SUBROUTINE ZLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT,
$ NV, WV, LDWV, WORK, LWORK )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
$ LDZ, LWORK, N, ND, NH, NS, NV, NW
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
COMPLEX*16 H( LDH, * ), SH( * ), T( LDT, * ), V( LDV, * ),
$ WORK( * ), WV( LDWV, * ), Z( LDZ, * )
* ..
*
* This subroutine is identical to ZLAQR3 except that it avoids
* recursion by calling ZLAHQR instead of ZLAQR4.
*
*
* ******************************************************************
* Aggressive early deflation:
*
* This subroutine accepts as input an upper Hessenberg matrix
* H and performs an unitary similarity transformation
* designed to detect and deflate fully converged eigenvalues from
* a trailing principal submatrix. On output H has been over-
* written by a new Hessenberg matrix that is a perturbation of
* an unitary similarity transformation of H. It is to be
* hoped that the final version of H has many zero subdiagonal
* entries.
*
* ******************************************************************
* WANTT (input) LOGICAL
* If .TRUE., then the Hessenberg matrix H is fully updated
* so that the triangular Schur factor may be
* computed (in cooperation with the calling subroutine).
* If .FALSE., then only enough of H is updated to preserve
* the eigenvalues.
*
* WANTZ (input) LOGICAL
* If .TRUE., then the unitary matrix Z is updated so
* so that the unitary Schur factor may be computed
* (in cooperation with the calling subroutine).
* If .FALSE., then Z is not referenced.
*
* N (input) INTEGER
* The order of the matrix H and (if WANTZ is .TRUE.) the
* order of the unitary matrix Z.
*
* KTOP (input) INTEGER
* It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
* KBOT and KTOP together determine an isolated block
* along the diagonal of the Hessenberg matrix.
*
* KBOT (input) INTEGER
* It is assumed without a check that either
* KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
* determine an isolated block along the diagonal of the
* Hessenberg matrix.
*
* NW (input) INTEGER
* Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1).
*
* H (input/output) COMPLEX*16 array, dimension (LDH,N)
* On input the initial N-by-N section of H stores the
* Hessenberg matrix undergoing aggressive early deflation.
* On output H has been transformed by a unitary
* similarity transformation, perturbed, and the returned
* to Hessenberg form that (it is to be hoped) has some
* zero subdiagonal entries.
*
* LDH (input) integer
* Leading dimension of H just as declared in the calling
* subroutine. N .LE. LDH
*
* ILOZ (input) INTEGER
* IHIZ (input) INTEGER
* Specify the rows of Z to which transformations must be
* applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N.
*
* Z (input/output) COMPLEX*16 array, dimension (LDZ,IHI)
* IF WANTZ is .TRUE., then on output, the unitary
* similarity transformation mentioned above has been
* accumulated into Z(ILOZ:IHIZ,ILO:IHI) from the right.
* If WANTZ is .FALSE., then Z is unreferenced.
*
* LDZ (input) integer
* The leading dimension of Z just as declared in the
* calling subroutine. 1 .LE. LDZ.
*
* NS (output) integer
* The number of unconverged (ie approximate) eigenvalues
* returned in SR and SI that may be used as shifts by the
* calling subroutine.
*
* ND (output) integer
* The number of converged eigenvalues uncovered by this
* subroutine.
*
* SH (output) COMPLEX*16 array, dimension KBOT
* On output, approximate eigenvalues that may
* be used for shifts are stored in SH(KBOT-ND-NS+1)
* through SR(KBOT-ND). Converged eigenvalues are
* stored in SH(KBOT-ND+1) through SH(KBOT).
*
* V (workspace) COMPLEX*16 array, dimension (LDV,NW)
* An NW-by-NW work array.
*
* LDV (input) integer scalar
* The leading dimension of V just as declared in the
* calling subroutine. NW .LE. LDV
*
* NH (input) integer scalar
* The number of columns of T. NH.GE.NW.
*
* T (workspace) COMPLEX*16 array, dimension (LDT,NW)
*
* LDT (input) integer
* The leading dimension of T just as declared in the
* calling subroutine. NW .LE. LDT
*
* NV (input) integer
* The number of rows of work array WV available for
* workspace. NV.GE.NW.
*
* WV (workspace) COMPLEX*16 array, dimension (LDWV,NW)
*
* LDWV (input) integer
* The leading dimension of W just as declared in the
* calling subroutine. NW .LE. LDV
*
* WORK (workspace) COMPLEX*16 array, dimension LWORK.
* On exit, WORK(1) is set to an estimate of the optimal value
* of LWORK for the given values of N, NW, KTOP and KBOT.
*
* LWORK (input) integer
* The dimension of the work array WORK. LWORK = 2*NW
* suffices, but greater efficiency may result from larger
* values of LWORK.
*
* If LWORK = -1, then a workspace query is assumed; ZLAQR2
* only estimates the optimal workspace size for the given
* values of N, NW, KTOP and KBOT. The estimate is returned
* in WORK(1). No error message related to LWORK is issued
* by XERBLA. Neither H nor Z are accessed.
