Provided by: liblapack-doc_3.12.0-3build1.1_all 
NAME
laqr5 - laqr5: step in hseqr
SYNOPSIS
Functions
subroutine claqr5 (wantt, wantz, kacc22, n, ktop, kbot, nshfts, s, h, ldh, iloz, ihiz, z,
ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh, ldwh)
CLAQR5 performs a single small-bulge multi-shift QR sweep.
subroutine dlaqr5 (wantt, wantz, kacc22, n, ktop, kbot, nshfts, sr, si, h, ldh, iloz,
ihiz, z, ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh, ldwh)
DLAQR5 performs a single small-bulge multi-shift QR sweep.
subroutine slaqr5 (wantt, wantz, kacc22, n, ktop, kbot, nshfts, sr, si, h, ldh, iloz,
ihiz, z, ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh, ldwh)
SLAQR5 performs a single small-bulge multi-shift QR sweep.
subroutine zlaqr5 (wantt, wantz, kacc22, n, ktop, kbot, nshfts, s, h, ldh, iloz, ihiz, z,
ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh, ldwh)
ZLAQR5 performs a single small-bulge multi-shift QR sweep.
Detailed Description
Function Documentation
subroutine claqr5 (logical wantt, logical wantz, integer kacc22, integer n, integer ktop,
integer kbot, integer nshfts, complex, dimension( * ) s, complex, dimension( ldh, * ) h,
integer ldh, integer iloz, integer ihiz, complex, dimension( ldz, * ) z, integer ldz,
complex, dimension( ldv, * ) v, integer ldv, complex, dimension( ldu, * ) u, integer ldu,
integer nv, complex, dimension( ldwv, * ) wv, integer ldwv, integer nh, complex,
dimension( ldwh, * ) wh, integer ldwh)
CLAQR5 performs a single small-bulge multi-shift QR sweep.
Purpose:
CLAQR5 called by CLAQR0 performs a
single small-bulge multi-shift QR sweep.
Parameters
WANTT
WANTT is LOGICAL
WANTT = .true. if the triangular Schur factor
is being computed. WANTT is set to .false. otherwise.
WANTZ
WANTZ is LOGICAL
WANTZ = .true. if the unitary Schur factor is being
computed. WANTZ is set to .false. otherwise.
KACC22
KACC22 is INTEGER with value 0, 1, or 2.
Specifies the computation mode of far-from-diagonal
orthogonal updates.
= 0: CLAQR5 does not accumulate reflections and does not
use matrix-matrix multiply to update far-from-diagonal
matrix entries.
= 1: CLAQR5 accumulates reflections and uses matrix-matrix
multiply to update the far-from-diagonal matrix entries.
= 2: Same as KACC22 = 1. This option used to enable exploiting
the 2-by-2 structure during matrix multiplications, but
this is no longer supported.
N
N is INTEGER
N is the order of the Hessenberg matrix H upon which this
subroutine operates.
KTOP
KTOP is INTEGER
KBOT
KBOT is INTEGER
These are the first and last rows and columns of an
isolated diagonal block upon which the QR sweep is to be
applied. It is assumed without a check that
either KTOP = 1 or H(KTOP,KTOP-1) = 0
and
either KBOT = N or H(KBOT+1,KBOT) = 0.
NSHFTS
NSHFTS is INTEGER
NSHFTS gives the number of simultaneous shifts. NSHFTS
must be positive and even.
S
S is COMPLEX array, dimension (NSHFTS)
S contains the shifts of origin that define the multi-
shift QR sweep. On output S may be reordered.
H
H is COMPLEX array, dimension (LDH,N)
On input H contains a Hessenberg matrix. On output a
multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
to the isolated diagonal block in rows and columns KTOP
through KBOT.
LDH
LDH is INTEGER
LDH is the leading dimension of H just as declared in the
calling procedure. LDH >= MAX(1,N).
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,IHIZ)
If WANTZ = .TRUE., then the QR Sweep unitary
similarity transformation is accumulated into
Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
If WANTZ = .FALSE., then Z is unreferenced.
LDZ
LDZ is INTEGER
LDA is the leading dimension of Z just as declared in
the calling procedure. LDZ >= N.
V
V is COMPLEX array, dimension (LDV,NSHFTS/2)
LDV
LDV is INTEGER
LDV is the leading dimension of V as declared in the
calling procedure. LDV >= 3.
U
U is COMPLEX array, dimension (LDU,2*NSHFTS)
LDU
LDU is INTEGER
LDU is the leading dimension of U just as declared in the
in the calling subroutine. LDU >= 2*NSHFTS.
NV
NV is INTEGER
NV is the number of rows in WV agailable for workspace.
NV >= 1.
