SUBROUTINE SLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
LOGICAL WANTT, WANTZ
* ..
* .. Array Arguments ..
REAL H( LDH, * ), WI( * ), WORK( * ), WR( * ),
$ Z( LDZ, * )
* ..
*
* This subroutine implements one level of recursion for SLAQR0.
* It is a complete implementation of the small bulge multi-shift
* QR algorithm. It may be called by SLAQR0 and, for large enough
* deflation window size, it may be called by SLAQR3. This
* subroutine is identical to SLAQR0 except that it calls SLAQR2
* instead of SLAQR3.
*
* Purpose
* =======
*
* SLAQR4 computes the eigenvalues of a Hessenberg matrix H
* and, optionally, the matrices T and Z from the Schur decomposition
* H = Z T Z**T, where T is an upper quasi-triangular matrix (the
* Schur form), and Z is the orthogonal matrix of Schur vectors.
*
* Optionally Z may be postmultiplied into an input orthogonal
* matrix Q so that this routine can give the Schur factorization
* of a matrix A which has been reduced to the Hessenberg form H
* by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
*
* Arguments
* =========
*
* WANTT (input) LOGICAL
* = .TRUE. : the full Schur form T is required;
* = .FALSE.: only eigenvalues are required.
*
* WANTZ (input) LOGICAL
* = .TRUE. : the matrix of Schur vectors Z is required;
* = .FALSE.: Schur vectors are not required.
*
* N (input) INTEGER
* The order of the matrix H. N .GE. 0.
*
* ILO (input) INTEGER
* IHI (input) INTEGER
* It is assumed that H is already upper triangular in rows
* and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,
* H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
* previous call to SGEBAL, and then passed to SGEHRD when the
* matrix output by SGEBAL is reduced to Hessenberg form.
* Otherwise, ILO and IHI should be set to 1 and N,
* respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
* If N = 0, then ILO = 1 and IHI = 0.
*
* H (input/output) REAL array, dimension (LDH,N)
* On entry, the upper Hessenberg matrix H.
* On exit, if INFO = 0 and WANTT is .TRUE., then H contains
* the upper quasi-triangular matrix T from the Schur
* decomposition (the Schur form); 2-by-2 diagonal blocks
* (corresponding to complex conjugate pairs of eigenvalues)
* are returned in standard form, with H(i,i) = H(i+1,i+1)
* and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
* .FALSE., then the contents of H are unspecified on exit.
* (The output value of H when INFO.GT.0 is given under the
* description of INFO below.)
*
* This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
* j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
*
* LDH (input) INTEGER
* The leading dimension of the array H. LDH .GE. max(1,N).
*
* WR (output) REAL array, dimension (IHI)
* WI (output) REAL array, dimension (IHI)
* The real and imaginary parts, respectively, of the computed
* eigenvalues of H(ILO:IHI,ILO:IHI) are stored WR(ILO:IHI)
* and WI(ILO:IHI). If two eigenvalues are computed as a
* complex conjugate pair, they are stored in consecutive
* elements of WR and WI, say the i-th and (i+1)th, with
* WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then
* the eigenvalues are stored in the same order as on the
* diagonal of the Schur form returned in H, with
* WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
* block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
* WI(i+1) = -WI(i).
*
* 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. ILO; IHI .LE. IHIZ .LE. N.
*
* Z (input/output) REAL array, dimension (LDZ,IHI)
* If WANTZ is .FALSE., then Z is not referenced.
* If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
* replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
* orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
* (The output value of Z when INFO.GT.0 is given under
* the description of INFO below.)
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. if WANTZ is .TRUE.
* then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1.
*
* WORK (workspace/output) REAL array, dimension LWORK
* On exit, if LWORK = -1, WORK(1) returns an estimate of
* the optimal value for LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK .GE. max(1,N)
* is sufficient, but LWORK typically as large as 6*N may
* be required for optimal performance. A workspace query
* to determine the optimal workspace size is recommended.
*
* If LWORK = -1, then SLAQR4 does a workspace query.
* In this case, SLAQR4 checks the input parameters and
* estimates the optimal workspace size for the given
* values of N, ILO and IHI. The estimate is returned
* in WORK(1). No error message related to LWORK is
* issued by XERBLA. Neither H nor Z are accessed.