*
* ================================================================
* Based on contributions by
* Karen Braman and Ralph Byers, Department of Mathematics,
* University of Kansas, USA
*
* ==================================================================
* .. Parameters ..
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 )
* ..
* .. Local Scalars ..
COMPLEX*16 BETA, CDUM, S, TAU
DOUBLE PRECISION FOO, SAFMAX, SAFMIN, SMLNUM, ULP
INTEGER I, IFST, ILST, INFO, INFQR, J, JW, KCOL, KLN,
$ KNT, KROW, KWTOP, LTOP, LWK1, LWK2, LWKOPT
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLABAD, ZCOPY, ZGEHRD, ZGEMM, ZLACPY, ZLAHQR,
$ ZLARF, ZLARFG, ZLASET, ZTREXC, ZUNGHR
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, INT, MAX, MIN
* ..
* .. Statement Functions ..
DOUBLE PRECISION CABS1
* ..
* .. Statement Function definitions ..
CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
* ..
* .. Executable Statements ..
*
* ==== Estimate optimal workspace. ====
*
JW = MIN( NW, KBOT-KTOP+1 )
IF( JW.LE.2 ) THEN
LWKOPT = 1
ELSE
*
* ==== Workspace query call to ZGEHRD ====
*
CALL ZGEHRD( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
LWK1 = INT( WORK( 1 ) )
*
* ==== Workspace query call to ZUNGHR ====
*
CALL ZUNGHR( JW, 1, JW-1, T, LDT, WORK, WORK, -1, INFO )
LWK2 = INT( WORK( 1 ) )
*
* ==== Optimal workspace ====
*
LWKOPT = JW + MAX( LWK1, LWK2 )
END IF
*
* ==== Quick return in case of workspace query. ====
*
IF( LWORK.EQ.-1 ) THEN
WORK( 1 ) = DCMPLX( LWKOPT, 0 )
RETURN
END IF
*
* ==== Nothing to do ...
* ... for an empty active block ... ====
NS = 0
ND = 0
IF( KTOP.GT.KBOT )
$ RETURN
* ... nor for an empty deflation window. ====
IF( NW.LT.1 )
$ RETURN
*
* ==== Machine constants ====
*
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
SAFMAX = RONE / SAFMIN
CALL DLABAD( SAFMIN, SAFMAX )
ULP = DLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( DBLE( N ) / ULP )
*
* ==== Setup deflation window ====
*
JW = MIN( NW, KBOT-KTOP+1 )
KWTOP = KBOT - JW + 1
IF( KWTOP.EQ.KTOP ) THEN
S = ZERO
ELSE
S = H( KWTOP, KWTOP-1 )
END IF
*
IF( KBOT.EQ.KWTOP ) THEN
*
* ==== 1-by-1 deflation window: not much to do ====
*
SH( KWTOP ) = H( KWTOP, KWTOP )
NS = 1
ND = 0
IF( CABS1( S ).LE.MAX( SMLNUM, ULP*CABS1( H( KWTOP,
$ KWTOP ) ) ) ) THEN
NS = 0
ND = 1
IF( KWTOP.GT.KTOP )
$ H( KWTOP, KWTOP-1 ) = ZERO
END IF
RETURN
END IF
*
* ==== Convert to spike-triangular form. (In case of a
* . rare QR failure, this routine continues to do
* . aggressive early deflation using that part of
* . the deflation window that converged using INFQR
* . here and there to keep track.) ====
*
CALL ZLACPY( 'U', JW, JW, H( KWTOP, KWTOP ), LDH, T, LDT )
CALL ZCOPY( JW-1, H( KWTOP+1, KWTOP ), LDH+1, T( 2, 1 ), LDT+1 )
*
CALL ZLASET( 'A', JW, JW, ZERO, ONE, V, LDV )
CALL ZLAHQR( .true., .true., JW, 1, JW, T, LDT, SH( KWTOP ), 1,
$ JW, V, LDV, INFQR )
*
* ==== Deflation detection loop ====
*
NS = JW
ILST = INFQR + 1
DO 10 KNT = INFQR + 1, JW
*
* ==== Small spike tip deflation test ====
*
FOO = CABS1( T( NS, NS ) )
IF( FOO.EQ.RZERO )
$ FOO = CABS1( S )
IF( CABS1( S )*CABS1( V( 1, NS ) ).LE.MAX( SMLNUM, ULP*FOO ) )
$ THEN
*
* ==== One more converged eigenvalue ====
*
NS = NS - 1
ELSE
*
* ==== One undflatable eigenvalue. Move it up out of the
* . way. (ZTREXC can not fail in this case.) ====
*
IFST = NS