WV
WV is COMPLEX array, dimension (LDWV,2*NSHFTS)
LDWV
LDWV is INTEGER
LDWV is the leading dimension of WV as declared in the
in the calling subroutine. LDWV >= NV.
NH
NH is INTEGER
NH is the number of columns in array WH available for
workspace. NH >= 1.
WH
WH is COMPLEX array, dimension (LDWH,NH)
LDWH
LDWH is INTEGER
Leading dimension of WH just as declared in the
calling procedure. LDWH >= 2*NSHFTS.
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
Lars Karlsson, Daniel Kressner, and Bruno Lang
Thijs Steel, Department of Computer science, KU Leuven, Belgium
References:
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining
Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume
23, pages 929--947, 2002.
Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed chains of bulges in
multishift QR algorithms. ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).
subroutine dlaqr5 (logical wantt, logical wantz, integer kacc22, integer n, integer ktop,
integer kbot, integer nshfts, double precision, dimension( * ) sr, double precision,
dimension( * ) si, double precision, dimension( ldh, * ) h, integer ldh, integer iloz,
integer ihiz, double precision, dimension( ldz, * ) z, integer ldz, double precision,
dimension( ldv, * ) v, integer ldv, double precision, dimension( ldu, * ) u, integer ldu,
integer nv, double precision, dimension( ldwv, * ) wv, integer ldwv, integer nh, double
precision, dimension( ldwh, * ) wh, integer ldwh)
DLAQR5 performs a single small-bulge multi-shift QR sweep.
Purpose:
DLAQR5, called by DLAQR0, performs a
single small-bulge multi-shift QR sweep.
Parameters
WANTT
WANTT is LOGICAL
WANTT = .true. if the quasi-triangular Schur factor
is being computed. WANTT is set to .false. otherwise.
WANTZ
WANTZ is LOGICAL
WANTZ = .true. if the orthogonal Schur factor is being
computed. WANTZ is set to .false. otherwise.
KACC22
KACC22 is INTEGER with value 0, 1, or 2.
Specifies the computation mode of far-from-diagonal
orthogonal updates.
= 0: DLAQR5 does not accumulate reflections and does not
use matrix-matrix multiply to update far-from-diagonal
matrix entries.
= 1: DLAQR5 accumulates reflections and uses matrix-matrix
multiply to update the far-from-diagonal matrix entries.
= 2: Same as KACC22 = 1. This option used to enable exploiting
the 2-by-2 structure during matrix multiplications, but
this is no longer supported.
N
N is INTEGER
N is the order of the Hessenberg matrix H upon which this
subroutine operates.
KTOP
KTOP is INTEGER
KBOT
KBOT is INTEGER
These are the first and last rows and columns of an
isolated diagonal block upon which the QR sweep is to be
applied. It is assumed without a check that
either KTOP = 1 or H(KTOP,KTOP-1) = 0
and
either KBOT = N or H(KBOT+1,KBOT) = 0.
NSHFTS
NSHFTS is INTEGER
NSHFTS gives the number of simultaneous shifts. NSHFTS
must be positive and even.
SR
SR is DOUBLE PRECISION array, dimension (NSHFTS)
SI
SI is DOUBLE PRECISION array, dimension (NSHFTS)
SR contains the real parts and SI contains the imaginary
parts of the NSHFTS shifts of origin that define the
multi-shift QR sweep. On output SR and SI may be
reordered.
H
H is DOUBLE PRECISION array, dimension (LDH,N)
On input H contains a Hessenberg matrix. On output a
multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
to the isolated diagonal block in rows and columns KTOP
through KBOT.
LDH
LDH is INTEGER
LDH is the leading dimension of H just as declared in the
calling procedure. LDH >= MAX(1,N).
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,IHIZ)
If WANTZ = .TRUE., then the QR Sweep orthogonal
similarity transformation is accumulated into
Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
If WANTZ = .FALSE., then Z is unreferenced.
LDZ
LDZ is INTEGER
LDA is the leading dimension of Z just as declared in
the calling procedure. LDZ >= N.
V
V is DOUBLE PRECISION array, dimension (LDV,NSHFTS/2)
LDV
LDV is INTEGER
LDV is the leading dimension of V as declared in the
calling procedure. LDV >= 3.
U
U is DOUBLE PRECISION array, dimension (LDU,2*NSHFTS)
LDU
LDU is INTEGER
LDU is the leading dimension of U just as declared in the
in the calling subroutine. LDU >= 2*NSHFTS.
NV
NV is INTEGER
NV is the number of rows in WV agailable for workspace.
NV >= 1.