*
*
* INFO (output) INTEGER
* = 0: successful exit
* .GT. 0: if INFO = i, SLAQR4 failed to compute all of
* the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
* and WI contain those eigenvalues which have been
* successfully computed. (Failures are rare.)
*
* If INFO .GT. 0 and WANT is .FALSE., then on exit,
* the remaining unconverged eigenvalues are the eigen-
* values of the upper Hessenberg matrix rows and
* columns ILO through INFO of the final, output
* value of H.
*
* If INFO .GT. 0 and WANTT is .TRUE., then on exit
*
* (*) (initial value of H)*U = U*(final value of H)
*
* where U is an orthogonal matrix. The final
* value of H is upper Hessenberg and quasi-triangular
* in rows and columns INFO+1 through IHI.
*
* If INFO .GT. 0 and WANTZ is .TRUE., then on exit
*
* (final value of Z(ILO:IHI,ILOZ:IHIZ)
* = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
*
* where U is the orthogonal matrix in (*) (regard-
* less of the value of WANTT.)
*
* If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
* accessed.
*
* ================================================================
* Based on contributions by
* Karen Braman and Ralph Byers, Department of Mathematics,
* University of Kansas, USA
*
* ================================================================
* 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.
*
* K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
* Algorithm Part II: Aggressive Early Deflation, SIAM Journal
* of Matrix Analysis, volume 23, pages 948--973, 2002.
*
* ================================================================
* .. Parameters ..
*
* ==== Matrices of order NTINY or smaller must be processed by
* . SLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
*
* ==== Exceptional deflation windows: try to cure rare
* . slow convergence by increasing the size of the
* . deflation window after KEXNW iterations. =====
*
* ==== Exceptional shifts: try to cure rare slow convergence
* . with ad-hoc exceptional shifts every KEXSH iterations.
* . The constants WILK1 and WILK2 are used to form the
* . exceptional shifts. ====
*
INTEGER NTINY
PARAMETER ( NTINY = 11 )
INTEGER KEXNW, KEXSH
PARAMETER ( KEXNW = 5, KEXSH = 6 )
REAL WILK1, WILK2
PARAMETER ( WILK1 = 0.75e0, WILK2 = -0.4375e0 )
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0e0, ONE = 1.0e0 )
* ..
* .. Local Scalars ..
REAL AA, BB, CC, CS, DD, SN, SS, SWAP
INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
$ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
$ LWKOPT, NDFL, NH, NHO, NIBBLE, NMIN, NS, NSMAX,
$ NSR, NVE, NW, NWMAX, NWR
LOGICAL NWINC, SORTED
CHARACTER JBCMPZ*2
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Local Arrays ..
REAL ZDUM( 1, 1 )
* ..
* .. External Subroutines ..
EXTERNAL SLACPY, SLAHQR, SLANV2, SLAQR2, SLAQR5
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, MAX, MIN, MOD, REAL
* ..
* .. Executable Statements ..
INFO = 0
*
* ==== Quick return for N = 0: nothing to do. ====
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = ONE
RETURN
END IF
*
* ==== Set up job flags for ILAENV. ====
*
IF( WANTT ) THEN
JBCMPZ( 1: 1 ) = 'S'
ELSE
JBCMPZ( 1: 1 ) = 'E'
END IF
IF( WANTZ ) THEN
JBCMPZ( 2: 2 ) = 'V'
ELSE
JBCMPZ( 2: 2 ) = 'N'
END IF
*
* ==== Tiny matrices must use SLAHQR. ====
*
IF( N.LE.NTINY ) THEN
*
* ==== Estimate optimal workspace. ====
*
LWKOPT = 1
IF( LWORK.NE.-1 )
$ CALL SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI,
$ ILOZ, IHIZ, Z, LDZ, INFO )
ELSE
*
* ==== Use small bulge multi-shift QR with aggressive early
* . deflation on larger-than-tiny matrices. ====
*
* ==== Hope for the best. ====
*
INFO = 0
*
* ==== NWR = recommended deflation window size. At this
* . point, N .GT. NTINY = 11, so there is enough
* . subdiagonal workspace for NWR.GE.2 as required.