CALL ZTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, INFO )
ILST = ILST + 1
END IF
10 CONTINUE
*
* ==== Return to Hessenberg form ====
*
IF( NS.EQ.0 )
$ S = ZERO
*
IF( NS.LT.JW ) THEN
*
* ==== sorting the diagonal of T improves accuracy for
* . graded matrices. ====
*
DO 30 I = INFQR + 1, NS
IFST = I
DO 20 J = I + 1, NS
IF( CABS1( T( J, J ) ).GT.CABS1( T( IFST, IFST ) ) )
$ IFST = J
20 CONTINUE
ILST = I
IF( IFST.NE.ILST )
$ CALL ZTREXC( 'V', JW, T, LDT, V, LDV, IFST, ILST, INFO )
30 CONTINUE
END IF
*
* ==== Restore shift/eigenvalue array from T ====
*
DO 40 I = INFQR + 1, JW
SH( KWTOP+I-1 ) = T( I, I )
40 CONTINUE
*
*
IF( NS.LT.JW .OR. S.EQ.ZERO ) THEN
IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
*
* ==== Reflect spike back into lower triangle ====
*
CALL ZCOPY( NS, V, LDV, WORK, 1 )
DO 50 I = 1, NS
WORK( I ) = DCONJG( WORK( I ) )
50 CONTINUE
BETA = WORK( 1 )
CALL ZLARFG( NS, BETA, WORK( 2 ), 1, TAU )
WORK( 1 ) = ONE
*
CALL ZLASET( 'L', JW-2, JW-2, ZERO, ZERO, T( 3, 1 ), LDT )
*
CALL ZLARF( 'L', NS, JW, WORK, 1, DCONJG( TAU ), T, LDT,
$ WORK( JW+1 ) )
CALL ZLARF( 'R', NS, NS, WORK, 1, TAU, T, LDT,
$ WORK( JW+1 ) )
CALL ZLARF( 'R', JW, NS, WORK, 1, TAU, V, LDV,
$ WORK( JW+1 ) )
*
CALL ZGEHRD( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
$ LWORK-JW, INFO )
END IF
*
* ==== Copy updated reduced window into place ====
*
IF( KWTOP.GT.1 )
$ H( KWTOP, KWTOP-1 ) = S*DCONJG( V( 1, 1 ) )
CALL ZLACPY( 'U', JW, JW, T, LDT, H( KWTOP, KWTOP ), LDH )
CALL ZCOPY( JW-1, T( 2, 1 ), LDT+1, H( KWTOP+1, KWTOP ),
$ LDH+1 )
*
* ==== Accumulate orthogonal matrix in order update
* . H and Z, if requested. (A modified version
* . of ZUNGHR that accumulates block Householder
* . transformations into V directly might be
* . marginally more efficient than the following.) ====
*
IF( NS.GT.1 .AND. S.NE.ZERO ) THEN
CALL ZUNGHR( JW, 1, NS, T, LDT, WORK, WORK( JW+1 ),
$ LWORK-JW, INFO )
CALL ZGEMM( 'N', 'N', JW, NS, NS, ONE, V, LDV, T, LDT, ZERO,
$ WV, LDWV )
CALL ZLACPY( 'A', JW, NS, WV, LDWV, V, LDV )
END IF
*
* ==== Update vertical slab in H ====
*
IF( WANTT ) THEN
LTOP = 1
ELSE
LTOP = KTOP
END IF
DO 60 KROW = LTOP, KWTOP - 1, NV
KLN = MIN( NV, KWTOP-KROW )
CALL ZGEMM( 'N', 'N', KLN, JW, JW, ONE, H( KROW, KWTOP ),
$ LDH, V, LDV, ZERO, WV, LDWV )
CALL ZLACPY( 'A', KLN, JW, WV, LDWV, H( KROW, KWTOP ), LDH )
60 CONTINUE
*
* ==== Update horizontal slab in H ====
*
IF( WANTT ) THEN
DO 70 KCOL = KBOT + 1, N, NH
KLN = MIN( NH, N-KCOL+1 )
CALL ZGEMM( 'C', 'N', JW, KLN, JW, ONE, V, LDV,
$ H( KWTOP, KCOL ), LDH, ZERO, T, LDT )
CALL ZLACPY( 'A', JW, KLN, T, LDT, H( KWTOP, KCOL ),
$ LDH )
70 CONTINUE
END IF
*
* ==== Update vertical slab in Z ====
*
IF( WANTZ ) THEN
DO 80 KROW = ILOZ, IHIZ, NV
KLN = MIN( NV, IHIZ-KROW+1 )
CALL ZGEMM( 'N', 'N', KLN, JW, JW, ONE, Z( KROW, KWTOP ),
$ LDZ, V, LDV, ZERO, WV, LDWV )
CALL ZLACPY( 'A', KLN, JW, WV, LDWV, Z( KROW, KWTOP ),
$ LDZ )
80 CONTINUE
END IF
END IF
*
* ==== Return the number of deflations ... ====
*
ND = JW - NS
*
* ==== ... and the number of shifts. (Subtracting
* . INFQR from the spike length takes care
* . of the case of a rare QR failure while
* . calculating eigenvalues of the deflation
* . window.) ====
*
NS = NS - INFQR
*
* ==== Return optimal workspace. ====
*
WORK( 1 ) = DCMPLX( LWKOPT, 0 )
*
* ==== End of ZLAQR2 ====
*
END