iparmq.f

          INTEGER FUNCTION IPARMQ( ISPEC, NAME, OPTS, N, ILO, IHI, LWORK )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            IHI, ILO, ISPEC, LWORK, N
          CHARACTER          NAME*( * ), OPTS*( * )
    *
    *  Purpose
    *  =======
    *
    *       This program sets problem and machine dependent parameters
    *       useful for xHSEQR and its subroutines. It is called whenever
    *       ILAENV is called with 12 <= ISPEC <= 16
    *
    *  Arguments
    *  =========
    *
    *       ISPEC  (input) integer scalar
    *              ISPEC specifies which tunable parameter IPARMQ should
    *              return.
    *
    *              ISPEC=12: (INMIN)  Matrices of order nmin or less
    *                        are sent directly to xLAHQR, the implicit
    *                        double shift QR algorithm.  NMIN must be
    *                        at least 11.
    *
    *              ISPEC=13: (INWIN)  Size of the deflation window.
    *                        This is best set greater than or equal to
    *                        the number of simultaneous shifts NS.
    *                        Larger matrices benefit from larger deflation
    *                        windows.
    *
    *              ISPEC=14: (INIBL) Determines when to stop nibbling and
    *                        invest in an (expensive) multi-shift QR sweep.
    *                        If the aggressive early deflation subroutine
    *                        finds LD converged eigenvalues from an order
    *                        NW deflation window and LD.GT.(NW*NIBBLE)/100,
    *                        then the next QR sweep is skipped and early
    *                        deflation is applied immediately to the
    *                        remaining active diagonal block.  Setting
    *                        IPARMQ(ISPEC=14) = 0 causes TTQRE to skip a
    *                        multi-shift QR sweep whenever early deflation
    *                        finds a converged eigenvalue.  Setting
    *                        IPARMQ(ISPEC=14) greater than or equal to 100
    *                        prevents TTQRE from skipping a multi-shift
    *                        QR sweep.
    *
    *              ISPEC=15: (NSHFTS) The number of simultaneous shifts in
    *                        a multi-shift QR iteration.
    *
    *              ISPEC=16: (IACC22) IPARMQ is set to 0, 1 or 2 with the
    *                        following meanings.
    *                        0:  During the multi-shift QR sweep,
    *                            xLAQR5 does not accumulate reflections and
    *                            does not use matrix-matrix multiply to
    *                            update the far-from-diagonal matrix
    *                            entries.
    *                        1:  During the multi-shift QR sweep,
    *                            xLAQR5 and/or xLAQRaccumulates reflections and uses
    *                            matrix-matrix multiply to update the
    *                            far-from-diagonal matrix entries.
    *                        2:  During the multi-shift QR sweep.
    *                            xLAQR5 accumulates reflections and takes
    *                            advantage of 2-by-2 block structure during
    *                            matrix-matrix multiplies.
    *                        (If xTRMM is slower than xGEMM, then
    *                        IPARMQ(ISPEC=16)=1 may be more efficient than
    *                        IPARMQ(ISPEC=16)=2 despite the greater level of
    *                        arithmetic work implied by the latter choice.)
    *
    *       NAME    (input) character string
    *               Name of the calling subroutine
    *
    *       OPTS    (input) character string
    *               This is a concatenation of the string arguments to
    *               TTQRE.
    *
    *       N       (input) integer scalar
    *               N is the order of the Hessenberg matrix H.
    *
    *       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.
    *
    *       LWORK   (input) integer scalar
    *               The amount of workspace available.
    *
    *  Further Details
    *  ===============
    *
    *       Little is known about how best to choose these parameters.
    *       It is possible to use different values of the parameters
    *       for each of CHSEQR, DHSEQR, SHSEQR and ZHSEQR.
    *
    *       It is probably best to choose different parameters for
    *       different matrices and different parameters at different
    *       times during the iteration, but this has not been
    *       implemented --- yet.
    *
    *
    *       The best choices of most of the parameters depend
    *       in an ill-understood way on the relative execution
    *       rate of xLAQR3 and xLAQR5 and on the nature of each
    *       particular eigenvalue problem.  Experiment may be the
    *       only practical way to determine which choices are most
    *       effective.
    *
    *       Following is a list of default values supplied by IPARMQ.
    *       These defaults may be adjusted in order to attain better
    *       performance in any particular computational environment.
    *
    *       IPARMQ(ISPEC=12) The xLAHQR vs xLAQR0 crossover point.
    *                        Default: 75. (Must be at least 11.)
    *
    *       IPARMQ(ISPEC=13) Recommended deflation window size.
    *                        This depends on ILO, IHI and NS, the
    *                        number of simultaneous shifts returned
    *                        by IPARMQ(ISPEC=15).  The default for
    *                        (IHI-ILO+1).LE.500 is NS.  The default
    *                        for (IHI-ILO+1).GT.500 is 3*NS/2.
