laqr5(3)

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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