DGELSY(3)

🌸 DGELSY(3) — the LAPACK page, but make it make sense 💅

okay so bestie, this one lives at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dgelsy.f and it's from the Library Functions Manual. Buckle up, because this routine is genuinely iconic — it solves least-squares problems when your matrix is being ✨complicated✨ (read: rank-deficient). No cap.

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✨ 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 — that's the whole vibe. 🦋

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👑 Function/Subroutine Documentation 👑

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. Main character energy fr fr. ⭐

🎀 Purpose:

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. 😭 (yes, it can be a whole mess and DGELSY will still handle it)

Several right hand side vectors b and solution vectors x can be handled in a single call — they're stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X. Efficiency. We love to see it. 💖

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. That's the number you actually care about, iykyk. 🔥

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. Literally just math slaying. 💅

This routine is basically identical to the original xGELSX except three differences:

• 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.
• Matrix B (the right hand side) is updated with Blas-3.
• The permutation of matrix B (the right hand side) is faster and more simple.

ngl those three bullet points are the whole glow-up from xGELSX. Blas-3 everywhere, we're thriving. 🌈

🌸 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 >= 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 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. 🙈 (yes bestie, your input matrix is gone — it got a makeover)
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• 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. 👑
• LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,M,N). Note the three-way max, don't get caught lacking.
• 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. The pivot 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. Tiny RCOND = spicy tolerance, choose wisely. 🔥
• 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. Zero. Nada. 💀
• WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. Free real estate info. ✨
• 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 -1 trick is your bestie, fr)

• INFO — INFO is INTEGER. Exit status, the vibe check at the end:
  • = 0: successful exit ✨ slay
  • < 0: If INFO = -i, the i-th argument had an illegal value. 😭 something you passed was not it

💜 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 204 of file dgelsy.f.

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🦋 little heads up from me to you: this page was generated automatically by Doxygen for LAPACK straight from the source code, so nobody hand-typed any of it. Also the header and footer paths show up abbreviated as /home/abui...RC/dgelsy.f(3) in the manual — that's the doc build being cute with long paths, not a typo in your reading. 🙈

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okay that's the whole page, bestie — go forth and solve those rank-deficient systems. 💖 LAPACK, Version 3.12.0 💖 — and that /home/abui...RC/dgelsy.f(3) footer? That's just the doc build signing off. periodt. 💅🎀