/home/abui...L/sgeqrf.f(3)

🎀 /home/abui...L/sgeqrf.f(3) — Library Functions Manual 🎀

okay bestie so this is a page from the LAPACK manual — Version 3.12.0 — and it's lowkey one of the most important routines in the whole factorization-verse. Let's get into it 💅

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✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/VARIANTS/qr/LL/sgeqrf.f

yeah that's the whole path, that's the name, she's that girl 💀 no short nickname, the path IS the brand. it's giving "main character energy" in a build directory.

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🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine SGEQRF (m, n, a, lda, tau, work, lwork, info)

SGEQRF VARIANT: left-looking Level 3 BLAS version of the algorithm. — periodt. this is the left-looking, Level 3 BLAS variant, not the regular one. iykyk 🔥

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🦋 Function/Subroutine Documentation 🦋

subroutine SGEQRF (integer m, integer n, real, dimension( lda, * ) a,
    integer lda, real, dimension( * ) tau, real, dimension( * ) work,
    integer lwork, integer info)

SGEQRF VARIANT: left-looking Level 3 BLAS version of the algorithm.

💖 Purpose 💖

okay so here's the vibe: SGEQRF computes a QR factorization of a real M-by-N matrix A: and the whole glow-up is

A = Q * R.

and like — this is the left-looking Level 3 BLAS version of the algorithm. 🫶 that's literally the whole point, she does the QR thing but left-looking, and she does it with Level 3 BLAS, slay.

👑 Parameters 👑

• M — M is INTEGER. the number of rows of the matrix A. requirement: M >= 0. no negative rows allowed bestie, we don't do that here ✨
• N — N is INTEGER. the number of columns of the matrix A. requirement: N >= 0. same energy, N >= 0 or nothing 🌈
• A — A is REAL array, dimension (LDA,N). on entry: it's the M-by-N matrix A. on exit: the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (and yes, R is upper triangular if m >= n); the elements below the diagonal, together with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details 👀). she transforms, she doesn't disappear 💅
• LDA — LDA is INTEGER. the leading dimension of the array A. requirement: LDA >= max(1,M). respect the stride, always 🎀
• TAU — TAU is REAL array, dimension (min(M,N)). the scalar factors of the elementary reflectors (again — see Further Details, it's all there). these lil scalars are the whole personality of Q ngl 💜
• WORK — WORK is REAL array, dimension (MAX(1,LWORK)). on exit: if INFO = 0, then WORK(1) returns the optimal LWORK. love a workspace that reports back 🫶
• LWORK — LWORK is INTEGER.

it's the dimension of the array WORK. the rules: LWORK >= 1 if MIN(M,N) = 0, otherwise the dimension can be divided into three parts. THREE parts. not two. not four. three. 🔥

1. ✨ the part for the triangular factor T. if the very last T is not bigger than any of the rest, then this part is NB x ceiling(K/NB), otherwise, NB x (K-NT), where K = min(M,N) and NT is the dimension of the very last T
2. ✨ the part for the very last T when T is bigger than any of the rest T. the size of this part is NT x NT, where NT = K - ceiling ((K-NX)/NB) x NB, where K = min(M,N), and NX is calculated by:

NX = MAX( 0, ILAENV( 3, 'SGEQRF', ' ', M, N, -1, -1 ) )

3. ✨ the part for dlarfb — size is max((N-M)*K, (N-M)*NB, K*NB, NB*NB)

so, and this is the moment, LWORK = part1 + part2 + part3 🧮 ate.

if LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. so don't come for her about LWORK when you asked for the query. she said what she said 💀

• INFO — INFO is INTEGER. the receipts:
  • = 0: successful exit 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value 🙈

🌟 Author 🌟

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

the whole squad, the founding mothers and fathers of this factorization fr fr 👑

📅 Date 📅

December 2016

(iconic, vintage, still serving 💅)

🧚 Further Details 🧚

so the matrix Q is represented as a product of elementary reflectors:

Q = H(1) H(2) . . . H(k), where k = min(m,n).

and each H(i) has the form:

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i), and tau in TAU(i). 🦋 look at her. structured. iconic. no cap.

⭐ Definition at line 151 of file sgeqrf.f. ⭐

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💫 Sign-off 💫

this page was Generated automatically by Doxygen for LAPACK from the source code — so like, a robot wrote it, but i'm here to make it cute 💖

LAPACK — Version 3.12.0 🎀

okay bye bestie, go factor some QR, drink water, and remember: LWORK = part1 + part2 + part3 🌈💜