gelss(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

🌸 NAME 🌸

gelss(3) — bestie, this is the Library Functions Manual page for gelss, and it's literally giving gelss: least squares using SVD, QR iteration. Lowkey iconic linear algebra behavior. 💅

✨ SYNOPSIS ✨

🎀 Functions 🎀

  • cgelss — CGELSS solves overdetermined or underdetermined systems for GE matrices
subroutine cgelss (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work,
           lwork, rwork, info)
  • dgelss — DGELSS solves overdetermined or underdetermined systems for GE matrices
subroutine dgelss (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work,
           lwork, info)
  • sgelss — SGELSS solves overdetermined or underdetermined systems for GE matrices
subroutine sgelss (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work,
           lwork, info)
  • zgelss — ZGELSS solves overdetermined or underdetermined systems for GE matrices
subroutine zgelss (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work,
           lwork, rwork, info)

💫 Detailed Description 💫

Okay so this is the detailed description corner, and the function documentation below is the main character. No cap, every parameter is about to get its moment. 🫶

🦋 Function Documentation 🦋

💅 subroutine cgelss — CGELSS solves overdetermined or underdetermined systems for GE matrices

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

Purpose:

Bestie, CGELSS computes the minimum norm solution to a complex linear least squares problem:

Minimize 2-norm(| b - A*x |).

It does this using the singular value decomposition (SVD) 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 effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value.

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 the matrices B and X. NRHS >= 0.
  • A — A is COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, the first min(m,n) rows of A are overwritten with its right singular vectors, stored rowwise.
  • LDA — LDA is INTEGER; the leading dimension of the 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, B is overwritten by the N-by-NRHS solution matrix X. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of the modulus of elements n+1:m in that column.
  • LDB — LDB is INTEGER; the leading dimension of the array B. LDB >= max(1,M,N).
  • S — S is REAL array, dimension (min(M,N)); the singular values of A in decreasing order. The condition number of A in the 2-norm = S(1)/S(min(m,n)).
  • RCOND — RCOND is REAL; RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead.
  • RANK — RANK is INTEGER; the effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1).
  • 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. LWORK >= 1, and also:
LWORK >=  2*min(M,N) + max(M,N,NRHS)

For good performance, LWORK should generally be larger.

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 (5*min(M,N)).
  • INFO — INFO is INTEGER:
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero.

Author:

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

Definition at line 176 of file cgelss.f.

💅 subroutine dgelss — DGELSS solves overdetermined or underdetermined systems for GE matrices

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

Purpose:

DGELSS computes the minimum norm solution to a real linear least squares problem:

Minimize 2-norm(| b - A*x |).

It uses the singular value decomposition (SVD) 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 effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value.

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 the 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, the first min(m,n) rows of A are overwritten with its right singular vectors, stored rowwise.
  • 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, B is overwritten by the N-by-NRHS solution matrix X. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements n+1:m in that column.
  • LDB — LDB is INTEGER; the leading dimension of the array B. LDB >= max(1,max(M,N)).
  • S — S is DOUBLE PRECISION array, dimension (min(M,N)); the singular values of A in decreasing order. The condition number of A in the 2-norm = S(1)/S(min(m,n)).
  • RCOND — RCOND is DOUBLE PRECISION; RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead.
  • RANK — RANK is INTEGER; the effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1).
  • WORK — WORK is DOUBLE PRECISION 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 >= 1, and also:
LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )

For good performance, LWORK should generally be larger.

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: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero.

Author:

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

Definition at line 170 of file dgelss.f.

💅 subroutine sgelss — SGELSS solves overdetermined or underdetermined systems for GE matrices

subroutine sgelss (integer m, integer n, integer nrhs, real, dimension( lda,
       * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real,
       dimension( * ) s, real rcond, integer rank, real, dimension( * ) work,
       integer lwork, integer info)

Purpose:

so like… SGELSS is out here computing the minimum norm solution to a real linear least squares problem, and honestly? iconic. The whole goal is:

Minimize 2-norm(| b - A*x |).

and it does that using the singular value decomposition (SVD) of A. Now A is an M-by-N matrix that may be rank-deficient — yes bestie, it can be messy like that, it handles it 💅

Also it's a multitasker: 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 queen 🎀

And the effective rank of A is figured out by treating as zero those singular values that are less than RCOND times the largest singular value — that's the whole ✨vibe check✨ for rank.