* . (In fact, there is enough subdiagonal space for
* . NWR.GE.3.) ====
*
NWR = ILAENV( 13, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
NW = NWR
*
* ==== NSR = recommended number of simultaneous shifts.
* . At this point N .GT. NTINY = 11, so there is at
* . enough subdiagonal workspace for NSR to be even
* . and greater than or equal to two as required. ====
*
NSR = ILAENV( 15, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
*
* ==== Estimate optimal workspace ====
*
* ==== Workspace query call to SLAQR2 ====
*
CALL SLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, LS, LD, WR, WI, H, LDH, N, H, LDH,
$ N, H, LDH, WORK, -1 )
*
* ==== Optimal workspace = MAX(SLAQR5, SLAQR2) ====
*
LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) )
*
* ==== Quick return in case of workspace query. ====
*
IF( LWORK.EQ.-1 ) THEN
WORK( 1 ) = REAL( LWKOPT )
RETURN
END IF
*
* ==== SLAHQR/SLAQR0 crossover point ====
*
NMIN = ILAENV( 12, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NMIN = MAX( NTINY, NMIN )
*
* ==== Nibble crossover point ====
*
NIBBLE = ILAENV( 14, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
NIBBLE = MAX( 0, NIBBLE )
*
* ==== Accumulate reflections during ttswp? Use block
* . 2-by-2 structure during matrix-matrix multiply? ====
*
KACC22 = ILAENV( 16, 'SLAQR4', JBCMPZ, N, ILO, IHI, LWORK )
KACC22 = MAX( 0, KACC22 )
KACC22 = MIN( 2, KACC22 )
*
* ==== NWMAX = the largest possible deflation window for
* . which there is sufficient workspace. ====
*
NWMAX = MIN( ( N-1 ) / 3, LWORK / 2 )
*
* ==== NSMAX = the Largest number of simultaneous shifts
* . for which there is sufficient workspace. ====
*
NSMAX = MIN( ( N+6 ) / 9, 2*LWORK / 3 )
NSMAX = NSMAX - MOD( NSMAX, 2 )
*
* ==== NDFL: an iteration count restarted at deflation. ====
*
NDFL = 1
*
* ==== ITMAX = iteration limit ====
*
ITMAX = MAX( 30, 2*KEXSH )*MAX( 10, ( IHI-ILO+1 ) )
*
* ==== Last row and column in the active block ====
*
KBOT = IHI
*
* ==== Main Loop ====
*
DO 80 IT = 1, ITMAX
*
* ==== Done when KBOT falls below ILO ====
*
IF( KBOT.LT.ILO )
$ GO TO 90
*
* ==== Locate active block ====
*
DO 10 K = KBOT, ILO + 1, -1
IF( H( K, K-1 ).EQ.ZERO )
$ GO TO 20
10 CONTINUE
K = ILO
20 CONTINUE
KTOP = K
*
* ==== Select deflation window size ====
*
NH = KBOT - KTOP + 1
IF( NDFL.LT.KEXNW .OR. NH.LT.NW ) THEN
*
* ==== Typical deflation window. If possible and
* . advisable, nibble the entire active block.
* . If not, use size NWR or NWR+1 depending upon
* . which has the smaller corresponding subdiagonal
* . entry (a heuristic). ====
*
NWINC = .TRUE.
IF( NH.LE.MIN( NMIN, NWMAX ) ) THEN
NW = NH
ELSE
NW = MIN( NWR, NH, NWMAX )
IF( NW.LT.NWMAX ) THEN
IF( NW.GE.NH-1 ) THEN
NW = NH
ELSE
KWTOP = KBOT - NW + 1
IF( ABS( H( KWTOP, KWTOP-1 ) ).GT.
$ ABS( H( KWTOP-1, KWTOP-2 ) ) )NW = NW + 1
END IF
END IF
END IF
ELSE
*
* ==== Exceptional deflation window. If there have
* . been no deflations in KEXNW or more iterations,
* . then vary the deflation window size. At first,
* . because, larger windows are, in general, more
* . powerful than smaller ones, rapidly increase the
* . window up to the maximum reasonable and possible.
* . Then maybe try a slightly smaller window. ====
*
IF( NWINC .AND. NW.LT.MIN( NWMAX, NH ) ) THEN
NW = MIN( NWMAX, NH, 2*NW )
ELSE
NWINC = .FALSE.