WV
WV is DOUBLE PRECISION array, dimension (LDWV,2*NSHFTS)
LDWV
LDWV is INTEGER
LDWV is the leading dimension of WV as declared in the
in the calling subroutine. LDWV >= NV.
NH
NH is INTEGER
NH is the number of columns in array WH available for
workspace. NH >= 1.
WH
WH is DOUBLE PRECISION array, dimension (LDWH,NH)
LDWH
LDWH is INTEGER
Leading dimension of WH just as declared in the
calling procedure. LDWH >= 2*NSHFTS.
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
Lars Karlsson, Daniel Kressner, and Bruno Lang
Thijs Steel, Department of Computer science, KU Leuven, Belgium
References:
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining
Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume
23, pages 929--947, 2002.
Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed chains of bulges in
multishift QR algorithms. ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).
subroutine slaqr5 (logical wantt, logical wantz, integer kacc22, integer n, integer ktop,
integer kbot, integer nshfts, real, dimension( * ) sr, real, dimension( * ) si, real,
dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, real, dimension( ldz, * )
z, integer ldz, real, dimension( ldv, * ) v, integer ldv, real, dimension( ldu, * ) u,
integer ldu, integer nv, real, dimension( ldwv, * ) wv, integer ldwv, integer nh, real,
dimension( ldwh, * ) wh, integer ldwh)
SLAQR5 performs a single small-bulge multi-shift QR sweep.
Purpose:
SLAQR5, called by SLAQR0, performs a
single small-bulge multi-shift QR sweep.
Parameters
WANTT
WANTT is LOGICAL
WANTT = .true. if the quasi-triangular Schur factor
is being computed. WANTT is set to .false. otherwise.
WANTZ
WANTZ is LOGICAL
WANTZ = .true. if the orthogonal Schur factor is being
computed. WANTZ is set to .false. otherwise.
KACC22
KACC22 is INTEGER with value 0, 1, or 2.
Specifies the computation mode of far-from-diagonal
orthogonal updates.
= 0: SLAQR5 does not accumulate reflections and does not
use matrix-matrix multiply to update far-from-diagonal
matrix entries.
= 1: SLAQR5 accumulates reflections and uses matrix-matrix
multiply to update the far-from-diagonal matrix entries.
= 2: Same as KACC22 = 1. This option used to enable exploiting
the 2-by-2 structure during matrix multiplications, but
this is no longer supported.
N
N is INTEGER
N is the order of the Hessenberg matrix H upon which this
subroutine operates.
KTOP
KTOP is INTEGER
KBOT
KBOT is INTEGER
These are the first and last rows and columns of an
isolated diagonal block upon which the QR sweep is to be
applied. It is assumed without a check that
either KTOP = 1 or H(KTOP,KTOP-1) = 0
and
either KBOT = N or H(KBOT+1,KBOT) = 0.
NSHFTS
NSHFTS is INTEGER
NSHFTS gives the number of simultaneous shifts. NSHFTS
must be positive and even.
SR
SR is REAL array, dimension (NSHFTS)
SI
SI is REAL array, dimension (NSHFTS)
SR contains the real parts and SI contains the imaginary
parts of the NSHFTS shifts of origin that define the
multi-shift QR sweep. On output SR and SI may be
reordered.
H
H is REAL array, dimension (LDH,N)
On input H contains a Hessenberg matrix. On output a
multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
to the isolated diagonal block in rows and columns KTOP
through KBOT.
LDH
LDH is INTEGER
LDH is the leading dimension of H just as declared in the
calling procedure. LDH >= MAX(1,N).
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,IHIZ)
If WANTZ = .TRUE., then the QR Sweep orthogonal
similarity transformation is accumulated into
Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
If WANTZ = .FALSE., then Z is unreferenced.
LDZ
LDZ is INTEGER
LDA is the leading dimension of Z just as declared in
the calling procedure. LDZ >= N.
V
V is REAL array, dimension (LDV,NSHFTS/2)
LDV
LDV is INTEGER
LDV is the leading dimension of V as declared in the
calling procedure. LDV >= 3.
U
U is REAL array, dimension (LDU,2*NSHFTS)
LDU
LDU is INTEGER
LDU is the leading dimension of U just as declared in the
in the calling subroutine. LDU >= 2*NSHFTS.
NV
NV is INTEGER
NV is the number of rows in WV agailable for workspace.
NV >= 1.
WV
WV is REAL array, dimension (LDWV,2*NSHFTS)
LDWV
LDWV is INTEGER
LDWV is the leading dimension of WV as declared in the
in the calling subroutine. LDWV >= NV.
NH
NH is INTEGER
NH is the number of columns in array WH available for
workspace. NH >= 1.
