DLAQR5(3)

💅 DLAQR5(3) — Library Functions Manual 💅

okay bestie, welcome to the DLAQR5(3) page — this is the LAPACK manual for dlaqr5.f, and we are literally about to get into a single small-bulge multi-shift QR sweep. No cap. 🫶

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dlaqr5.f

🦋 SYNOPSIS 🦋

Functions/Subroutines 🧚

subroutine DLAQR5 (wantt, wantz, kacc22, n, ktop, kbot, nshfts, sr, si,
    h, ldh, iloz, ihiz, z, ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh,
    ldwh)

DLAQR5 performs a single small-bulge multi-shift QR sweep. Iconic. 💅

✨ Function/Subroutine Documentation ✨

🎀 subroutine DLAQR5 🎀

subroutine DLAQR5 (logical wantt, logical wantz, integer kacc22, integer n,
    integer ktop, integer kbot, integer nshfts, double precision, dimension(
    * ) sr, double precision, dimension( * ) si, double precision,
    dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, double
    precision, dimension( ldz, * ) z, integer ldz, double precision,
    dimension( ldv, * ) v, integer ldv, double precision, dimension( ldu, *
    ) u, integer ldu, integer nv, double precision, dimension( ldwv, * ) wv,
    integer ldwv, integer nh, double precision, dimension( ldwh, * ) wh,
    integer ldwh)

DLAQR5 performs a single small-bulge multi-shift QR sweep. Yes, she said it twice. Main character energy. 👑

Purpose: 💖

DLAQR5, called by DLAQR0, performs a single small-bulge multi-shift QR sweep. That's the whole assignment, bestie — she's a helper called by DLAQR0 and she's doing one very specific sweep. Periodt. 🌈

Parameters 🎀

• WANTT — LOGICAL. WANTT = .true. if the quasi-triangular Schur factor is being computed. WANTT is set to .false. otherwise. 🦋
• WANTZ — LOGICAL. WANTZ = .true. if the orthogonal Schur factor is being computed. WANTZ is set to .false. otherwise. 🌸
• KACC22 — INTEGER with value 0, 1, or 2. Specifies the computation mode of far-from-diagonal orthogonal updates.
  • = 0: DLAQR5 does not accumulate reflections and does not use matrix-matrix multiply to update far-from-diagonal matrix entries.
  • = 1: DLAQR5 accumulates reflections and uses matrix-matrix multiply to update the far-from-diagonal matrix entries.
  • = 2: Same as KACC22 = 1. This option used to enable exploiting the 2-by-2 structure during matrix multiplications, but this is no longer supported. 💀
• N — INTEGER. N is the order of the Hessenberg matrix H upon which this subroutine operates. 📐
• KTOP — INTEGER.
• KBOT — INTEGER. These are the first and last rows and columns of an isolated diagonal block upon which the QR sweep is to be applied. It is assumed without a check that either KTOP = 1 or H(KTOP,KTOP-1) = 0 and either KBOT = N or H(KBOT+1,KBOT) = 0.
• NSHFTS — INTEGER. NSHFTS gives the number of simultaneous shifts. NSHFTS must be positive and even. ⭐
• SR — DOUBLE PRECISION array, dimension (NSHFTS).
• SI — DOUBLE PRECISION array, dimension (NSHFTS). SR contains the real parts and SI contains the imaginary parts of the NSHFTS shifts of origin that define the multi-shift QR sweep. On output SR and SI may be reordered. 🔥
• H — DOUBLE PRECISION array, dimension (LDH,N). On input H contains a Hessenberg matrix. On output a multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied to the isolated diagonal block in rows and columns KTOP through KBOT.
• LDH — INTEGER. LDH is the leading dimension of H just as declared in the calling procedure. LDH >= MAX(1,N).
• ILOZ — INTEGER.
• IHIZ — INTEGER. Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N. 🎀
• Z — DOUBLE PRECISION array, dimension (LDZ,IHIZ). If WANTZ = .TRUE., then the QR Sweep orthogonal similarity transformation is accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. If WANTZ = .FALSE., then Z is unreferenced. 💜
• LDZ — INTEGER. LDA is the leading dimension of Z just as declared in the calling procedure. LDZ >= N. (yes, the source says LDA here — iconic, we keep it 💅)
• V — DOUBLE PRECISION array, dimension (LDV,NSHFTS/2).
• LDV — INTEGER. LDV is the leading dimension of V as declared in the calling procedure. LDV >= 3.
• U — DOUBLE PRECISION array, dimension (LDU,2*NSHFTS).
• LDU — INTEGER. LDU is the leading dimension of U just as declared in the in the calling subroutine. LDU >= 2*NSHFTS. (the double "in the" is canon, bestie 🫶)
• NV — INTEGER. NV is the number of rows in WV agailable for workspace. NV >= 1. (typo "agailable" stays — it's part of the lore)
• WV — DOUBLE PRECISION array, dimension (LDWV,2*NSHFTS).
• LDWV — INTEGER. LDWV is the leading dimension of WV as declared in the in the calling subroutine. LDWV >= NV.
• NH — INTEGER. NH is the number of columns in array WH available for workspace. NH >= 1.
• WH — DOUBLE PRECISION array, dimension (LDWH,NH).
• LDWH — INTEGER. Leading dimension of WH just as declared in the calling procedure. LDWH >= 2*NSHFTS.

Heads up: KACC22 = 2 is the same as KACC22 = 1; the 2-by-2 structure exploitation during matrix multiplications used to be a thing, but this is no longer supported. Not me delivering the news, just the manpage. 💀

Also: it is assumed without a check that either KTOP = 1 or H(KTOP,KTOP-1) = 0, and either KBOT = N or H(KBOT+1,KBOT) = 0.

Author 🌈

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

Contributors: ✨

• Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
• Lars Karlsson, Daniel Kressner, and Bruno Lang
• Thijs Steel, Department of Computer science, KU Leuven, Belgium

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.
• Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed chains of bulges in multishift QR algorithms. ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).

Definition at line 262 of file dlaqr5.f 📍

Definition at line 262 of file dlaqr5.f.

Author (Doxygen's version) 👑

Generated automatically by Doxygen for LAPACK from the source code.

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okay bestie, that's the whole DLAQR5(3) page — mwah 💋. This beauty is from LAPACK, Version 3.12.0, and the source/version footer points at /home/abui...RC/dlaqr5.f(3). Periodt. 💅