SRC/DEPRECATED/zgelsx.f(3)

💖 NAME 💖

SRC/DEPRECATED/zgelsx.f — that's the file, bestie. This is its page in the Library Functions Manual, and it's all about the deprecated LAPACK routine zgelsx. 🫶

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✨ SYNOPSIS ✨

🌸 Functions/Subroutines 🌸

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

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🎀 Function/Subroutine Documentation 🎀

💅 subroutine zgelsx (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, double precision, dimension( * ) rwork, integer info)

subroutine zgelsx (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, double precision,
    dimension( * ) rwork, integer info)

ZGELSX solves overdetermined or underdetermined systems for GE matrices — that's the whole vibe of this icon. 👑

💖 Purpose 💖

🚨 Deprecated alert, bestie: This routine is deprecated and has been replaced by routine ZGELSY. So like… know her, respect her, but maybe use the replacement for new stuff.

ZGELSX 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. No cap. ✨

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. That's a lot of columns but she's built for it. 🫶

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. The math is mathing. 🌈

🌸 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. 🎀
• 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 >= 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,M,N). 🌟
• JPVT — JPVT is INTEGER array, dimension (N). On entry, if JPVT(i) .ne. 0, the i-th column of A is an initial column, otherwise it is a free column. Before the QR factorization of A, all initial columns are permuted to the leading positions; only the remaining free columns are moved as a result of column pivoting during the factorization. On exit, if JPVT(i) = k, then the i-th column of A*P was the k-th column of A. 🧚
• 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. 👑
• WORK — WORK is COMPLEX*16 array, dimension (min(M,N) + max( N, 2*min(M,N)+NRHS )). ⭐
• 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 😭

👑 Author 👑

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

Definition at line 182 of file zgelsx.f. 📌

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🌈 Author 🌈

Generated automatically by Doxygen for LAPACK from the source code. 🤖💅

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💖 That's the whole page, bestie! Source/version: SRC/DEPRECATED/zgelsx.f(3) and LAPACK Version 3.12.0. Date? The footer doesn't list one, so I'm not inventing a date — no cap, periodt. 🎀✨