SRC/cgels.f(3)

Bestie, this is SRC/cgels.f(3) from the Library Functions Manual — welcome to the LAPACK corner of the group chat. The whole page is about cgels.f, a Fortran subroutine with main-character energy: it solves overdetermined or underdetermined complex linear systems for GE matrices. 💖

🌸 NAME 🌸

SRC/cgels.f — that's the name. Short, iconic, and ready to solve.

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

subroutine cgels (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
     CGELS solves overdetermined or underdetermined systems for GE
    matrices

🫶 Function/Subroutine Documentation 🫶

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

CGELS solves overdetermined or underdetermined systems for GE matrices — that's the whole mood, and honestly it ate. 💅

💖 Purpose 💖

Okay so CGELS solves overdetermined or underdetermined complex linear systems involving an M-by-N matrix A, or its conjugate-transpose, using a QR or LQ factorization of A. It's giving "linear algebra but make it efficient." ✨

Important caveat, bestie: A is assumed to have full rank, and the protection against rank-deficient matrices is, like, rudimentary. This subroutine only detects exact rank-deficiency — that's when a diagonal element of the triangular factor of A is exactly zero.

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 those almost-rank-deficient matrices may be less accurate due to a loss of numerical precision. No cap, that's the warning. 😭

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 = 'C' and m >= n: find the minimum norm solution of an underdetermined system A**H * X = B.
4. If TRANS = 'C' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem

minimize || B - A**H * 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. Slay, batch processing. 🦋

🎀 Parameters 🎀

TRANS

TRANS is CHARACTER*1.

• = 'N': the linear system involves A;
• = 'C': the linear system involves A**H.

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.

• if M >= N, A is overwritten by details of its QR factorization as returned by CGEQRF;
• if M < N, A is overwritten by details of its LQ factorization as returned by CGELQF.

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 matrix B of right hand side vectors, stored columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS if TRANS = 'C'.

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 the modulus 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 = 'C' and m >= n, rows 1 to M of B contain the minimum norm solution vectors;
• if TRANS = 'C' 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 the modulus 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 COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. Iconic little workspace receipt. 🧾

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.

Workspace query moment: 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.

👑 Author 👑

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

Definition at line 190 of file cgels.f.

🌈 Author 🌈

Generated automatically by Doxygen for LAPACK from the source code.

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That's the whole page, bestie — every parameter, every caveat, every little workspace detail. Cute sign-off and footer receipt: LAPACK Version 3.12.0, file SRC/cgels.f(3). Date: not shown in the original footer, so no invented date, no cap. Auto-generated by Doxygen for LAPACK, now fully translated into group-chat energy. 💅✨🎀