ZGELSY(3)

💅 ZGELSY(3) 💅

okay bestie, welcome to the Library Functions Manual page for ZGELSY(3) — header path and all: /home/abui...RC/zgelsy.f(3). grab a strawberry drink 🍓 because we’re about to make a rank-deficient least-squares solver feel like main character energy.

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🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zgelsy.f

that’s it, that’s the NAME section — the full source path, no cap. 💅

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🦋 SYNOPSIS 🦋

🧁 Functions/Subroutines 🧁

subroutine ZGELSY (m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work,
    lwork, rwork, info)
     ZGELSY solves overdetermined or underdetermined systems for GE
    matrices

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💖 Function/Subroutine Documentation 💖

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)

ZGELSY solves overdetermined or underdetermined systems for GE matrices. literally what it says on the tin. ✨

🎀 Purpose 🎀

okay so ZGELSY 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. yes bestie, rank-deficient is allowed, we’re not judging. 🫶

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 batch energy, honestly. 💅

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. slay. 👑

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. iconic linear algebra behavior. 🌈

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.

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🧚 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*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. it really said “your old A? gone, but make it factorization details.” 😭

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, 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. so it’s giving “where did this column come from?” receipts. 🧾

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.

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*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. 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 ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR, and ZUNMRZ.

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 the “ask nicely, don’t yell” mode. 🙈

RWORK

RWORK is DOUBLE PRECISION array, dimension (2*N).

INFO

INFO is INTEGER.

• = 0: successful exit
• < 0: if INFO = -i, the i-th argument had an illegal value

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👑 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 zgelsy.f 📍

Definition at line 210 of file zgelsy.f.

🤖 Author 🤖

Generated automatically by Doxygen for LAPACK from the source code.

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💖 cute sign-off: source/version is LAPACK Version 3.12.0, and the footer path is /home/abui...RC/zgelsy.f(3). slay, and may your rank always be effective. ✨