SRC/sgeqrt3.f(3)

📦
version 3.12.1 · 7 versions
, in arch rolling

💅 SRC/sgeqrt3.f 💅

okay so bestie, welcome to the glow-up zone ✨ — this here's a page from the Library Functions Manual, all about a lil routine called sgeqrt3 that lives in SRC/sgeqrt3.f. she's recursive, she's real, she does QR factorization, and she's about to eat. 🍓

✨ SYNOPSIS ✨

🎀 Functions/Subroutines

  • recursive subroutine sgeqrt3 (m, n, a, lda, t, ldt, info) — SGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.

👑 Function/Subroutine Documentation 👑

🌸 recursive subroutine sgeqrt3

recursive subroutine sgeqrt3 (integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldt, * ) t, integer ldt, integer info)

SGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q. periodt. 💅

💖 Purpose:

!>
!> SGEQRT3 recursively computes a QR factorization of a real M-by-N
!> matrix A, using the compact WY representation of Q.
!>
!> Based on the algorithm of Elmroth and Gustavson,
!> IBM J. Res. Develop. Vol 44 No. 4 July 2000.
!>

so like, it takes a real M-by-N matrix A and does QR factorization on it, using the compact WY representation of Q. no big deal, just iconic behavior. and credit where it's due — this is based on the algorithm of Elmroth and Gustavson, published in IBM J. Res. Develop. Vol 44 No. 4 July 2000. 🧚

🦋 Parameters

  • M — M is INTEGER. it's the number of rows of the matrix A, and the vibe check says M >= N. 🔥
!>          M is INTEGER
!>          The number of rows of the matrix A.  M >= N.
!>
  • N — N is INTEGER. the number of columns of the matrix A. N >= 0, no negativity allowed bestie. 😭
!>          N is INTEGER
!>          The number of columns of the matrix A.  N >= 0.
!>
  • A — A is REAL array, dimension (LDA,N). on entry, it's the real M-by-N matrix A. on exit, the elements on and above the diagonal contain the N-by-N upper triangular matrix R; the elements below the diagonal are the columns of V. see below for further details, iykyk. 👀
!>          A is REAL array, dimension (LDA,N)
!>          On entry, the real M-by-N matrix A.  On exit, the elements on and
!>          above the diagonal contain the N-by-N upper triangular matrix R; the
!>          elements below the diagonal are the columns of V.  See below for
!>          further details.
!>
  • LDA — LDA is INTEGER. the leading dimension of the array A. needs LDA >= max(1,M). it's giving "respect the dimensions" energy. 💀
!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,M).
!>
  • T — T is REAL array, dimension (LDT,N). the N-by-N upper triangular factor of the block reflector. the elements on and above the diagonal contain the block reflector T; the elements below the diagonal are not used (they're just there, living their truth, unbothered 🙈). see below for further details.
!>          T is REAL array, dimension (LDT,N)
!>          The N-by-N upper triangular factor of the block reflector.
!>          The elements on and above the diagonal contain the block
!>          reflector T; the elements below the diagonal are not used.
!>          See below for further details.
!>
  • LDT — LDT is INTEGER. the leading dimension of the array T. needs LDT >= max(1,N). 🌈
!>          LDT is INTEGER
!>          The leading dimension of the array T.  LDT >= max(1,N).
!>
  • INFO — INFO is INTEGER. the whole mood report:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value — not me getting called out by an index 😭
!>          INFO is INTEGER
!>          = 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 collaborated, we love a group project that actually delivers 🫶

⭐ Further Details:

okay this is the good part, the lore drop:

!>
!>  The matrix V stores the elementary reflectors H(i) in the i-th column
!>  below the diagonal. For example, if M=5 and N=3, the matrix V is
!>
!>               V = (  1       )
!>                   ( v1  1    )
!>                   ( v1 v2  1 )
!>                   ( v1 v2 v3 )
!>                   ( v1 v2 v3 )
!>
!>  where the vi's represent the vectors which define H(i), which are returned
!>  in the matrix A.  The 1's along the diagonal of V are not stored in A.  The
!>  block reflector H is then given by
!>
!>               H = I - V * T * V**T
!>
!>  where V**T is the transpose of V.
!>
!>  For details of the algorithm, see Elmroth and Gustavson (cited above).
!>

so like — the matrix V stores the elementary reflectors H(i) in the i-th column, below the diagonal. for the example where M=5 and N=3, you get that whole triangle vibe up there 👆. the vi's are the vectors defining H(i), and they get returned in the matrix A. but heads up: the 1's along the diagonal of V are not stored in A — she's implied, she's mysterious, she doesn't need a storage slot. ✨

then the block reflector H is given by H = I - V * T * V**T, where V**T is the transpose of V. simple as that, no cap. 🦋

and for the full algorithm tea, go read Elmroth and Gustavson (cited above, bestie, scroll up 💜).

💡 also: Definition at line 131 of file sgeqrt3.f. — so if you're out here reading the actual source, that's where she lives.

🧚 Author

Generated automatically by Doxygen for LAPACK from the source code. yes, a robot wrote this page, and honestly? she ate. 🤖💅

and that's the whole page, bestie! 🎀 this was LAPACK, Version 3.12.0, for SRC/sgeqrt3.f(3) — slay responsibly, and remember: LDA >= max(1,M), always. 💖🌸

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