SRC/dgels.f(3)

🌸 SRC/dgels.f 🌸

okay so this page is from the Library Functions Manual, and the main character is SRC/dgels.f. Buckle up, bestie, because this is some classic LAPACK tea. 💖

✨ NAME ✨

SRC/dgels.f — yep, that's literally the whole name section. Short, sweet, iconic. 💅

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

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

💖 Function/Subroutine Documentation 💖

🌟 subroutine dgels 🌟

   subroutine dgels (character trans, integer m, integer n, integer nrhs,
       double precision, dimension( lda, * ) a, integer lda, double precision,
       dimension( ldb, * ) b, integer ldb, double precision, dimension( * )
       work, integer lwork, integer info)

DGELS solves overdetermined or underdetermined systems for GE matrices — and yes, that's the whole vibe. ✨

🎯 Purpose 🎯

okay so DGELS is out here solving overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ factorization of A. Slay. 🦋

🚨 Real talk, bestie: 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. No cap.

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. 😭

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. Group chat energy: multiple besties, one call. 🫶

🧸 Parameters 🧸

TRANS 🌈

TRANS is CHARACTER*1

• = 'N': the linear system involves A;
• = 'T': the linear system involves A**T.

Pick your fighter. 💅

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:

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

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 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 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 >= 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.

🫶 Author 🫶

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

Definition at line 191 of file dgels.f.

🫶 Author 🫶

Generated automatically by Doxygen for LAPACK from the source code.

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And that's the whole page, bestie: this is SRC/dgels.f(3) from the Library Functions Manual, and the footer situation is LAPACK Version 3.12.0 — generated automatically by Doxygen for LAPACK from the source code, no cap. 💅✨