iparmq(3)

🌸 NAME 🌸 — iparmq(3)

okay so bestie, welcome to iparmq(3) from the Library Functions Manual 💖 This page is literally about iparmq — aka iparmq: set parameters for hseqr. That's the whole vibe: it's the little parameter-picker for the eigenvalue girlies ✨

✨ SYNOPSIS ✨

🧚 Functions

integer function iparmq (ispec, name, opts, n, ilo, ihi, lwork)
    IPARMQ

🌈 Detailed Description 🌈

This is where the docs roll into the function documentation, bestie — keep scrolling, the details are right there 🫶

💅 Function Documentation 💅

integer function iparmq (integer ispec, character, dimension( * ) name,
    character, dimension( * ) opts, integer n, integer ilo, integer ihi,
    integer lwork)
    IPARMQ

🎯 Purpose

okay so this program sets problem and machine dependent parameters that are useful for xHSEQR and related subroutines for eigenvalue problems. It is called whenever IPARMQ is called with 12 <= ISPEC <= 16 👑

🧩 Parameters

• ISPEC — ISPEC is INTEGER. It specifies which tunable parameter IPARMQ should return. Here's the menu, bestie:
  • 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 — no cap, that's the rule 💅
  • 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 > (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/QZ sweep, blocked eigenvalue reordering, blocked Hessenberg-triangular reduction, reflections and/or rotations are not accumulated when updating the far-from-diagonal matrix entries.
    • 1: during the multi-shift QR/QZ sweep, blocked eigenvalue reordering, blocked Hessenberg-triangular reduction, reflections and/or rotations are accumulated, and matrix-matrix multiplication is used to update the far-from-diagonal matrix entries.
    • 2: during the multi-shift QR/QZ sweep, blocked eigenvalue reordering, blocked Hessenberg-triangular reduction, reflections and/or rotations are accumulated, and 2-by-2 block structure is exploited 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.)
  • ISPEC=17 (ICOST): an estimate of the relative cost of flops within the near-the-diagonal shift chase compared to flops within the BLAS calls of a QZ sweep.
• NAME — NAME is CHARACTER string. It's the name of the calling subroutine.
• OPTS — OPTS is CHARACTER string. This is a concatenation of the string arguments to TTQRE.
• N — N is INTEGER. N is the order of the Hessenberg matrix H.
• ILO — ILO is INTEGER.
• IHI — IHI is INTEGER. It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N.
• LWORK — LWORK is INTEGER. The amount of workspace available.

💖 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌟 Further Details

okay so the tea is: 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. periodt.

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. fr fr 🔥

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) <= 500 is NS. The default for (IHI-ILO+1) > 500 is 3*NS/2.
• IPARMQ(ISPEC=14) — nibble crossover point. Default: 14.
• IPARMQ(ISPEC=15) — number of simultaneous shifts, NS, for a multi-shift QR iteration.

If IHI-ILO+1 is ...

                        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.
• IPARMQ(ISPEC=17) — relative cost heuristic for blocksize selection. Expressed as a percentage. Default: 10.

Definition at line 229 of file iparmq.f.

🤖 Author

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