SRC/zgelst.f(3)

🦋 SRC/zgelst.f(3) 🦋

okay so bestie, welcome to the Library Functions Manual page for SRC/zgelst.f. This one is all about ZGELST, and it is lowkey iconic: it solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q. ✨💅

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🌸 NAME 🌸

SRC/zgelst.f

💅 SYNOPSIS 💅

✨ Functions/Subroutines ✨

subroutine zgelst (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
     ZGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q.

🎀 Function/Subroutine Documentation 🎀

🌈 subroutine zgelst 🌈

subroutine zgelst (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)

ZGELST solves overdetermined or underdetermined systems for GE matrices using QR or LQ factorization with compact WY representation of Q. That's the whole assignment, and it ate. 💖

🎯 Purpose 🎯

okay so here's the lore: ZGELST solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its conjugate-transpose, using a QR or LQ factorization of A with compact WY representation of Q.

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**T * 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**T * 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.
  • On exit:
    • if M >= N, A is overwritten by details of its QR factorization as returned by ZGEQRT;
    • if M < N, A is overwritten by details of its LQ factorization as returned by ZGELQT.
• 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 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.

🧚 Contributors: 🧚

• November 2022, Igor Kozachenko,
• Computer Science Division,
• University of California, Berkeley

📌 Definition 📌

Definition at line 201 of file zgelst.f.

💫 Author 💫

Generated automatically by Doxygen for LAPACK from the source code.

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xoxo, bestie 💖 — footer moment: LAPACK Version 3.12.0, page SRC/zgelst.f(3). There's no date in this footer, just the version and file, and that's periodt. 🎀