    *
    *       IPARMQ(ISPEC=14) Nibble crossover point.  Default: 14.
    *
    *       IPARMQ(ISPEC=15) Number of simultaneous shifts, NS.
    *                        a multi-shift QR iteration.
    *
    *                        If IHI-ILO+1 is ...
    *
    *                        greater than      ...but less    ... the
    *                        or equal to ...      than        default is
    *
    *                                0               30       NS =   2+
    *                               30               60       NS =   4+
    *                               60              150       NS =  10
    *                              150              590       NS =  **
    *                              590             3000       NS =  64
    *                             3000             6000       NS = 128
    *                             6000             infinity   NS = 256
    *
    *                    (+)  By default matrices of this order are
    *                         passed to the implicit double shift routine
    *                         xLAHQR.  See IPARMQ(ISPEC=12) above.   These
    *                         values of NS are used only in case of a rare
    *                         xLAHQR failure.
    *
    *                    (**) The asterisks (**) indicate an ad-hoc
    *                         function increasing from 10 to 64.
    *
    *       IPARMQ(ISPEC=16) Select structured matrix multiply.
    *                        (See ISPEC=16 above for details.)
    *                        Default: 3.
    *
    *     ================================================================
    *     .. Parameters ..
          INTEGER            INMIN, INWIN, INIBL, ISHFTS, IACC22
          PARAMETER          ( INMIN = 12, INWIN = 13, INIBL = 14,
         $                   ISHFTS = 15, IACC22 = 16 )
          INTEGER            NMIN, K22MIN, KACMIN, NIBBLE, KNWSWP
          PARAMETER          ( NMIN = 75, K22MIN = 14, KACMIN = 14,
         $                   NIBBLE = 14, KNWSWP = 500 )
          REAL               TWO
          PARAMETER          ( TWO = 2.0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            NH, NS
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          LOG, MAX, MOD, NINT, REAL
    *     ..
    *     .. Executable Statements ..
          IF( ( ISPEC.EQ.ISHFTS ) .OR. ( ISPEC.EQ.INWIN ) .OR.
         $    ( ISPEC.EQ.IACC22 ) ) THEN
    *
    *        ==== Set the number simultaneous shifts ====
    *
             NH = IHI - ILO + 1
             NS = 2
             IF( NH.GE.30 )
         $      NS = 4
             IF( NH.GE.60 )
         $      NS = 10
             IF( NH.GE.150 )
         $      NS = MAX( 10, NH / NINT( LOG( REAL( NH ) ) / LOG( TWO ) ) )
             IF( NH.GE.590 )
         $      NS = 64
             IF( NH.GE.3000 )
         $      NS = 128
             IF( NH.GE.6000 )
         $      NS = 256
             NS = MAX( 2, NS-MOD( NS, 2 ) )
          END IF
    *
          IF( ISPEC.EQ.INMIN ) THEN
    *
    *
    *        ===== Matrices of order smaller than NMIN get sent
    *        .     to xLAHQR, the classic double shift algorithm.
    *        .     This must be at least 11. ====
    *
             IPARMQ = NMIN
    *
          ELSE IF( ISPEC.EQ.INIBL ) THEN
    *
    *        ==== INIBL: skip a multi-shift qr iteration and
    *        .    whenever aggressive early deflation finds
    *        .    at least (NIBBLE*(window size)/100) deflations. ====
    *
             IPARMQ = NIBBLE
    *
          ELSE IF( ISPEC.EQ.ISHFTS ) THEN
    *
    *        ==== NSHFTS: The number of simultaneous shifts =====
    *
             IPARMQ = NS
    *
          ELSE IF( ISPEC.EQ.INWIN ) THEN
    *
    *        ==== NW: deflation window size.  ====
    *
             IF( NH.LE.KNWSWP ) THEN
                IPARMQ = NS
             ELSE
                IPARMQ = 3*NS / 2
             END IF
    *
          ELSE IF( ISPEC.EQ.IACC22 ) THEN
    *
    *        ==== IACC22: Whether to accumulate reflections
    *        .     before updating the far-from-diagonal elements
    *        .     and whether to use 2-by-2 block structure while
    *        .     doing it.  A small amount of work could be saved
    *        .     by making this choice dependent also upon the
    *        .     NH=IHI-ILO+1.
    *
             IPARMQ = 0
             IF( NS.GE.KACMIN )
         $      IPARMQ = 1
             IF( NS.GE.K22MIN )
         $      IPARMQ = 2
    *
          ELSE
    *        ===== invalid value of ispec =====
             IPARMQ = -1
    *
          END IF
    *
    *     ==== End of IPARMQ ====
    *
          END