SLAQR5, called by SLAQR0, performs a


    single small-bulge multi-shift QR sweep.

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: 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: SLAQR5 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 REAL array of size (NSHFTS)

SI

          SI is REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

September 2012

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

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.