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