SRC/slaqr5.f(3)

NAME
SRC/slaqr5.f

SYNOPSIS

  Functions/Subroutines
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.

Function/Subroutine Documentation

  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).

Definition at line 262 of file slaqr5.f.

Author

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