Ubuntu Manpage: laqr2 - laqr2: step in hseqr

Provided by: liblapack-doc_3.12.0-3build1_all

NAME

       laqr2 - laqr2: step in hseqr

SYNOPSIS

   Functions
       subroutine claqr2 (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)
           CLAQR2 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 dlaqr2 (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)
           DLAQR2 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 slaqr2 (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)
           SLAQR2 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 zlaqr2 (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)
           ZLAQR2 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 claqr2 (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)
CLAQR2 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:

CLAQR2 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.

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 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 is INTEGER
Deflation window size. 1 <= NW <= (KBOT-KTOP+1).

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 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 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 is INTEGER
The number of converged eigenvalues uncovered by this
subroutine.

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 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 is INTEGER
The number of columns of T. NH >= NW.

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 is INTEGER
The number of rows of work array WV available for
workspace. NV >= NW.

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; 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.

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 dlaqr2 (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)
DLAQR2 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:

DLAQR2 is identical to DLAQR3 except that it avoids
recursion by calling DLAHQR instead of DLAQR4.

Aggressive early deflation:

This subroutine 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 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 is INTEGER
Deflation window size. 1 <= NW <= (KBOT-KTOP+1).

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 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 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 is INTEGER
The number of converged eigenvalues uncovered by this
subroutine.

SR is DOUBLE PRECISION array, dimension (KBOT)

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 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 is INTEGER
The number of columns of T. NH >= NW.

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 is INTEGER
The number of rows of work array WV available for
workspace. NV >= NW.

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; DLAQR2
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 slaqr2 (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)
SLAQR2 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:

SLAQR2 is identical to SLAQR3 except that it avoids
recursion by calling SLAHQR instead of SLAQR4.

Aggressive early deflation:

Parameters
WANTT

WANTZ

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 is INTEGER
Deflation window size. 1 <= NW <= (KBOT-KTOP+1).

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 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 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 is INTEGER
The number of converged eigenvalues uncovered by this
subroutine.

SR is REAL array, dimension (KBOT)

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 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 is INTEGER
The number of columns of T. NH >= NW.

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 is INTEGER
The number of rows of work array WV available for
workspace. NV >= NW.

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; SLAQR2
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 zlaqr2 (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)
ZLAQR2 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:

ZLAQR2 is identical to ZLAQR3 except that it avoids
recursion by calling ZLAHQR instead of ZLAQR4.

Aggressive early deflation:

ZLAQR2 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

WANTZ

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 is INTEGER
Deflation window size. 1 <= NW <= (KBOT-KTOP+1).

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 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 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 is INTEGER
The number of converged eigenvalues uncovered by this
subroutine.

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 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 is INTEGER
The number of columns of T. NH >= NW.

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 is INTEGER
The number of rows of work array WV available for
workspace. NV >= NW.

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; ZLAQR2
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

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