SLAQR5, called by SLAQR0, performs a


    single small-bulge multi-shift QR sweep.

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: 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


             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.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, 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.GE.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.GE.3.

U

          U is REAL array, dimension (LDU,3*NSHFTS-3)

LDU

          LDU is INTEGER


             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


             NH is the number of columns in array WH available for


             workspace. NH.GE.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.GE.3*NSHFTS-3.

NV

          NV is INTEGER


             NV is the number of rows in WV agailable for workspace.


             NV.GE.1.

WV

          WV is REAL array, dimension (LDWV,3*NSHFTS-3)

LDWV

          LDWV is INTEGER


             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.

June 2016

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.