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SUBROUTINE <a name="ZLAQR5.1"></a><a href="zlaqr5.f.html#ZLAQR5.1">ZLAQR5</a>( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, S,
$ H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U, LDU, NV,
$ WV, LDWV, NH, WH, LDWH )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> INTEGER IHIZ, ILOZ, KACC22, KBOT, KTOP, LDH, LDU, LDV,
$ LDWH, LDWV, LDZ, N, NH, NSHFTS, NV
LOGICAL WANTT, WANTZ
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> COMPLEX*16 H( LDH, * ), S( * ), U( LDU, * ), V( LDV, * ),
$ WH( LDWH, * ), WV( LDWV, * ), Z( LDZ, * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This auxiliary subroutine called by <a name="ZLAQR0.19"></a><a href="zlaqr0.f.html#ZLAQR0.1">ZLAQR0</a> performs a
</span><span class="comment">*</span><span class="comment"> single small-bulge multi-shift QR sweep.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WANTT (input) logical scalar
</span><span class="comment">*</span><span class="comment"> WANTT = .true. if the triangular Schur factor
</span><span class="comment">*</span><span class="comment"> is being computed. WANTT is set to .false. otherwise.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WANTZ (input) logical scalar
</span><span class="comment">*</span><span class="comment"> WANTZ = .true. if the unitary Schur factor is being
</span><span class="comment">*</span><span class="comment"> computed. WANTZ is set to .false. otherwise.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> KACC22 (input) integer with value 0, 1, or 2.
</span><span class="comment">*</span><span class="comment"> Specifies the computation mode of far-from-diagonal
</span><span class="comment">*</span><span class="comment"> orthogonal updates.
</span><span class="comment">*</span><span class="comment"> = 0: <a name="ZLAQR5.33"></a><a href="zlaqr5.f.html#ZLAQR5.1">ZLAQR5</a> does not accumulate reflections and does not
</span><span class="comment">*</span><span class="comment"> use matrix-matrix multiply to update far-from-diagonal
</span><span class="comment">*</span><span class="comment"> matrix entries.
</span><span class="comment">*</span><span class="comment"> = 1: <a name="ZLAQR5.36"></a><a href="zlaqr5.f.html#ZLAQR5.1">ZLAQR5</a> accumulates reflections and uses matrix-matrix
</span><span class="comment">*</span><span class="comment"> multiply to update the far-from-diagonal matrix entries.
</span><span class="comment">*</span><span class="comment"> = 2: <a name="ZLAQR5.38"></a><a href="zlaqr5.f.html#ZLAQR5.1">ZLAQR5</a> accumulates reflections, uses matrix-matrix
</span><span class="comment">*</span><span class="comment"> multiply to update the far-from-diagonal matrix entries,
</span><span class="comment">*</span><span class="comment"> and takes advantage of 2-by-2 block structure during
</span><span class="comment">*</span><span class="comment"> matrix multiplies.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) integer scalar
</span><span class="comment">*</span><span class="comment"> N is the order of the Hessenberg matrix H upon which this
</span><span class="comment">*</span><span class="comment"> subroutine operates.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> KTOP (input) integer scalar
</span><span class="comment">*</span><span class="comment"> KBOT (input) integer scalar
</span><span class="comment">*</span><span class="comment"> These are the first and last rows and columns of an
</span><span class="comment">*</span><span class="comment"> isolated diagonal block upon which the QR sweep is to be
</span><span class="comment">*</span><span class="comment"> applied. It is assumed without a check that
</span><span class="comment">*</span><span class="comment"> either KTOP = 1 or H(KTOP,KTOP-1) = 0
</span><span class="comment">*</span><span class="comment"> and
</span><span class="comment">*</span><span class="comment"> either KBOT = N or H(KBOT+1,KBOT) = 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> NSHFTS (input) integer scalar
</span><span class="comment">*</span><span class="comment"> NSHFTS gives the number of simultaneous shifts. NSHFTS
</span><span class="comment">*</span><span class="comment"> must be positive and even.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> S (input) COMPLEX*16 array of size (NSHFTS)
</span><span class="comment">*</span><span class="comment"> S contains the shifts of origin that define the multi-
</span><span class="comment">*</span><span class="comment"> shift QR sweep.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> H (input/output) COMPLEX*16 array of size (LDH,N)
</span><span class="comment">*</span><span class="comment"> On input H contains a Hessenberg matrix. On output a
</span><span class="comment">*</span><span class="comment"> multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
</span><span class="comment">*</span><span class="comment"> to the isolated diagonal block in rows and columns KTOP
</span><span class="comment">*</span><span class="comment"> through KBOT.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDH (input) integer scalar
</span><span class="comment">*</span><span class="comment"> LDH is the leading dimension of H just as declared in the
</span><span class="comment">*</span><span class="comment"> calling procedure. LDH.GE.MAX(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ILOZ (input) INTEGER
</span><span class="comment">*</span><span class="comment"> IHIZ (input) INTEGER
</span><span class="comment">*</span><span class="comment"> Specify the rows of Z to which transformations must be
</span><span class="comment">*</span><span class="comment"> applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Z (input/output) COMPLEX*16 array of size (LDZ,IHI)
</span><span class="comment">*</span><span class="comment"> If WANTZ = .TRUE., then the QR Sweep unitary
</span><span class="comment">*</span><span class="comment"> similarity transformation is accumulated into
</span><span class="comment">*</span><span class="comment"> Z(ILOZ:IHIZ,ILO:IHI) from the right.