IF( NW.EQ.NH .AND. NH.GT.2 )
$ NW = NH - 1
END IF
END IF
*
* ==== Aggressive early deflation:
* . split workspace under the subdiagonal into
* . - an nw-by-nw work array V in the lower
* . left-hand-corner,
* . - an NW-by-at-least-NW-but-more-is-better
* . (NW-by-NHO) horizontal work array along
* . the bottom edge,
* . - an at-least-NW-but-more-is-better (NHV-by-NW)
* . vertical work array along the left-hand-edge.
* . ====
*
KV = N - NW + 1
KT = NW + 1
NHO = ( N-NW-1 ) - KT + 1
KWV = NW + 2
NVE = ( N-NW ) - KWV + 1
*
* ==== Aggressive early deflation ====
*
CALL SLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ,
$ IHIZ, Z, LDZ, LS, LD, WR, WI, H( KV, 1 ), LDH,
$ NHO, H( KV, KT ), LDH, NVE, H( KWV, 1 ), LDH,
$ WORK, LWORK )
*
* ==== Adjust KBOT accounting for new deflations. ====
*
KBOT = KBOT - LD
*
* ==== KS points to the shifts. ====
*
KS = KBOT - LS + 1
*
* ==== Skip an expensive QR sweep if there is a (partly
* . heuristic) reason to expect that many eigenvalues
* . will deflate without it. Here, the QR sweep is
* . skipped if many eigenvalues have just been deflated
* . or if the remaining active block is small.
*
IF( ( LD.EQ.0 ) .OR. ( ( 100*LD.LE.NW*NIBBLE ) .AND. ( KBOT-
$ KTOP+1.GT.MIN( NMIN, NWMAX ) ) ) ) THEN
*
* ==== NS = nominal number of simultaneous shifts.
* . This may be lowered (slightly) if SLAQR2
* . did not provide that many shifts. ====
*
NS = MIN( NSMAX, NSR, MAX( 2, KBOT-KTOP ) )
NS = NS - MOD( NS, 2 )
*
* ==== If there have been no deflations
* . in a multiple of KEXSH iterations,
* . then try exceptional shifts.
* . Otherwise use shifts provided by
* . SLAQR2 above or from the eigenvalues
* . of a trailing principal submatrix. ====
*
IF( MOD( NDFL, KEXSH ).EQ.0 ) THEN
KS = KBOT - NS + 1
DO 30 I = KBOT, MAX( KS+1, KTOP+2 ), -2
SS = ABS( H( I, I-1 ) ) + ABS( H( I-1, I-2 ) )
AA = WILK1*SS + H( I, I )
BB = SS
CC = WILK2*SS
DD = AA
CALL SLANV2( AA, BB, CC, DD, WR( I-1 ), WI( I-1 ),
$ WR( I ), WI( I ), CS, SN )
30 CONTINUE
IF( KS.EQ.KTOP ) THEN
WR( KS+1 ) = H( KS+1, KS+1 )
WI( KS+1 ) = ZERO
WR( KS ) = WR( KS+1 )
WI( KS ) = WI( KS+1 )
END IF
ELSE
*
* ==== Got NS/2 or fewer shifts? Use SLAHQR
* . on a trailing principal submatrix to
* . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
* . there is enough space below the subdiagonal
* . to fit an NS-by-NS scratch array.) ====
*
IF( KBOT-KS+1.LE.NS / 2 ) THEN
KS = KBOT - NS + 1
KT = N - NS + 1
CALL SLACPY( 'A', NS, NS, H( KS, KS ), LDH,
$ H( KT, 1 ), LDH )
CALL SLAHQR( .false., .false., NS, 1, NS,
$ H( KT, 1 ), LDH, WR( KS ), WI( KS ),
$ 1, 1, ZDUM, 1, INF )
KS = KS + INF
*
* ==== In case of a rare QR failure use
* . eigenvalues of the trailing 2-by-2
* . principal submatrix. ====
*
IF( KS.GE.KBOT ) THEN
AA = H( KBOT-1, KBOT-1 )
CC = H( KBOT, KBOT-1 )
BB = H( KBOT-1, KBOT )
DD = H( KBOT, KBOT )
CALL SLANV2( AA, BB, CC, DD, WR( KBOT-1 ),
$ WI( KBOT-1 ), WR( KBOT ),
$ WI( KBOT ), CS, SN )
KS = KBOT - 1
END IF
END IF
*
IF( KBOT-KS+1.GT.NS ) THEN
*
* ==== Sort the shifts (Helps a little)
* . Bubble sort keeps complex conjugate
* . pairs together. ====
*
SORTED = .false.
