SRC/zgelsy.f(3)

✨ SRC/zgelsy.f(3) ✨

okay bestie, welcome to the Library Functions Manual page for SRC/zgelsy.f — yes it's a Fortran file, yes it's iconic, let's go 💅

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🌸 NAME 🌸

SRC/zgelsy.f

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🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine zgelsy (m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work,
    lwork, rwork, info)

ZGELSY solves overdetermined or underdetermined systems for GE matrices — that's the whole vibe, no cap 🔥

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💖 Function/Subroutine Documentation 💖

subroutine zgelsy (integer m, integer n, integer nrhs, complex*16,
    dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b,
    integer ldb, integer, dimension( * ) jpvt, double precision rcond,
    integer rank, complex*16, dimension( * ) work, integer lwork, double
    precision, dimension( * ) rwork, integer info)

ZGELSY solves overdetermined or underdetermined systems for GE matrices — main character energy fr fr 👑

✨ Purpose ✨

okay so here's the tea 🍓:

!>
!> ZGELSY computes the minimum-norm solution to a complex 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 unitary 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**H [ inv(T11)*Q1**H*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 permutation of matrix B (the right hand side) is faster and
!>     more simple.
!>   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.
!>

so basically it does the least-squares thing, handles rank-deficiency like a boss, and does it with Blas-3 because we're efficient in this house 💅 lowkey obsessed with how it just annihilates R12 with unitary transformations from the right 😭

🎀 Parameters 🎀

M

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

number of rows of A, and it's M >= 0 — no negatives bestie, we don't do that here 🙈

N

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

number of columns of A, N >= 0 periodt 💜

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 — i.e. how many columns B and X have. NRHS >= 0, obviously ✨

A

!>          A is COMPLEX*16 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.
!>

it's a COMPLEX*16 array, dimension (LDA,N). on entry it's the M-by-N matrix A — on exit it's been overwritten by details of its complete orthogonal factorization 👀 like it goes in as A and comes out as lore

LDA

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

leading dimension of A, LDA >= max(1,M) 🫶

B

!>          B is COMPLEX*16 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.
!>

COMPLEX*16 array, dimension (LDB,NRHS). goes in as the M-by-NRHS right hand side, comes out as the N-by-NRHS solution matrix X — glow-up of the century ngl ⭐ and if M = 0 or N = 0, B is not referenced — she doesn't even get looked at 💀

LDB

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

leading dimension of B, LDB >= max(1,M,N) — note it's max of three things, don't get it twisted 🌈

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 A*P
!>          was the k-th column of A.
!>

INTEGER array, dimension (N). on entry: if JPVT(i) .ne. 0, the i-th column of A gets 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 A*P was the k-th column of A — she keeps receipts 💅

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. used to determine 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 < 1/RCOND ✨ iykyk

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, i.e. the order of the submatrix R11. same as the order of the submatrix T11 in the complete orthogonal factorization of A 🔥 and hey, if NRHS = 0, then RANK = 0 on output — zero rights, zero rank, we love a clean slate 🎀

WORK

!>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>

COMPLEX*16 array, dimension (MAX(1,LWORK)). on exit, if INFO = 0, WORK(1) returns the optimal LWORK — she's helpful like that 💖

LWORK

!>          LWORK is INTEGER
!>          The dimension of the array WORK.
!>          The unblocked strategy requires that:
!>            LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )
!>          where MN = min(M,N).
!>          The block algorithm requires that:
!>            LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )
!>          where NB is an upper bound on the blocksize returned
!>          by ILAENV for the routines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR,
!>          and ZUNMRZ.
!>
!>          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.
!>

INTEGER, the dimension of the array WORK. ok so there's two strategies and you gotta meet the right bar bestie:

• unblocked strategy requires LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS ) where MN = min(M,N)
• block algorithm requires LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS ) where NB is an upper bound on the blocksize returned by ILAENV for the routines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR, and ZUNMRZ

and if LWORK = -1, 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 😌 she's just peeking, not committing

RWORK

!>          RWORK is DOUBLE PRECISION array, dimension (2*N)
!>

DOUBLE PRECISION array, dimension (2*N) — short and sweet 🌸

INFO

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

INTEGER:

• = 0 — successful exit, we ate and left no crumbs 👑
• < 0 — if INFO = -i, the i-th argument had an illegal value 😭

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🌈 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
• E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
• G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

Definition at line 210 of file zgelsy.f 🎯

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🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code. (yes the robot did the typing, we just did the vibes ✨)

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okay that's the whole page bestie, you're now officially fluent in ZGELSY 🫶💅

— with love, LAPACK Version 3.12.0, SRC/zgelsy.f(3) 🦋⭐💖