NAME

       dlaqr5.f -

SYNOPSIS

   Functions/Subroutines
       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.

Function/Subroutine Documentation

   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 scalar
                        WANTT = .true. if the quasi-triangular Schur factor
                        is being computed.  WANTT is set to .false. otherwise.

           WANTZ

                     WANTZ is logical scalar
                        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: DLAQR5 accumulates reflections, uses matrix-matrix
                        multiply to update the far-from-diagonal matrix entries,
                        and takes advantage of 2-by-2 block structure during
                        matrix multiplies.

           N

                     N is integer scalar
                        N is the order of the Hessenberg matrix H upon which this
                        subroutine operates.

           KTOP

                     KTOP is integer scalar

           KBOT

                     KBOT is integer scalar
                        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 scalar
                        NSHFTS gives the number of simultaneous shifts.  NSHFTS
                        must be positive and even.

           SR

                     SR is DOUBLE PRECISION array of size (NSHFTS)

           SI

                     SI is DOUBLE PRECISION array of size (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 of size (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 scalar
                        LDH is the leading dimension of H just as declared in the
                        calling procedure.  LDH.GE.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 .LE. ILOZ .LE. IHIZ .LE. N

           Z

                     Z is DOUBLE PRECISION array of size (LDZ,IHI)
                        If WANTZ = .TRUE., then the QR Sweep orthogonal
                        similarity transformation is accumulated into
                        Z(ILOZ:IHIZ,ILO:IHI) from the right.
                        If WANTZ = .FALSE., then Z is unreferenced.

           LDZ

                     LDZ is integer scalar
                        LDA is the leading dimension of Z just as declared in
                        the calling procedure. LDZ.GE.N.

           V

                     V is DOUBLE PRECISION array of size (LDV,NSHFTS/2)

           LDV

                     LDV is integer scalar
                        LDV is the leading dimension of V as declared in the
                        calling procedure.  LDV.GE.3.

           U

                     U is DOUBLE PRECISION array of size
                        (LDU,3*NSHFTS-3)

           LDU

                     LDU is integer scalar
                        LDU is the leading dimension of U just as declared in the
                        in the calling subroutine.  LDU.GE.3*NSHFTS-3.

           NH

                     NH is integer scalar
                        NH is the number of columns in array WH available for
                        workspace. NH.GE.1.

           WH

                     WH is DOUBLE PRECISION array of size (LDWH,NH)

           LDWH

                     LDWH is integer scalar
                        Leading dimension of WH just as declared in the
                        calling procedure.  LDWH.GE.3*NSHFTS-3.

           NV

                     NV is integer scalar
                        NV is the number of rows in WV agailable for workspace.
                        NV.GE.1.

           WV

                     WV is DOUBLE PRECISION array of size
                        (LDWV,3*NSHFTS-3)

           LDWV

                     LDWV is integer scalar
                        LDWV is the leading dimension of WV as declared in the
                        in the calling subroutine.  LDWV.GE.NV.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           September 2012

       Contributors:
           Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

       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.

Author

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