table of contents
| laqr3(3) | LAPACK | laqr3(3) |
NAME¶
laqr3 - laqr3: step in hseqr
SYNOPSIS¶
Functions¶
subroutine claqr3 (wantt, wantz, n, ktop, kbot, nw, h, ldh,
iloz, ihiz, z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work,
lwork)
CLAQR3 performs the unitary similarity transformation of a Hessenberg
matrix to detect and deflate fully converged eigenvalues from a trailing
principal submatrix (aggressive early deflation). subroutine dlaqr3
(wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sr,
si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork)
DLAQR3 performs the orthogonal similarity transformation of a
Hessenberg matrix to detect and deflate fully converged eigenvalues from a
trailing principal submatrix (aggressive early deflation). subroutine
slaqr3 (wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz,
ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork)
SLAQR3 performs the orthogonal similarity transformation of a
Hessenberg matrix to detect and deflate fully converged eigenvalues from a
trailing principal submatrix (aggressive early deflation). subroutine
zlaqr3 (wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz,
ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork)
ZLAQR3 performs the unitary similarity transformation of a Hessenberg
matrix to detect and deflate fully converged eigenvalues from a trailing
principal submatrix (aggressive early deflation).
Detailed Description¶
Function Documentation¶
subroutine claqr3 (logical wantt, logical wantz, integer n, integer ktop, integer kbot, integer nw, complex, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, complex, dimension( ldz, * ) z, integer ldz, integer ns, integer nd, complex, dimension( * ) sh, complex, dimension( ldv, * ) v, integer ldv, integer nh, complex, dimension( ldt, * ) t, integer ldt, integer nv, complex, dimension( ldwv, * ) wv, integer ldwv, complex, dimension( * ) work, integer lwork)¶
CLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
Purpose:
Aggressive early deflation:
CLAQR3 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.
Parameters
WANTT
WANTT is 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
WANTZ is 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
N is INTEGER
The order of the matrix H and (if WANTZ is .TRUE.) the
order of the unitary matrix Z.
KTOP
KTOP is 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
KBOT is 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
NW is INTEGER
Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
H
H is 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
LDH is INTEGER
Leading dimension of H just as declared in the calling
subroutine. N <= LDH
ILOZ
ILOZ is INTEGER
IHIZ
IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
Z
Z is 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,ILOZ:IHIZ) from the right.
If WANTZ is .FALSE., then Z is unreferenced.
LDZ
LDZ is INTEGER
The leading dimension of Z just as declared in the
calling subroutine. 1 <= LDZ.
NS
NS is INTEGER
The number of unconverged (ie approximate) eigenvalues
returned in SR and SI that may be used as shifts by the
calling subroutine.
ND
ND is INTEGER
The number of converged eigenvalues uncovered by this
subroutine.
SH
SH is 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
V is COMPLEX array, dimension (LDV,NW)
An NW-by-NW work array.
LDV
LDV is INTEGER
The leading dimension of V just as declared in the
calling subroutine. NW <= LDV
NH
NH is INTEGER
The number of columns of T. NH >= NW.
T
T is COMPLEX array, dimension (LDT,NW)
LDT
LDT is INTEGER
The leading dimension of T just as declared in the
calling subroutine. NW <= LDT
NV
NV is INTEGER
The number of rows of work array WV available for
workspace. NV >= NW.
WV
WV is COMPLEX array, dimension (LDWV,NW)
LDWV
LDWV is INTEGER
The leading dimension of W just as declared in the
calling subroutine. NW <= LDV
WORK
WORK is 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
LWORK is 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; CLAQR3
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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics,
University of Kansas, USA
subroutine dlaqr3 (logical wantt, logical wantz, integer n, integer ktop, integer kbot, integer nw, double precision, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, double precision, dimension( ldz, * ) z, integer ldz, integer ns, integer nd, double precision, dimension( * ) sr, double precision, dimension( * ) si, double precision, dimension( ldv, * ) v, integer ldv, integer nh, double precision, dimension( ldt, * ) t, integer ldt, integer nv, double precision, dimension( ldwv, * ) wv, integer ldwv, double precision, dimension( * ) work, integer lwork)¶
DLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
Purpose:
Aggressive early deflation:
DLAQR3 accepts as input an upper Hessenberg matrix
H and performs an orthogonal 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 orthogonal similarity transformation of H. It is to be
hoped that the final version of H has many zero subdiagonal
entries.