DO 50 K = KBOT, KS + 1, -1
IF( SORTED )
$ GO TO 60
SORTED = .true.
DO 40 I = KS, K - 1
IF( ABS( WR( I ) )+ABS( WI( I ) ).LT.
$ ABS( WR( I+1 ) )+ABS( WI( I+1 ) ) ) THEN
SORTED = .false.
*
SWAP = WR( I )
WR( I ) = WR( I+1 )
WR( I+1 ) = SWAP
*
SWAP = WI( I )
WI( I ) = WI( I+1 )
WI( I+1 ) = SWAP
END IF
40 CONTINUE
50 CONTINUE
60 CONTINUE
END IF
*
* ==== Shuffle shifts into pairs of real shifts
* . and pairs of complex conjugate shifts
* . assuming complex conjugate shifts are
* . already adjacent to one another. (Yes,
* . they are.) ====
*
DO 70 I = KBOT, KS + 2, -2
IF( WI( I ).NE.-WI( I-1 ) ) THEN
*
SWAP = WR( I )
WR( I ) = WR( I-1 )
WR( I-1 ) = WR( I-2 )
WR( I-2 ) = SWAP
*
SWAP = WI( I )
WI( I ) = WI( I-1 )
WI( I-1 ) = WI( I-2 )
WI( I-2 ) = SWAP
END IF
70 CONTINUE
END IF
*
* ==== If there are only two shifts and both are
* . real, then use only one. ====
*
IF( KBOT-KS+1.EQ.2 ) THEN
IF( WI( KBOT ).EQ.ZERO ) THEN
IF( ABS( WR( KBOT )-H( KBOT, KBOT ) ).LT.
$ ABS( WR( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN
WR( KBOT-1 ) = WR( KBOT )
ELSE
WR( KBOT ) = WR( KBOT-1 )
END IF
END IF
END IF
*
* ==== Use up to NS of the the smallest magnatiude
* . shifts. If there aren't NS shifts available,
* . then use them all, possibly dropping one to
* . make the number of shifts even. ====
*
NS = MIN( NS, KBOT-KS+1 )
NS = NS - MOD( NS, 2 )
KS = KBOT - NS + 1
*
* ==== Small-bulge multi-shift QR sweep:
* . split workspace under the subdiagonal into
* . - a KDU-by-KDU work array U in the lower
* . left-hand-corner,
* . - a KDU-by-at-least-KDU-but-more-is-better
* . (KDU-by-NHo) horizontal work array WH along
* . the bottom edge,
* . - and an at-least-KDU-but-more-is-better-by-KDU
* . (NVE-by-KDU) vertical work WV arrow along
* . the left-hand-edge. ====
*
KDU = 3*NS - 3
KU = N - KDU + 1
KWH = KDU + 1
NHO = ( N-KDU+1-4 ) - ( KDU+1 ) + 1
KWV = KDU + 4
NVE = N - KDU - KWV + 1
*
* ==== Small-bulge multi-shift QR sweep ====
*
CALL SLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS,
$ WR( KS ), WI( KS ), H, LDH, ILOZ, IHIZ, Z,
$ LDZ, WORK, 3, H( KU, 1 ), LDH, NVE,
$ H( KWV, 1 ), LDH, NHO, H( KU, KWH ), LDH )
END IF
*
* ==== Note progress (or the lack of it). ====
*
IF( LD.GT.0 ) THEN
NDFL = 1
ELSE
NDFL = NDFL + 1
END IF
*
* ==== End of main loop ====
80 CONTINUE
*
* ==== Iteration limit exceeded. Set INFO to show where
* . the problem occurred and exit. ====
*
INFO = KBOT
90 CONTINUE
END IF
*
* ==== Return the optimal value of LWORK. ====
*
WORK( 1 ) = REAL( LWKOPT )
*
* ==== End of SLAQR4 ====
*
END