CLAQR2
Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..
This subroutine is identical to CLAQR3 except that it avoids
recursion by calling CLAHQR instead of CLAQR4.
******************************************************************
Aggressive early deflation:
This subroutine accepts as input an upper Hessenberg matrix
H and performs an unitary similarity transformation
designed to detect and deflate fully converged eigenvalues from
a trailing principal submatrix. On output H has been over-
written by a new Hessenberg matrix that is a perturbation of
an unitary similarity transformation of H. It is to be
hoped that the final version of H has many zero subdiagonal
entries.
******************************************************************
WANTT (input) LOGICAL
If .TRUE., then the Hessenberg matrix H is fully updated
so that the triangular Schur factor may be
computed (in cooperation with the calling subroutine).
If .FALSE., then only enough of H is updated to preserve
the eigenvalues.
WANTZ (input) LOGICAL
If .TRUE., then the unitary matrix Z is updated so
so that the unitary Schur factor may be computed
(in cooperation with the calling subroutine).
If .FALSE., then Z is not referenced.
N (input) INTEGER
The order of the matrix H and (if WANTZ is .TRUE.) the
order of the unitary matrix Z.
KTOP (input) INTEGER
It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
KBOT and KTOP together determine an isolated block
along the diagonal of the Hessenberg matrix.
KBOT (input) INTEGER
It is assumed without a check that either
KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
determine an isolated block along the diagonal of the
Hessenberg matrix.
NW (input) INTEGER
Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
H (input/output) COMPLEX array, dimension (LDH,N)
On input the initial N-by-N section of H stores the
Hessenberg matrix undergoing aggressive early deflation.
On output H has been transformed by a unitary
similarity transformation, perturbed, and the returned
to Hessenberg form that (it is to be hoped) has some
zero subdiagonal entries.
LDH (input) integer
Leading dimension of H just as declared in the calling
subroutine. N .LE. LDH
ILOZ (input) INTEGER
IHIZ (input) INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
Z (input/output) COMPLEX array, dimension (LDZ,N)
IF WANTZ is .TRUE., then on output, the unitary
similarity transformation mentioned above has been
accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
If WANTZ is .FALSE., then Z is unreferenced.
LDZ (input) integer
The leading dimension of Z just as declared in the
calling subroutine. 1 .LE. LDZ.
NS (output) integer
The number of unconverged (ie approximate) eigenvalues
returned in SR and SI that may be used as shifts by the
calling subroutine.
ND (output) integer
The number of converged eigenvalues uncovered by this
subroutine.
SH (output) COMPLEX array, dimension KBOT
On output, approximate eigenvalues that may
be used for shifts are stored in SH(KBOT-ND-NS+1)
through SR(KBOT-ND). Converged eigenvalues are
stored in SH(KBOT-ND+1) through SH(KBOT).
V (workspace) COMPLEX array, dimension (LDV,NW)
An NW-by-NW work array.
LDV (input) integer scalar
The leading dimension of V just as declared in the
calling subroutine. NW .LE. LDV
NH (input) integer scalar
The number of columns of T. NH.GE.NW.
T (workspace) COMPLEX array, dimension (LDT,NW)
LDT (input) integer
The leading dimension of T just as declared in the
calling subroutine. NW .LE. LDT
NV (input) integer
The number of rows of work array WV available for
workspace. NV.GE.NW.
WV (workspace) COMPLEX array, dimension (LDWV,NW)
LDWV (input) integer
The leading dimension of W just as declared in the
calling subroutine. NW .LE. LDV
WORK (workspace) COMPLEX array, dimension LWORK.
On exit, WORK(1) is set to an estimate of the optimal value
of LWORK for the given values of N, NW, KTOP and KBOT.
LWORK (input) integer
The dimension of the work array WORK. LWORK = 2*NW
suffices, but greater efficiency may result from larger
values of LWORK.
If LWORK = -1, then a workspace query is assumed; CLAQR2
only estimates the optimal workspace size for the given
values of N, NW, KTOP and KBOT. The estimate is returned
in WORK(1). No error message related to LWORK is issued
by XERBLA. Neither H nor Z are accessed.