Parameters
WANTT
WANTT is LOGICAL
If .TRUE., then the Hessenberg matrix H is fully updated
so that the quasi-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
WANTZ is LOGICAL
If .TRUE., then the orthogonal matrix Z is updated so
so that the orthogonal Schur factor may be computed
(in cooperation with the calling subroutine).
If .FALSE., then Z is not referenced.
N
N is INTEGER
The order of the matrix H and (if WANTZ is .TRUE.) the
order of the orthogonal matrix Z.
KTOP
KTOP is 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
KBOT is 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
NW is INTEGER
Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
H
H is DOUBLE PRECISION 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 an orthogonal
similarity transformation, perturbed, and the returned
to Hessenberg form that (it is to be hoped) has some
zero subdiagonal entries.
LDH
LDH is INTEGER
Leading dimension of H just as declared in the calling
subroutine. N <= LDH
ILOZ
ILOZ is INTEGER
IHIZ
IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
Z
Z is DOUBLE PRECISION array, dimension (LDZ,N)
IF WANTZ is .TRUE., then on output, the orthogonal
similarity transformation mentioned above has been
accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
If WANTZ is .FALSE., then Z is unreferenced.
LDZ
LDZ is INTEGER
The leading dimension of Z just as declared in the
calling subroutine. 1 <= LDZ.
NS
NS is INTEGER
The number of unconverged (ie approximate) eigenvalues
returned in SR and SI that may be used as shifts by the
calling subroutine.
ND
ND is INTEGER
The number of converged eigenvalues uncovered by this
subroutine.
SR
SR is DOUBLE PRECISION array, dimension (KBOT)
SI
SI is DOUBLE PRECISION array, dimension (KBOT)
On output, the real and imaginary parts of approximate
eigenvalues that may be used for shifts are stored in
SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
The real and imaginary parts of converged eigenvalues
are stored in SR(KBOT-ND+1) through SR(KBOT) and
SI(KBOT-ND+1) through SI(KBOT), respectively.
V
V is DOUBLE PRECISION array, dimension (LDV,NW)
An NW-by-NW work array.
LDV
LDV is INTEGER
The leading dimension of V just as declared in the
calling subroutine. NW <= LDV
NH
NH is INTEGER
The number of columns of T. NH >= NW.
T
T is DOUBLE PRECISION array, dimension (LDT,NW)
LDT
LDT is INTEGER
The leading dimension of T just as declared in the
calling subroutine. NW <= LDT
NV
NV is INTEGER
The number of rows of work array WV available for
workspace. NV >= NW.
WV
WV is DOUBLE PRECISION array, dimension (LDWV,NW)
LDWV
LDWV is INTEGER
The leading dimension of W just as declared in the
calling subroutine. NW <= LDV
WORK
WORK is DOUBLE PRECISION 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
LWORK is 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; DLAQR3
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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics,
University of Kansas, USA
subroutine slaqr3 (logical wantt, logical wantz, integer n, integer ktop, integer kbot, integer nw, real, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, real, dimension( ldz, * ) z, integer ldz, integer ns, integer nd, real, dimension( * ) sr, real, dimension( * ) si, real, dimension( ldv, * ) v, integer ldv, integer nh, real, dimension( ldt, * ) t, integer ldt, integer nv, real, dimension( ldwv, * ) wv, integer ldwv, real, dimension( * ) work, integer lwork)¶
SLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
Purpose:
Aggressive early deflation:
SLAQR3 accepts as input an upper Hessenberg matrix
H and performs an orthogonal 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 orthogonal similarity transformation of H. It is to be
hoped that the final version of H has many zero subdiagonal
entries.
Parameters
WANTT
WANTT is LOGICAL
If .TRUE., then the Hessenberg matrix H is fully updated
so that the quasi-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
WANTZ is LOGICAL
If .TRUE., then the orthogonal matrix Z is updated so
so that the orthogonal Schur factor may be computed
(in cooperation with the calling subroutine).
If .FALSE., then Z is not referenced.
N
N is INTEGER
The order of the matrix H and (if WANTZ is .TRUE.) the
order of the orthogonal matrix Z.
KTOP
KTOP is 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
KBOT is 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
NW is INTEGER
Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
H
H is REAL 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 an orthogonal
similarity transformation, perturbed, and the returned
to Hessenberg form that (it is to be hoped) has some
zero subdiagonal entries.
LDH
LDH is INTEGER
Leading dimension of H just as declared in the calling
subroutine. N <= LDH
ILOZ
ILOZ is INTEGER
IHIZ
IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
Z
Z is REAL array, dimension (LDZ,N)
IF WANTZ is .TRUE., then on output, the orthogonal
similarity transformation mentioned above has been
accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
If WANTZ is .FALSE., then Z is unreferenced.
