SRC/zgels.f(3)

💖 SRC/zgels.f 💖

okay so this is the SRC/zgels.f page from the Library Functions Manual — yes bestie, we’re deep in the LAPACK corner of the group chat, and this one is lowkey iconic. no cap. 💅

✨ NAME ✨

SRC/zgels.f

🌸 SYNOPSIS 🌸

🧚 Functions/Subroutines 🧚

subroutine zgels (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)

ZGELS solves overdetermined or underdetermined systems for GE matrices 🫶

🛠️ Function/Subroutine Documentation 🛠️

subroutine zgels (character trans, integer m, integer n, integer nrhs,
    complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension(
    ldb, * ) b, integer ldb, complex*16, dimension( * ) work, integer lwork,
    integer info)

ZGELS solves overdetermined or underdetermined systems for GE matrices

🎯 Purpose:

okay so ZGELS solves overdetermined or underdetermined complex linear systems involving an M-by-N matrix A, or its conjugate-transpose, using a QR or LQ factorization of A. ✨

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. 🚨

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 = 'C' and m >= n: find the minimum norm solution of an underdetermined system A**H * X = B. 🦋
4. If TRANS = 'C' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem

minimize || B - A**H * 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. 🌈

🌟 Parameters 🌟

• TRANS — TRANS is CHARACTER*1
  • = 'N': the linear system involves A;
  • = 'C': the linear system involves A**H.
• 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 COMPLEX*16 array, dimension (LDA,N)
  • On entry, the M-by-N matrix A.
  • if M >= N, A is overwritten by details of its QR factorization as returned by ZGEQRF;
  • if M < N, A is overwritten by details of its LQ factorization as returned by ZGELQF.
• 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 matrix B of right hand side vectors, stored columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS if TRANS = 'C'.
  • 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 the modulus 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 = 'C' and m >= n, rows 1 to M of B contain the minimum norm solution vectors;
  • if TRANS = 'C' 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 the modulus 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 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, 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 190 of file zgels.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

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and that’s the whole vibe, bestie — the footer says LAPACK, Version 3.12.0, and this is SRC/zgels.f(3). the footer doesn’t print a separate date here, just so we’re clear, but that’s the source/version info. slay 💅✨🎀