💖 Parameters (the sgelss edition, real numbers only bestie)

  • M — M is INTEGER. The number of rows of the matrix A. M >= 0. That's it, that's the row count 🙈
  • 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. Multi-RHS slay 🔥
  • A — A is REAL array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, the first min(m,n) rows of A are overwritten with its right singular vectors, stored rowwise. Like, it moves out and the SVD vectors move in 😭
  • 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 M-by-NRHS right hand side matrix B. On exit, B is overwritten by the N-by-NRHS solution matrix X. Oh and heads up: if m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements n+1:m in that column. Free leftovers, no cap 🍓
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,max(M,N)).
  • S — S is REAL array, dimension (min(M,N)). The singular values of A in decreasing order. The condition number of A in the 2-norm = S(1)/S(min(m,n)). Sorted, bestie, descending, the way nature intended ⭐
  • RCOND — RCOND is REAL. RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead. Negative rcond? machine precision. It said "I'll pick," and honestly same 💅
  • RANK — RANK is INTEGER. The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1). Main character number 👑
  • WORK — WORK is REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. It tells you what it wanted, king behavior 🫶
  • LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= 1, and also:
LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )

For good performance, LWORK should generally be larger. And this part is important bestie: 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. It's just peeking, not committing 👀

  • INFO — INFO is INTEGER:
    • = 0: successful exit ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value. Rude.
    • > 0: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero. Not us crying at the bidiagonal 💀

🌸 Author

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

Definition at line 170 of file sgelss.f.

🦋 subroutine zgelss

okay bestie, now the complex one. Deep breath. Here's the whole signature verbatim, do not touch:

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

ZGELSS solves overdetermined or underdetermined systems for GE matrices

🎀 Purpose

ZGELSS computes the minimum norm solution to a complex linear least squares problem:

Minimize 2-norm(| b - A*x |).

using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient — same energy as the real one, just with imaginary parts living their truth 🧚

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 effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value. Yes, exact same rule, it's a family trait 💜

✨ Parameters (zgelss, now with complex flavor)

  • 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 COMPLEX*16 array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, the first min(m,n) rows of A are overwritten with its right singular vectors, stored rowwise.
  • LDA — LDA is INTEGER. The leading dimension of the array 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, B is overwritten by the N-by-NRHS solution matrix X. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of the modulus of elements n+1:m in that column. Modulus! complex edition! 😭
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,M,N).
  • S — S is DOUBLE PRECISION array, dimension (min(M,N)). The singular values of A in decreasing order. The condition number of A in the 2-norm = S(1)/S(min(m,n)).
  • RCOND — RCOND is DOUBLE PRECISION. RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead.
  • RANK — RANK is INTEGER. The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1).
  • WORK — WORK is COMPLEX*16 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 >= 1, and also:
LWORK >=  2*min(M,N) + max(M,N,NRHS)

For good performance, LWORK should generally be larger. 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. Just window shopping, no commitment 💅

  • RWORK — RWORK is DOUBLE PRECISION array, dimension (5*min(M,N)). Yeah it needs a whole real scratchpad, the complex life is expensive 🙈
  • INFO — INFO is INTEGER:
    • = 0: successful exit 🌈
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero. Again with the bidiagonal. It's giving drama 💀

💖 Author

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

Definition at line 176 of file zgelss.f.

⭐ Author

Generated automatically by Doxygen for LAPACK from the source code.

and that's the whole page bestie 🫶 this was auto-generated by Doxygen for LAPACK straight from the source code, and the version tag on the bottom reads LAPACK Version 3.12.0 with gelss(3) hanging out on the far right — so like, cite it cute when you quote it, mwah 💋✨

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