SRC/zlaqr5.f(3)            Library Functions Manual            SRC/zlaqr5.f(3)

NAME
       SRC/zlaqr5.f

SYNOPSIS
   Functions/Subroutines
       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.

Function/Subroutine Documentation
   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).

       Definition at line 254 of file zlaqr5.f.

Author
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LAPACK                          Version 3.12.0                 SRC/zlaqr5.f(3)