SRC/dgelsy.f(3)

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version 3.12.1 · 7 versions
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💅 SRC/dgelsy.f — from the Library Functions Manual, bestie

okay so picture this: it's a Fortran file, it lives in SRC/, and it's literally the moment dgelsy became a whole legend. this is the manual page for it, and we're gonna get into every single detail together. no skimming, no "you get the gist" — we're doing this properly 🫶

✨ NAME ✨

SRC/dgelsy.f — that's the file, that's the name, that's the whole vibe. periodt.

🌸 SYNOPSIS 🌸

Functions/Subroutines

  • subroutine dgelsy (m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work, lwork, info) — DGELSY solves overdetermined or underdetermined systems for GE matrices 💖

🦋 Function/Subroutine Documentation 🦋

subroutine dgelsy — the full glow-up, signature and all

subroutine dgelsy (integer m, integer n, integer nrhs, double precision,
    dimension( lda, * ) a, integer lda, double precision, dimension( ldb, *
    ) b, integer ldb, integer, dimension( * ) jpvt, double precision rcond,
    integer rank, double precision, dimension( * ) work, integer lwork,
    integer info)

DGELSY solves overdetermined or underdetermined systems for GE matrices. iconic behaviour, honestly ✨

🎀 Purpose 🎀

bestie, here's the actual mission statement, copied exactly because I would never do you dirty:

!>
!> DGELSY computes the minimum-norm solution to a real linear least
!> squares problem:
!>     minimize || A * X - B ||
!> using a complete orthogonal factorization of A.  A is an M-by-N
!> matrix which may be rank-deficient.
!>
!> Several right hand side vectors b and solution vectors x can be
!> handled in a single call; they are stored as the columns of the
!> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
!> matrix X.
!>
!> The routine first computes a QR factorization with column pivoting:
!>     A * P = Q * [ R11 R12 ]
!>                 [  0  R22 ]
!> with R11 defined as the largest leading submatrix whose estimated
!> condition number is less than 1/RCOND.  The order of R11, RANK,
!> is the effective rank of A.
!>
!> Then, R22 is considered to be negligible, and R12 is annihilated
!> by orthogonal transformations from the right, arriving at the
!> complete orthogonal factorization:
!>    A * P = Q * [ T11 0 ] * Z
!>                [  0  0 ]
!> The minimum-norm solution is then
!>    X = P * Z**T [ inv(T11)*Q1**T*B ]
!>                 [        0         ]
!> where Q1 consists of the first RANK columns of Q.
!>
!> This routine is basically identical to the original xGELSX except
!> three differences:
!>   o The call to the subroutine xGEQPF has been substituted by the
!>     the call to the subroutine xGEQP3. This subroutine is a Blas-3
!>     version of the QR factorization with column pivoting.
!>   o Matrix B (the right hand side) is updated with Blas-3.
!>   o The permutation of matrix B (the right hand side) is faster and
!>     more simple.
!>

the way it does a QR factorization with column pivoting and then a whole complete orthogonal factorization… main character energy fr fr 👑

🌈 Parameters 🌈

M

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

how many rows A has. M >= 0, no exceptions, we don't do negative rows here 💀

N

!>          N is INTEGER
!>          The number of columns of the matrix A.  N >= 0.
!>

how many columns A has. also N >= 0. we stay non-negative, it's a whole boundary thing 🎀

NRHS

!>          NRHS is INTEGER
!>          The number of right hand sides, i.e., the number of
!>          columns of matrices B and X. NRHS >= 0.
!>

the number of right hand sides — like, the number of columns of matrices B and X. and yes, NRHS >= 0. slay.