This subroutine is identical to CLAQR3 except that it avoids
recursion by calling CLAHQR instead of CLAQR4.
******************************************************************
Aggressive early deflation:
This subroutine accepts as input an upper Hessenberg matrix
H and performs an unitary similarity transformation
designed to detect and deflate fully converged eigenvalues from
a trailing principal submatrix. On output H has been over-
written by a new Hessenberg matrix that is a perturbation of
an unitary similarity transformation of H. It is to be
hoped that the final version of H has many zero subdiagonal
entries.
******************************************************************
WANTT (input) LOGICAL
If .TRUE., then the Hessenberg matrix H is fully updated
so that the triangular Schur factor may be
computed (in cooperation with the calling subroutine).
If .FALSE., then only enough of H is updated to preserve
the eigenvalues.
WANTZ (input) LOGICAL
If .TRUE., then the unitary matrix Z is updated so
so that the unitary Schur factor may be computed
(in cooperation with the calling subroutine).
If .FALSE., then Z is not referenced.
N (input) INTEGER
The order of the matrix H and (if WANTZ is .TRUE.) the
order of the unitary matrix Z.
KTOP (input) INTEGER
It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
KBOT and KTOP together determine an isolated block
along the diagonal of the Hessenberg matrix.
KBOT (input) INTEGER
It is assumed without a check that either
KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
determine an isolated block along the diagonal of the
Hessenberg matrix.
NW (input) INTEGER
Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
H (input/output) COMPLEX array, dimension (LDH,N)
On input the initial N-by-N section of H stores the
Hessenberg matrix undergoing aggressive early deflation.
On output H has been transformed by a unitary
similarity transformation, perturbed, and the returned
to Hessenberg form that (it is to be hoped) has some
zero subdiagonal entries.
LDH (input) integer
Leading dimension of H just as declared in the calling
subroutine. N .LE. LDH
ILOZ (input) INTEGER
IHIZ (input) INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
Z (input/output) COMPLEX array, dimension (LDZ,N)
IF WANTZ is .TRUE., then on output, the unitary
similarity transformation mentioned above has been
accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
If WANTZ is .FALSE., then Z is unreferenced.
LDZ (input) integer
The leading dimension of Z just as declared in the
calling subroutine. 1 .LE. LDZ.
NS (output) integer
The number of unconverged (ie approximate) eigenvalues
returned in SR and SI that may be used as shifts by the
calling subroutine.
ND (output) integer
The number of converged eigenvalues uncovered by this
subroutine.
SH (output) COMPLEX array, dimension KBOT
On output, approximate eigenvalues that may
be used for shifts are stored in SH(KBOT-ND-NS+1)
through SR(KBOT-ND). Converged eigenvalues are
stored in SH(KBOT-ND+1) through SH(KBOT).
V (workspace) COMPLEX array, dimension (LDV,NW)
An NW-by-NW work array.
LDV (input) integer scalar
The leading dimension of V just as declared in the
calling subroutine. NW .LE. LDV
NH (input) integer scalar
The number of columns of T. NH.GE.NW.
T (workspace) COMPLEX array, dimension (LDT,NW)
LDT (input) integer
The leading dimension of T just as declared in the
calling subroutine. NW .LE. LDT
NV (input) integer
The number of rows of work array WV available for
workspace. NV.GE.NW.
WV (workspace) COMPLEX array, dimension (LDWV,NW)
LDWV (input) integer
The leading dimension of W just as declared in the
calling subroutine. NW .LE. LDV
WORK (workspace) COMPLEX array, dimension LWORK.
On exit, WORK(1) is set to an estimate of the optimal value
of LWORK for the given values of N, NW, KTOP and KBOT.
LWORK (input) integer
The dimension of the work array WORK. LWORK = 2*NW
suffices, but greater efficiency may result from larger
values of LWORK.
If LWORK = -1, then a workspace query is assumed; CLAQR2
only estimates the optimal workspace size for the given
values of N, NW, KTOP and KBOT. The estimate is returned
in WORK(1). No error message related to LWORK is issued
by XERBLA. Neither H nor Z are accessed.