SRC/dlaqr5.f(3)

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SRC/dlaqr5.f(3) 💅

hey bestie, welcome to the Library Functions Manual page for SRC/dlaqr5.f 🫶 This is the source page for dlaqr5, and she is literally here to perform a single small-bulge multi-shift QR sweep. No cap. ✨

🌸 NAME 🌸

SRC/dlaqr5.f — yes, that's the name, and she's the file behind dlaqr5, the routine that performs a single small-bulge multi-shift QR sweep. Periodt. 💖

🦋 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.

💖 Function/Subroutine Documentation 💖

Okay so here's the full signature in all her typed-out glory:

   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.

✨ Purpose: ✨

The source literally says:

!>
!>    DLAQR5, called by DLAQR0, performs a
!>    single small-bulge multi-shift QR sweep.
!>

So yeah: called by DLAQR0, does one small-bulge multi-shift QR sweep. Iconic. 💅

🎀 Parameters 🎀

Every single one, no skips, bestie. 💖

WANTT

WANTT is LOGICAL. It's .true. if the quasi-triangular Schur factor is being computed, and .false. otherwise. 🦋

WANTZ

WANTZ is LOGICAL. It's .true. if the orthogonal Schur factor is being computed, and .false. otherwise. ✨

KACC22

KACC22 is INTEGER with value 0, 1, or 2. It 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.
⚠️ so KACC22 = 2 is same as 1 now, but that old 2-by-2 structure exploit is no longer supported — noted, bestie.

N

N is INTEGER. N is the order of the Hessenberg matrix H upon which this subroutine operates. 🌸

KTOP

KTOP is INTEGER. 💜

KBOT

KBOT is 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

NSHFTS is INTEGER. NSHFTS gives the number of simultaneous shifts. NSHFTS must be positive and even. ✨

SR

SR is DOUBLE PRECISION array, dimension (NSHFTS).

SI

SI is 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

H is 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

LDH is INTEGER. LDH is the leading dimension of H just as declared in the calling procedure. LDH >= MAX(1,N). 💖

ILOZ

ILOZ is INTEGER.

IHIZ

IHIZ is INTEGER.

Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N. 🦋

Z

Z is 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

LDZ is INTEGER. The source literally says LDA is the leading dimension of Z just as declared in the calling procedure — yes, LDA, not LDZ, iykyk 💀 — and LDZ >= N.

V

V is DOUBLE PRECISION array, dimension (LDV,NSHFTS/2).

LDV

LDV is INTEGER. LDV is the leading dimension of V as declared in the calling procedure. LDV >= 3. ✨

U

U is DOUBLE PRECISION array, dimension (LDU,2*NSHFTS).

LDU

LDU is INTEGER. LDU is the leading dimension of U just as declared in the in the calling subroutine. LDU >= 2*NSHFTS. (yes, "in the in the" — we preserve the source's little quirks 🫶)

NV

NV is INTEGER. NV is the number of rows in WV agailable for workspace. NV >= 1. (source says agailable, bestie, not me 😭)

WV

WV is DOUBLE PRECISION array, dimension (LDWV,2*NSHFTS).

LDWV

LDWV is INTEGER. LDWV is the leading dimension of WV as declared in the in the calling subroutine. LDWV >= NV. 💜

NH

NH is INTEGER. NH is the number of columns in array WH available for workspace. NH >= 1.

WH

WH is DOUBLE PRECISION array, dimension (LDWH,NH).

LDWH

LDWH is INTEGER. Leading dimension of WH just as declared in the calling procedure. LDWH >= 2*NSHFTS. 🎀

👑 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.

💀 Author 💀

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xoxo bestie, that's the footer tea: LAPACK Version 3.12.0, page SRC/dlaqr5.f(3) 💋🫶
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