SUBROUTINE CLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, 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 .. COMPLEX H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * ) * .. * * This subroutine implements one level of recursion for CLAQR0. * It is a complete implementation of the small bulge multi-shift * QR algorithm. It may be called by CLAQR0 and, for large enough * deflation window size, it may be called by CLAQR3. This * subroutine is identical to CLAQR0 except that it calls CLAQR2 * instead of CLAQR3. * * Purpose * ======= * * CLAQR4 computes the eigenvalues of a Hessenberg matrix H * and, optionally, the matrices T and Z from the Schur decomposition * H = Z T Z**H, where T is an upper triangular matrix (the * Schur form), and Z is the unitary matrix of Schur vectors. * * Optionally Z may be postmultiplied into an input unitary * 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 unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H. * * 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 CGEBAL, and then passed to CGEHRD when the * matrix output by CGEBAL 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) COMPLEX 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 triangular matrix T from the Schur * decomposition (the Schur form). If INFO = 0 and WANT 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). * * W (output) COMPLEX array, dimension (N) * The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored * in W(ILO:IHI). 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 W(i) = H(i,i). * * Z (input/output) COMPLEX 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) COMPLEX 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 CLAQR4 does a workspace query. * In this case, CLAQR4 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, CLAQR4 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 a unitary matrix. The final * value of H is upper Hessenberg and 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 unitary 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 * . CLAHQR 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 PARAMETER ( WILK1 = 0.75e0 ) COMPLEX ZERO, ONE PARAMETER ( ZERO = ( 0.0e0, 0.0e0 ), $ ONE = ( 1.0e0, 0.0e0 ) ) REAL TWO PARAMETER ( TWO = 2.0e0 ) * .. * .. Local Scalars .. COMPLEX AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2 REAL S 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 .. COMPLEX ZDUM( 1, 1 ) * .. * .. External Subroutines .. EXTERNAL CLACPY, CLAHQR, CLAQR2, CLAQR5 * .. * .. Intrinsic Functions .. INTRINSIC ABS, AIMAG, CMPLX, INT, MAX, MIN, MOD, REAL, $ SQRT * .. * .. Statement Functions .. REAL CABS1 * .. * .. Statement Function definitions .. CABS1( CDUM ) = ABS( REAL( CDUM ) ) + ABS( AIMAG( CDUM ) ) * .. * .. 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 CLAHQR. ==== * IF( N.LE.NTINY ) THEN * * ==== Estimate optimal workspace. ==== * LWKOPT = 1 IF( LWORK.NE.-1 ) $ CALL CLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, 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, 'CLAQR4', 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, 'CLAQR4', 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 CLAQR2 ==== * CALL CLAQR2( WANTT, WANTZ, N, ILO, IHI, NWR+1, H, LDH, ILOZ, $ IHIZ, Z, LDZ, LS, LD, W, H, LDH, N, H, LDH, N, H, $ LDH, WORK, -1 ) * * ==== Optimal workspace = MAX(CLAQR5, CLAQR2) ==== * LWKOPT = MAX( 3*NSR / 2, INT( WORK( 1 ) ) ) * * ==== Quick return in case of workspace query. ==== * IF( LWORK.EQ.-1 ) THEN WORK( 1 ) = CMPLX( LWKOPT, 0 ) RETURN END IF * * ==== CLAHQR/CLAQR0 crossover point ==== * NMIN = ILAENV( 12, 'CLAQR4', JBCMPZ, N, ILO, IHI, LWORK ) NMIN = MAX( NTINY, NMIN ) * * ==== Nibble crossover point ==== * NIBBLE = ILAENV( 14, 'CLAQR4', 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, 'CLAQR4', 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 70 IT = 1, ITMAX * * ==== Done when KBOT falls below ILO ==== * IF( KBOT.LT.ILO ) $ GO TO 80 * * ==== 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( CABS1( H( KWTOP, KWTOP-1 ) ).GT. $ CABS1( 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 CLAQR2( WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, $ IHIZ, Z, LDZ, LS, LD, W, 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 CLAQR2 * . 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 * . CLAQR2 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, KS + 1, -2 W( I ) = H( I, I ) + WILK1*CABS1( H( I, I-1 ) ) W( I-1 ) = W( I ) 30 CONTINUE ELSE * * ==== Got NS/2 or fewer shifts? Use CLAHQR * . 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 CLACPY( 'A', NS, NS, H( KS, KS ), LDH, $ H( KT, 1 ), LDH ) CALL CLAHQR( .false., .false., NS, 1, NS, $ H( KT, 1 ), LDH, W( 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. Scale to avoid * . overflows, underflows and subnormals. * . (The scale factor S can not be zero, * . because H(KBOT,KBOT-1) is nonzero.) ==== * IF( KS.GE.KBOT ) THEN S = CABS1( H( KBOT-1, KBOT-1 ) ) + $ CABS1( H( KBOT, KBOT-1 ) ) + $ CABS1( H( KBOT-1, KBOT ) ) + $ CABS1( H( KBOT, KBOT ) ) AA = H( KBOT-1, KBOT-1 ) / S CC = H( KBOT, KBOT-1 ) / S BB = H( KBOT-1, KBOT ) / S DD = H( KBOT, KBOT ) / S TR2 = ( AA+DD ) / TWO DET = ( AA-TR2 )*( DD-TR2 ) - BB*CC RTDISC = SQRT( -DET ) W( KBOT-1 ) = ( TR2+RTDISC )*S W( KBOT ) = ( TR2-RTDISC )*S * KS = KBOT - 1 END IF END IF * IF( KBOT-KS+1.GT.NS ) THEN * * ==== Sort the shifts (Helps a little) ==== * SORTED = .false. DO 50 K = KBOT, KS + 1, -1 IF( SORTED ) $ GO TO 60 SORTED = .true. DO 40 I = KS, K - 1 IF( CABS1( W( I ) ).LT.CABS1( W( I+1 ) ) ) $ THEN SORTED = .false. SWAP = W( I ) W( I ) = W( I+1 ) W( I+1 ) = SWAP END IF 40 CONTINUE 50 CONTINUE 60 CONTINUE END IF END IF * * ==== If there are only two shifts, then use * . only one. ==== * IF( KBOT-KS+1.EQ.2 ) THEN IF( CABS1( W( KBOT )-H( KBOT, KBOT ) ).LT. $ CABS1( W( KBOT-1 )-H( KBOT, KBOT ) ) ) THEN W( KBOT-1 ) = W( KBOT ) ELSE W( KBOT ) = W( KBOT-1 ) 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 CLAQR5( WANTT, WANTZ, KACC22, N, KTOP, KBOT, NS, $ W( 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 ==== 70 CONTINUE * * ==== Iteration limit exceeded. Set INFO to show where * . the problem occurred and exit. ==== * INFO = KBOT 80 CONTINUE END IF * * ==== Return the optimal value of LWORK. ==== * WORK( 1 ) = CMPLX( LWKOPT, 0 ) * * ==== End of CLAQR4 ==== * END