A

!>          A is DOUBLE PRECISION array, dimension (LDA,N)
!>          On entry, the M-by-N matrix A.
!>          On exit, A has been overwritten by details of its
!>          complete orthogonal factorization.
!>

on entry it's the M-by-N matrix A. on exit? it's been overwritten by details of its complete orthogonal factorization. like, she changed. she grew. she's not the same matrix she was going in 🙈 (so don't come crying to me when your original A is gone, that's the deal)

LDA

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

leading dimension of the array A. LDA >= max(1,M) — and yes we say max(1,M) and not just M, because zero-sized things exist and we respect them 💅

B

!>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
!>          On entry, the M-by-NRHS right hand side matrix B.
!>          On exit, the N-by-NRHS solution matrix X.
!>          If M = 0 or N = 0, B is not referenced.
!>

on entry: the M-by-NRHS right hand side matrix B. on exit: the N-by-NRHS solution matrix X. glow-up of the century, honestly ✨ but listen — if M = 0 or N = 0, B is not referenced at all. she's off duty that day 🫶

LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B. LDB >= max(1,M,N).
!>

leading dimension of the array B. LDB >= max(1,M,N) — it's max of three things, not two. read it again. max(1,M,N). iykyk 😌

JPVT

!>          JPVT is INTEGER array, dimension (N)
!>          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
!>          to the front of AP, otherwise column i is a free column.
!>          On exit, if JPVT(i) = k, then the i-th column of AP
!>          was the k-th column of A.
!>

an INTEGER array, dimension (N). on entry: if JPVT(i) .ne. 0, the i-th column of A is permuted to the front of AP, otherwise column i is a free column. on exit: if JPVT(i) = k, then the i-th column of AP was the k-th column of A. so it's both the instruction manual and the receipt 🧾💖

RCOND

!>          RCOND is DOUBLE PRECISION
!>          RCOND is used to determine the effective rank of A, which
!>          is defined as the order of the largest leading triangular
!>          submatrix R11 in the QR factorization with pivoting of A,
!>          whose estimated condition number < 1/RCOND.
!>

DOUBLE PRECISION. it's the whole vibe check for the effective rank of A — defined as the order of the largest leading triangular submatrix R11 in the QR factorization with pivoting of A, whose estimated condition number is < 1/RCOND. so essentially it decides how much of A gets to sit at the cool table 💅

RANK

!>          RANK is INTEGER
!>          The effective rank of A, i.e., the order of the submatrix
!>          R11.  This is the same as the order of the submatrix T11
!>          in the complete orthogonal factorization of A.
!>          If NRHS = 0, RANK = 0 on output.
!>

INTEGER. the effective rank of A, aka the order of the submatrix R11. and this is the same as the order of the submatrix T11 in the complete orthogonal factorization of A. and — important — if NRHS = 0, then RANK = 0 on output. no right hand sides, no rank. rules are rules 💀

WORK

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

DOUBLE PRECISION array, dimension (MAX(1,LWORK)). on exit, if INFO = 0, then WORK(1) returns the optimal LWORK. she's telling you what she really needed all along 🥺

LWORK

!>          LWORK is INTEGER
!>          The dimension of the array WORK.
!>          The unblocked strategy requires that:
!>             LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
!>          where MN = min( M, N ).
!>          The block algorithm requires that:
!>             LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
!>          where NB is an upper bound on the blocksize returned
!>          by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR,
!>          and DORMRZ.
!>
!>          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.
!>

the dimension of the array WORK. two whole strategies, bestie, and they have different requirements:

  • the unblocked strategy requires LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ), where MN = min( M, N )
  • the block algorithm requires LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ), where NB is an upper bound on the blocksize returned by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR, and DORMRZ

and the fun part: 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 -1 is not a bug, it's a question 🧚✨

fr fr do the math on your LWORK before you call her. she will notice. and yes, those MAX calls are real, and MN = min( M, N ) is doing the heavy lifting.

INFO

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

INTEGER. the report card:

  • = 0: successful exit — we ate and left no crumbs 🔥
  • < 0: if INFO = -i, the i-th argument had an illegal value. so read the number, count to that argument, apologize to her 🙈

👑 Author 👑

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

the whole squad, credited, as they should be 💜

🍓 Contributors 🍓

  • A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
  • E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
  • G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

giving credit where credit is due, always 🫶

⭐ Definition at line 204 of file dgelsy.f. ⭐

that's the exact spot, no guessing 💅

🌸 Author 🌸

Generated automatically by Doxygen for LAPACK from the source code. (yes, a robot wrote the original page, but this rewrite is all me, bestie 😌)

💖 little sign-off before you go: this whole thing is LAPACK, Version 3.12.0, and this particular page is SRC/dgelsy.f(3). that's the version, that's the file, that's the tea. now go forth and solve those least squares problems, you absolute icon 🎀✨🦋
📦 where this page comes from