LDZ
LDZ is INTEGER
The leading dimension of Z just as declared in the
calling subroutine. 1 <= LDZ.
NS
NS is INTEGER
The number of unconverged (ie approximate) eigenvalues
returned in SR and SI that may be used as shifts by the
calling subroutine.
ND
ND is INTEGER
The number of converged eigenvalues uncovered by this
subroutine.
SR
SR is REAL array, dimension (KBOT)
SI
SI is REAL array, dimension (KBOT)
On output, the real and imaginary parts of approximate
eigenvalues that may be used for shifts are stored in
SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
The real and imaginary parts of converged eigenvalues
are stored in SR(KBOT-ND+1) through SR(KBOT) and
SI(KBOT-ND+1) through SI(KBOT), respectively.
V
V is REAL array, dimension (LDV,NW)
An NW-by-NW work array.
LDV
LDV is INTEGER
The leading dimension of V just as declared in the
calling subroutine. NW <= LDV
NH
NH is INTEGER
The number of columns of T. NH >= NW.
T
T is REAL array, dimension (LDT,NW)
LDT
LDT is INTEGER
The leading dimension of T just as declared in the
calling subroutine. NW <= LDT
NV
NV is INTEGER
The number of rows of work array WV available for
workspace. NV >= NW.
WV
WV is REAL array, dimension (LDWV,NW)
LDWV
LDWV is INTEGER
The leading dimension of W just as declared in the
calling subroutine. NW <= LDV
WORK
WORK is REAL 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
LWORK is 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; SLAQR3
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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics,
University of Kansas, USA
subroutine zlaqr3 (logical wantt, logical wantz, integer n, integer ktop, integer kbot, integer nw, complex*16, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, complex*16, dimension( ldz, * ) z, integer ldz, integer ns, integer nd, complex*16, dimension( * ) sh, complex*16, dimension( ldv, * ) v, integer ldv, integer nh, complex*16, dimension( ldt, * ) t, integer ldt, integer nv, complex*16, dimension( ldwv, * ) wv, integer ldwv, complex*16, dimension( * ) work, integer lwork)¶
ZLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
Purpose:
Aggressive early deflation:
ZLAQR3 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.
Parameters
WANTT
WANTT is 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
WANTZ is 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
N is INTEGER
The order of the matrix H and (if WANTZ is .TRUE.) the
order of the unitary matrix Z.
KTOP
KTOP is 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
KBOT is 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
NW is INTEGER
Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
H
H is COMPLEX*16 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
LDH is INTEGER
Leading dimension of H just as declared in the calling
subroutine. N <= LDH
ILOZ
ILOZ is INTEGER
IHIZ
IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
Z
Z is COMPLEX*16 array, dimension (LDZ,N)
IF WANTZ is .TRUE., then on output, the unitary
similarity transformation mentioned above has been
accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
If WANTZ is .FALSE., then Z is unreferenced.
LDZ
LDZ is INTEGER
The leading dimension of Z just as declared in the
calling subroutine. 1 <= LDZ.
NS
NS is INTEGER
The number of unconverged (ie approximate) eigenvalues
returned in SR and SI that may be used as shifts by the
calling subroutine.
ND
ND is INTEGER
The number of converged eigenvalues uncovered by this
subroutine.
SH
SH is COMPLEX*16 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
V is COMPLEX*16 array, dimension (LDV,NW)
An NW-by-NW work array.
LDV
LDV is INTEGER
The leading dimension of V just as declared in the
calling subroutine. NW <= LDV
NH
NH is INTEGER
The number of columns of T. NH >= NW.
T
T is COMPLEX*16 array, dimension (LDT,NW)
LDT
LDT is INTEGER
The leading dimension of T just as declared in the
calling subroutine. NW <= LDT
NV
NV is INTEGER
The number of rows of work array WV available for
workspace. NV >= NW.
WV
WV is COMPLEX*16 array, dimension (LDWV,NW)
LDWV
LDWV is INTEGER
The leading dimension of W just as declared in the
calling subroutine. NW <= LDV
WORK
WORK is COMPLEX*16 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
LWORK is 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; ZLAQR3
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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics,
University of Kansas, USA
Author¶
Generated automatically by Doxygen for LAPACK from the source code.
| Wed Feb 7 2024 11:30:40 | Version 3.12.0 |