</span><span class="comment">*</span><span class="comment"> If WANTZ = .FALSE., then Z is unreferenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDZ (input) integer scalar
</span><span class="comment">*</span><span class="comment"> LDA is the leading dimension of Z just as declared in
</span><span class="comment">*</span><span class="comment"> the calling procedure. LDZ.GE.N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> V (workspace) COMPLEX*16 array of size (LDV,NSHFTS/2)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDV (input) integer scalar
</span><span class="comment">*</span><span class="comment"> LDV is the leading dimension of V as declared in the
</span><span class="comment">*</span><span class="comment"> calling procedure. LDV.GE.3.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> U (workspace) COMPLEX*16 array of size
</span><span class="comment">*</span><span class="comment"> (LDU,3*NSHFTS-3)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDU (input) integer scalar
</span><span class="comment">*</span><span class="comment"> LDU is the leading dimension of U just as declared in the
</span><span class="comment">*</span><span class="comment"> in the calling subroutine. LDU.GE.3*NSHFTS-3.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> NH (input) integer scalar
</span><span class="comment">*</span><span class="comment"> NH is the number of columns in array WH available for
</span><span class="comment">*</span><span class="comment"> workspace. NH.GE.1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WH (workspace) COMPLEX*16 array of size (LDWH,NH)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDWH (input) integer scalar
</span><span class="comment">*</span><span class="comment"> Leading dimension of WH just as declared in the
</span><span class="comment">*</span><span class="comment"> calling procedure. LDWH.GE.3*NSHFTS-3.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> NV (input) integer scalar
</span><span class="comment">*</span><span class="comment"> NV is the number of rows in WV agailable for workspace.
</span><span class="comment">*</span><span class="comment"> NV.GE.1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WV (workspace) COMPLEX*16 array of size
</span><span class="comment">*</span><span class="comment"> (LDWV,3*NSHFTS-3)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDWV (input) integer scalar
</span><span class="comment">*</span><span class="comment"> LDWV is the leading dimension of WV as declared in the
</span><span class="comment">*</span><span class="comment"> in the calling subroutine. LDWV.GE.NV.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ================================================================
</span><span class="comment">*</span><span class="comment"> Based on contributions by
</span><span class="comment">*</span><span class="comment"> Karen Braman and Ralph Byers, Department of Mathematics,
</span><span class="comment">*</span><span class="comment"> University of Kansas, USA
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ============================================================
</span><span class="comment">*</span><span class="comment"> Reference:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
</span><span class="comment">*</span><span class="comment"> Algorithm Part I: Maintaining Well Focused Shifts, and
</span><span class="comment">*</span><span class="comment"> Level 3 Performance, SIAM Journal of Matrix Analysis,
</span><span class="comment">*</span><span class="comment"> volume 23, pages 929--947, 2002.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ============================================================
</span><span class="comment">*</span><span class="comment"> .. Parameters ..
</span> COMPLEX*16 ZERO, ONE
PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ),
$ ONE = ( 1.0d0, 0.0d0 ) )
DOUBLE PRECISION RZERO, RONE
PARAMETER ( RZERO = 0.0d0, RONE = 1.0d0 )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Scalars ..
</span> COMPLEX*16 ALPHA, BETA, CDUM, REFSUM
DOUBLE PRECISION H11, H12, H21, H22, SAFMAX, SAFMIN, SCL,
$ SMLNUM, TST1, TST2, ULP
INTEGER I2, I4, INCOL, J, J2, J4, JBOT, JCOL, JLEN,
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