CGELSY(3)

okay bestie, this is CGELSY(3) from the Library Functions Manual — the LAPACK page for the complex least-squares overachiever. 💅

🌈 NAME

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/cgelsy.f

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✨ SYNOPSIS ✨

🦋 Functions/Subroutines

subroutine CGELSY (m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work,
    lwork, rwork, info)
     CGELSY solves overdetermined or underdetermined systems for GE
    matrices

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💅 Function/Subroutine Documentation 💅

subroutine CGELSY (integer m, integer n, integer nrhs, complex, dimension(
    lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb,
    integer, dimension( * ) jpvt, real rcond, integer rank, complex,
    dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer
    info)

CGELSY solves overdetermined or underdetermined systems for GE matrices — that's the whole vibe, bestie.

🎯 Purpose

okay so here's the Purpose block in its full glory, untouched, because the math is literally the main character:

!>
!> CGELSY 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.
!>

Translation: it's giving minimum-norm least-squares with a complete orthogonal factorization, and CGELSY handles rank-deficient matrices too. It's basically xGELSX with exactly those three differences. No cap. ✨

💖 Parameters

M

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

N

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

NRHS

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

A

A is COMPLEX 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.

LDA

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

B

B is COMPLEX 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.

LDB

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

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.

RCOND

RCOND is REAL. 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.

RANK

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

WORK

WORK is COMPLEX 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.

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 CGEQP3, CTZRZF, CTZRQF, CUNMQR, and CUNMRZ.

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.

RWORK

RWORK is REAL array, dimension (2*N).

INFO

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.

💜 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 cgelsy.f.

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🫶 Author

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xoxo bestie, footer sign-off: LAPACK Version 3.12.0, source file /home/abui...RC/cgelsy.f(3). 💖