SRC/stzrzf.f(3)

🌸 SRC/stzrzf.f — like, the library functions manual page for stzrzf (iykyk) 🌸

okay bestie so this is the SRC/stzrzf.f(3) page from the Library Functions Manual and it's literally all about one little Fortran subroutine that does one very specific slay. buckle up 💅

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✨ NAME ✨

SRC/stzrzf.f — that's it, that's the name, she's lowkey iconic for it 🎀

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✨ SYNOPSIS ✨

Functions/Subroutines

• subroutine stzrzf (m, n, a, lda, tau, work, lwork, info) — STZRZF 🦋

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✨ Function/Subroutine Documentation ✨

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

STZRZF 💖

🧚 Purpose 🧚

okay so here's the vibe, straight up:

!>
!> STZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A
!> to upper triangular form by means of orthogonal transformations.
!>
!> The upper trapezoidal matrix A is factored as
!>
!>    A = ( R  0 ) * Z,
!>
!> where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
!> triangular matrix.
!>

so she takes your real upper trapezoidal matrix A (M-by-N, and M<=N, no exceptions bestie) and reduces it to upper triangular form using orthogonal transformations. the glow-up is real ✨

🫶 Parameters 🫶

• M

!>          M is INTEGER
!>          The number of rows of the matrix A.  M >= 0.
!>

• N

!>          N is INTEGER
!>          The number of columns of the matrix A.  N >= M.
!>

• A

!>          A is REAL array, dimension (LDA,N)
!>          On entry, the leading M-by-N upper trapezoidal part of the
!>          array A must contain the matrix to be factorized.
!>          On exit, the leading M-by-M upper triangular part of A
!>          contains the upper triangular matrix R, and elements M+1 to
!>          N of the first M rows of A, with the array TAU, represent the
!>          orthogonal matrix Z as a product of M elementary reflectors.
!>

she goes in as your trapezoidal situation and comes out holding R in the leading M-by-M upper triangular part 💅 and the rest? elements M+1 to N of the first M rows of A, together with TAU, are out here representing the orthogonal matrix Z as a product of M elementary reflectors. main character behavior fr fr.

• LDA

!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,M).
!>

• TAU

!>          TAU is REAL array, dimension (M)
!>          The scalar factors of the elementary reflectors.
!>

• WORK

!>          WORK is REAL array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>

• LWORK

!>          LWORK is INTEGER
!>          The dimension of the array WORK.  LWORK >= max(1,M).
!>          For optimum performance LWORK >= M*NB, where NB is
!>          the optimal blocksize.
!>
!>          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.
!>

ngl the LWORK = -1 thing is a whole mood — you're not actually asking her to do the work, you're just asking her how much space she'd want. she answers in WORK(1) and XERBLA stays completely quiet about it. no drama. 🎀

• INFO

!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>

= 0 means successful exit, we ate and left no crumbs 🔥. < 0 means INFO = -i and the i-th argument had an illegal value — someone did NOT read the constraints, and honestly? embarrassing for them 🙈.

👑 Author 👑

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

🍓 Contributors: 🍓

• A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

💜 Further Details: 💜

okay this part is the lore, stay with me:

!>
!>  The N-by-N matrix Z can be computed by
!>
!>     Z =  Z(1)*Z(2)* ... *Z(M)
!>
!>  where each N-by-N Z(k) is given by
!>
!>     Z(k) = I - tau(k)*v(k)*v(k)**T
!>
!>  with v(k) is the kth row vector of the M-by-N matrix
!>
!>     V = ( I   A(:,M+1:N) )
!>
!>  I is the M-by-M identity matrix, A(:,M+1:N)
!>  is the output stored in A on exit from STZRZF,
!>  and tau(k) is the kth element of the array TAU.
!>
!>

so yeah, Z is just Z(1)*Z(2)* ... *Z(M), each Z(k) = I - tau(k)*v(k)*v(k)**T, with v(k) being the kth row vector of V = ( I A(:,M+1:N) ). I is the M-by-M identity matrix (yes that identity matrix 💅), A(:,M+1:N) is the output stored in A on exit from STZRZF, and tau(k) is the kth element of TAU. it's giving choreography. it's giving coordinated. periodt. 🌈

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Definition at line 150 of file stzrzf.f. 📍

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✨ Author ✨

Generated automatically by Doxygen for LAPACK from the source code. (yes she was literally born this way, no cap 🤖✨)

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and that's the whole page bestie!! 🫶 this tea was served by LAPACK, Version 3.12.0, dated SRC/stzrzf.f(3) — mwah, go reduce something trapezoidal, ily, byeeee 💖🎀🦋