA(j) and BETA(j) should be
correct for j=INFO+1,...,N.
> N: =N+1: other then QZ iteration failed in DHGEQZ,
=N+2: error return from DTGEVC.
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
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 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 California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
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.
Generated automatically by Doxygen for LAPACK from the source code.
| Fri Jun 23 2023 | Version 3.9.0 |