SRC/sgelst.f(3)

💖 SRC/sgelst.f 💖

okay so this is the SRC/sgelst.f(3) page from the Library Functions Manual ✨ bestie, we’re getting into some real linear-systems math, but I promise it’s gonna slay once we break it down 💅

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✨ NAME ✨

SRC/sgelst.f

that’s it. that’s the NAME section. minimal, iconic, no notes 🎀

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🌸 SYNOPSIS 🌸

Functions/Subroutines 🧚

• subroutine sgelst (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info) — SGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.

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🦋 Function/Subroutine Documentation 🦋

🌟 subroutine sgelst 🌟

subroutine sgelst (character trans, integer m, integer n, integer nrhs,
    real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b,
    integer ldb, real, dimension( * ) work, integer lwork, integer info)

SGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.

🎯 Purpose 🎯

okay so bestie, here’s the whole vibe of what SGELST is doing:

SGELST solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ factorization of A with compact WY representation of Q 🔥

heads up: it is assumed that A has full rank, and only a rudimentary protection against rank-deficient matrices is provided. This subroutine only detects exact rank-deficiency, where a diagonal element of the triangular factor of A is exactly zero. 😭

also lowkey important: it is conceivable for one (or more) of the diagonal elements of the triangular factor of A to be subnormally tiny numbers without this subroutine signalling an error. The solutions computed for such almost-rank-deficient matrices may be less accurate due to a loss of numerical precision. so like… keep that in mind, bestie 🫶

The following options are provided:

1. If TRANS = 'N' and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - A*X ||.
2. If TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system A * X = B.
3. If TRANS = 'T' and m >= n: find the minimum norm solution of an underdetermined system A**T * X = B.
4. If TRANS = 'T' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - A**T * X ||.

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. periodt ✨

📏 Parameters 📏

• TRANS — TRANS is CHARACTER*1
  • = 'N': the linear system involves A;
  • = 'T': the linear system involves A**T. 🌀
• 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 the matrices B and X. NRHS >=0.
• A — A is REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit:
  • if M >= N, A is overwritten by details of its QR factorization as returned by SGEQRT;
  • if M < N, A is overwritten by details of its LQ factorization as returned by SGELQT. 🦋
• LDA — LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).
• B — B is REAL array, dimension (LDB,NRHS) On entry, the matrix B of right hand side vectors, stored columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS if TRANS = 'T'. On exit, if INFO = 0, B is overwritten by the solution vectors, stored columnwise:
  • if TRANS = 'N' and m >= n, rows 1 to n of B contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements N+1 to M in that column;
  • if TRANS = 'N' and m < n, rows 1 to N of B contain the minimum norm solution vectors;
  • if TRANS = 'T' and m >= n, rows 1 to M of B contain the minimum norm solution vectors;
  • if TRANS = 'T' and m < n, rows 1 to M of B contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements M+1 to N in that column. 💅
• LDB — LDB is INTEGER The leading dimension of the array B. LDB >= MAX(1,M,N).
• 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, MN + max( MN, NRHS ) ). For optimal performance, LWORK >= max( 1, (MN + max( MN, NRHS ))*NB ). where MN = min(M,N) and NB is the optimum block size. 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. 🔍
• INFO — INFO is INTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, the i-th diagonal element of the triangular factor of A is exactly zero, so that A does not have full rank; the least squares solution could not be computed. 💀

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✨ Author ✨

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

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🌈 Contributors 🌈

• November 2022, Igor Kozachenko, Computer Science Division, University of California, Berkeley 🧚

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Definition at line 201 of file sgelst.f.

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📝 Author 📝

Generated automatically by Doxygen for LAPACK from the source code.

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and that’s the whole page, bestie 🫶 LAPACK Version 3.12.0, source page SRC/sgelst.f(3), signed off with maximum main-character energy 💖✨🎀