gelsy(3)

📦

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, in arch rolling

🎀 gelsy(3) 🎀 — from the Library Functions Manual, section 3, and yes bestie we're doing linear least squares today 💅

🌸 NAME 🌸

gelsy — gelsy: least squares using complete orthogonal factor

✨ SYNOPSIS ✨

🦋 Functions

cgelsy — CGELSY solves overdetermined or underdetermined systems for GE matrices

subroutine cgelsy (m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work, lwork, rwork, info)

dgelsy — DGELSY solves overdetermined or underdetermined systems for GE matrices

subroutine dgelsy (m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work, lwork, info)

sgelsy — SGELSY solves overdetermined or underdetermined systems for GE matrices

subroutine sgelsy (m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work, lwork, info)

zgelsy — ZGELSY solves overdetermined or underdetermined systems for GE matrices

subroutine zgelsy (m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work, lwork, rwork, info)

🌈 Detailed Description 🌈

okay so this is the overview page for all four flavors of the routine — same energy, four different precisions and number types. the actual details live in the Function Documentation section right below, so let's get into it 👑💖

💜 Function Documentation 💜

🧚 subroutine cgelsy

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

CGELSY solves overdetermined or underdetermined systems for GE matrices

Purpose:

💖 CGELSY computes the minimum-norm solution to a complex 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 (which, real talk, is the whole reason this routine exists).

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.

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 ⭐

Then, R22 is considered to be negligible, and R12 is annihilated by unitary 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**H [ inv(T11)*Q1**H*B ]
             [        0         ]

where Q1 consists of the first RANK columns of Q.

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

The permutation of matrix B (the right hand side) is faster and more simple.
The call to the subroutine xGEQPF has been substituted by 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.

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 COMPLEX 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, it eats your input, that's the deal).
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, 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).
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 A*P was the k-th column of A.
RCOND — RCOND is REAL. 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.
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.
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. The unblocked strategy requires that:

LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )

where MN = min(M,N). The block algorithm requires that:

LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )

where NB is an upper bound on the blocksize returned by ILAENV for the routines CGEQP3, CTZRZF, CTZRQF, CUNMQR, and CUNMRZ.

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. (iconic behaviour tbh ✨)

RWORK — RWORK is REAL array, dimension (2*N).
INFO — INFO is INTEGER.
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value 💀

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 210 of file cgelsy.f.

🧚 subroutine dgelsy

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

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.

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 ⭐

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.

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

The call to the subroutine xGEQPF has been substituted by 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.

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 🙈
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).

and bestie the parameter list keeps going — JPVT is up next, catch you in the next part 💅🦋

JPVT — bestie, JPVT is an 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. 💅
RCOND — RCOND is DOUBLE PRECISION. It's 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. This is the rank vibe check, no cap. 🔥
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. 🎀
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. 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.

INFO — INFO is INTEGER. = 0: successful exit. < 0: If INFO = -i, the i-th argument had an illegal value. 💀

🌟 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.

🧮 `subroutine sgelsy` — SGELSY solves overdetermined or underdetermined systems for GE matrices

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

📜 Purpose

SGELSY 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. ✨

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 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. 👑

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. 🌈

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.

🎛️ 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 REAL 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. 💅
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, 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).
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. 🎀
RCOND — RCOND is REAL. 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.
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.
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. 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 SGEQP3, STZRZF, STZRQF, SORMQR, and SORMRZ.

INFO — INFO is INTEGER. = 0: successful exit. < 0: If INFO = -i, the i-th argument had an illegal value. 🙈

🌟 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 sgelsy.f.

🧚 `subroutine zgelsy` — ZGELSY solves overdetermined or underdetermined systems for GE matrices

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

📜 Purpose

🌸 Parameters 🌸

okay bestie, here's the whole cast — every single parameter, no skips, periodt. 💖

M — okay so M is INTEGER. it's the number of rows of matrix A, and the vibe check is M >= 0. literally that's it. 🌈
N — N is INTEGER. number of columns of matrix A. rule? N >= 0. slay. ✨
NRHS — NRHS is INTEGER. the number of right hand sides, i.e., the number of columns of matrices B and X. and yes, NRHS >= 0. 💅
A — A is COMPLEX*16 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. iconic but also: don't expect your original A back. 🫶
LDA — LDA is INTEGER. the leading dimension of the array A. requirement: LDA >= max(1,M). no cap. 🌸
B — B is COMPLEX*16 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. noted. 💖
LDB — LDB is INTEGER. the leading dimension of the array B. and it's giving LDB >= max(1,M,N). periodt. 🔥
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 A*P was the k-th column of A. iykyk. 🦋
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. bestie, this is the rank-deciding tea. ☕
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. no surprises. 👑
WORK — WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)). on exit, if INFO = 0, WORK(1) returns the optimal LWORK. cute. 🎀
LWORK — LWORK is INTEGER. the dimension of the array WORK. the unblocked strategy requires:

LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )

where MN = min(M,N). the block algorithm requires:

LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )

where NB is an upper bound on the blocksize returned by ILAENV for the routines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR, and ZUNMRZ.

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. lowkey important. 💀

RWORK — RWORK is DOUBLE PRECISION array, dimension (2*N). simple. 🌈
INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. so if it's negative, check your args, bestie. 🙈

👑 Author 👑

the legends behind this routine:

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

periodt. 💅

🫶 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

they ate. 🔥

Definition at line 210 of file zgelsy.f.

🤖 Author (Doxygen's version) 🤖

okay so this part is literally:

Generated automatically by Doxygen for LAPACK from the source code.

no cap. 🧚

💖 that's the whole page, bestie! this is LAPACK Version 3.12.0, and you've been reading the gelsy(3) manual page. slay! ✨

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

gelsy(3)

🎀 gelsy(3) 🎀 — from the Library Functions Manual, section 3, and yes bestie we're doing linear least squares today 💅

🌸 NAME 🌸

✨ SYNOPSIS ✨

🦋 Functions

🌈 Detailed Description 🌈

💜 Function Documentation 💜

🧚 subroutine cgelsy

Purpose:

Parameters

Author

Contributors:

🧚 subroutine dgelsy

Purpose:

Parameters

🌟 Author

💖 Contributors:

🧮 subroutine sgelsy — SGELSY solves overdetermined or underdetermined systems for GE matrices

📜 Purpose

🎛️ Parameters

🌟 Author

💖 Contributors:

🧚 subroutine zgelsy — ZGELSY solves overdetermined or underdetermined systems for GE matrices

📜 Purpose

🌸 Parameters 🌸

👑 Author 👑

🫶 Contributors: 🫶

🤖 Author (Doxygen's version) 🤖

🧮 `subroutine sgelsy` — SGELSY solves overdetermined or underdetermined systems for GE matrices

🧚 `subroutine zgelsy` — ZGELSY solves overdetermined or underdetermined systems for GE matrices