SRC/zgglse.f(3)

💅 SRC/zgglse.f(3) 💅

Okay bestie, welcome to the Library Functions Manual — today's main character is SRC/zgglse.f, and the vibe is "complex linear algebra but make it cute." 🫶✨

💖 NAME 💖

SRC/zgglse.f — that's the file, that's the legend, no cap. 💅

✨ SYNOPSIS ✨

🌟 Functions/Subroutines 🌟

       subroutine zgglse (m, n, p, a, lda, b, ldb, c, d, x, work, lwork, info)

ZGGLSE solves overdetermined or underdetermined systems for OTHER matrices — yes, OTHER is literally in all caps, bestie. 😭

🦋 Function/Subroutine Documentation 🦋

   subroutine zgglse (integer m, integer n, integer p, complex*16, dimension(
       lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb,
       complex*16, dimension( * ) c, complex*16, dimension( * ) d, complex*16,
       dimension( * ) x, complex*16, dimension( * ) work, integer lwork,
       integer info)

ZGGLSE solves overdetermined or underdetermined systems for OTHER matrices — she's back, still iconic, still solving. 💖

🎀 Purpose 🎀

Okay so ZGGLSE solves the linear equality-constrained least squares (LSE) problem:

minimize || c - A*x ||_2   subject to   B*x = d

where A is an M-by-N matrix, B is a P-by-N matrix, c is a given M-vector, and d is a given P-vector. It is assumed that

P <= N <= M+P

and

rank(B) = P and  rank( (A) ) = N.
                             ( (B) )

These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized RQ factorization of the matrices (B, A) given by

B = (0 R)*Q,   A = Z*T*Q.

🚨 Caveat time, bestie: callers of this subroutine should note that the singularity/rank-deficiency checks implemented in this subroutine are rudimentary. The ZTRTRS subroutine called by this subroutine only signals a failure due to singularity if the problem is exactly singular.

It is conceivable for one (or more) of the factors involved in the generalized RQ factorization of the pair (B, A) to be subnormally close to singularity without this subroutine signalling an error. The solutions computed for such almost-rank-deficient problems may be less accurate due to a loss of numerical precision. So like... proceed with the vibe check, okay? 😭

🌸 Parameters 🌸

• M — M is INTEGER; it's the number of rows of the matrix A. M >= 0. 💅
• N — N is INTEGER; it's the number of columns of the matrices A and B. N >= 0. ✨
• P — P is INTEGER; it's the number of rows of the matrix B. 0 <= P <= N <= M+P. 🎀
• A — A is COMPLEX*16 array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix T. 🦋
• LDA — LDA is INTEGER; the leading dimension of the array A. LDA >= max(1,M). 🌟
• B — B is COMPLEX*16 array, dimension (LDB,N). On entry, the P-by-N matrix B. On exit, the upper triangle of the subarray B(1:P,N-P+1:N) contains the P-by-P upper triangular matrix R. 🔥
• LDB — LDB is INTEGER; the leading dimension of the array B. LDB >= max(1,P). 💜
• C — C is COMPLEX*16 array, dimension (M). On entry, C contains the right hand side vector for the least squares part of the LSE problem. On exit, the residual sum of squares for the solution is given by the sum of squares of elements N-P+1 to M of vector C. 🍓
• D — D is COMPLEX*16 array, dimension (P). On entry, D contains the right hand side vector for the constrained equation. On exit, D is destroyed. 💀 (RIP, but make it linear algebra.)
• X — X is COMPLEX*16 array, dimension (N). On exit, X is the solution of the LSE problem. 👑
• 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. LWORK >= max(1,M+N+P). For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, where NB is an upper bound for the optimal blocksizes for ZGEQRF, CGERQF, ZUNMQR and CUNMRQ. 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. 🙈
  • = 1: the upper triangular factor R associated with B in the generalized RQ factorization of the pair (B, A) is exactly singular, so that rank(B) < P; the least squares solution could not be computed. 😭
  • = 2: the (N-P) by (N-P) part of the upper trapezoidal factor T associated with A in the generalized RQ factorization of the pair (B, A) is exactly singular, so that the original's stacked notation says rank( (A) ) < N with ( (B) ) underneath; 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 188 of file zgglse.f.

💜 Author 💜

Generated automatically by Doxygen for LAPACK from the source code. — Doxygen did its little auto-gen thing, and honestly? Slay. ✨

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Signed, sealed, delivered with LAPACK Version 3.12.0 energy — footer source SRC/zgglse.f(3), footer date not printed in the source text (so no invented dates, bestie), periodt. 💖🎀🫶