WH
WH is REAL array, dimension (LDWH,NH)
LDWH
LDWH is INTEGER
Leading dimension of WH just as declared in the
calling procedure. LDWH >= 2*NSHFTS.
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
Lars Karlsson, Daniel Kressner, and Bruno Lang
Thijs Steel, Department of Computer science, KU Leuven, Belgium
References:
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining
Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume
23, pages 929--947, 2002.
Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed chains of bulges in
multishift QR algorithms. ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).
subroutine zlaqr5 (logical wantt, logical wantz, integer kacc22, integer n, integer ktop,
integer kbot, integer nshfts, complex*16, dimension( * ) s, complex*16, dimension( ldh, *
) h, integer ldh, integer iloz, integer ihiz, complex*16, dimension( ldz, * ) z, integer
ldz, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldu, * ) u,
integer ldu, integer nv, complex*16, dimension( ldwv, * ) wv, integer ldwv, integer nh,
complex*16, dimension( ldwh, * ) wh, integer ldwh)
ZLAQR5 performs a single small-bulge multi-shift QR sweep.
Purpose:
ZLAQR5, called by ZLAQR0, performs a
single small-bulge multi-shift QR sweep.
Parameters
WANTT
WANTT is LOGICAL
WANTT = .true. if the triangular Schur factor
is being computed. WANTT is set to .false. otherwise.
WANTZ
WANTZ is LOGICAL
WANTZ = .true. if the unitary Schur factor is being
computed. WANTZ is set to .false. otherwise.
KACC22
KACC22 is INTEGER with value 0, 1, or 2.
Specifies the computation mode of far-from-diagonal
orthogonal updates.
= 0: ZLAQR5 does not accumulate reflections and does not
use matrix-matrix multiply to update far-from-diagonal
matrix entries.
= 1: ZLAQR5 accumulates reflections and uses matrix-matrix
multiply to update the far-from-diagonal matrix entries.
= 2: Same as KACC22 = 1. This option used to enable exploiting
the 2-by-2 structure during matrix multiplications, but
this is no longer supported.
N
N is INTEGER
N is the order of the Hessenberg matrix H upon which this
subroutine operates.
KTOP
KTOP is INTEGER
KBOT
KBOT is INTEGER
These are the first and last rows and columns of an
isolated diagonal block upon which the QR sweep is to be
applied. It is assumed without a check that
either KTOP = 1 or H(KTOP,KTOP-1) = 0
and
either KBOT = N or H(KBOT+1,KBOT) = 0.
NSHFTS
NSHFTS is INTEGER
NSHFTS gives the number of simultaneous shifts. NSHFTS
must be positive and even.
S
S is COMPLEX*16 array, dimension (NSHFTS)
S contains the shifts of origin that define the multi-
shift QR sweep. On output S may be reordered.
H
H is COMPLEX*16 array, dimension (LDH,N)
On input H contains a Hessenberg matrix. On output a
multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
to the isolated diagonal block in rows and columns KTOP
through KBOT.
LDH
LDH is INTEGER
LDH is the leading dimension of H just as declared in the
calling procedure. LDH >= MAX(1,N).
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,IHIZ)
If WANTZ = .TRUE., then the QR Sweep unitary
similarity transformation is accumulated into
Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
If WANTZ = .FALSE., then Z is unreferenced.
LDZ
LDZ is INTEGER
LDA is the leading dimension of Z just as declared in
the calling procedure. LDZ >= N.
V
V is COMPLEX*16 array, dimension (LDV,NSHFTS/2)
LDV
LDV is INTEGER
LDV is the leading dimension of V as declared in the
calling procedure. LDV >= 3.
U
U is COMPLEX*16 array, dimension (LDU,2*NSHFTS)
LDU
LDU is INTEGER
LDU is the leading dimension of U just as declared in the
in the calling subroutine. LDU >= 2*NSHFTS.
NV
NV is INTEGER
NV is the number of rows in WV agailable for workspace.
NV >= 1.
WV
WV is COMPLEX*16 array, dimension (LDWV,2*NSHFTS)
LDWV
LDWV is INTEGER
LDWV is the leading dimension of WV as declared in the
in the calling subroutine. LDWV >= NV.
NH
NH is INTEGER
NH is the number of columns in array WH available for
workspace. NH >= 1.
WH
WH is COMPLEX*16 array, dimension (LDWH,NH)
LDWH
LDWH is INTEGER
Leading dimension of WH just as declared in the
calling procedure. LDWH >= 2*NSHFTS.
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
Lars Karlsson, Daniel Kressner, and Bruno Lang
Thijs Steel, Department of Computer science, KU Leuven, Belgium
References:
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining
Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume
23, pages 929--947, 2002.
Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed chains of bulges in
multishift QR algorithms. ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).
Author
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