ZGGGLM(3)

💅 ZGGGLM(3) — bestie edition

okay so you've stumbled into zggglm.f from the LAPACK 3.12.0 source tree, and honestly? this one's kind of a baddie. it lives at:

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

it's a Library Functions Manual situation, and the whole file is just here to serve up one single icon of a routine. slay.

---

✨ SYNOPSIS ✨

Functions/Subroutines 🫶

• subroutine ZGGGLM (n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info) — the one, the only, ZGGGLM 🎀

---

🌸 Function/Subroutine Documentation 🌸

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

ZGGGLM. that's her name. 💜

✨ Purpose ✨

!>
!> ZGGGLM solves a general Gauss-Markov linear model (GLM) problem:
!>
!>         minimize || y ||_2   subject to   d = A*x + B*y
!>             x
!>
!> where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
!> given N-vector. It is assumed that M <= N <= M+P, and
!>
!>            rank(A) = M    and    rank( A B ) = N.
!>
!> Under these assumptions, the constrained equation is always
!> consistent, and there is a unique solution x and a minimal 2-norm
!> solution y, which is obtained using a generalized QR factorization
!> of the matrices (A, B) given by
!>
!>    A = Q*(R),   B = Q*T*Z.
!>          (0)
!>
!> In particular, if matrix B is square nonsingular, then the problem
!> GLM is equivalent to the following weighted linear least squares
!> problem
!>
!>              minimize || inv(B)*(d-A*x) ||_2
!>                  x
!>
!> where inv(B) denotes the inverse of B.
!>

so basically: you're minimizing || y ||_2 over x, subject to d = A*x + B*y, no cap. 🔥

💖 Parameters 💖

N

!>          N is INTEGER
!>          The number of rows of the matrices A and B.  N >= 0.
!>

M

!>          M is INTEGER
!>          The number of columns of the matrix A.  0 <= M <= N.
!>

P

!>          P is INTEGER
!>          The number of columns of the matrix B.  P >= N-M.
!>

A

!>          A is COMPLEX*16 array, dimension (LDA,M)
!>          On entry, the N-by-M matrix A.
!>          On exit, the upper triangular part of the array A contains
!>          the M-by-M upper triangular matrix R.
!>

LDA

!>          LDA is INTEGER
!>          The leading dimension of the array A. LDA >= max(1,N).
!>

B

!>          B is COMPLEX*16 array, dimension (LDB,P)
!>          On entry, the N-by-P matrix B.
!>          On exit, if N <= P, the upper triangle of the subarray
!>          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
!>          if N > P, the elements on and above the (N-P)th subdiagonal
!>          contain the N-by-P upper trapezoidal matrix T.
!>

LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B. LDB >= max(1,N).
!>

D

!>          D is COMPLEX*16 array, dimension (N)
!>          On entry, D is the left hand side of the GLM equation.
!>          On exit, D is destroyed.
!>

⚠️ bestie listen. D gets destroyed on exit. she does not survive. do not come looking for her later, she's gone 💀

X

!>          X is COMPLEX*16 array, dimension (M)
!>

Y

!>          Y is COMPLEX*16 array, dimension (P)
!>
!>          On exit, X and Y are the solutions of the GLM 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,N+M+P).
!>          For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
!>          where NB is an upper bound for the optimal blocksizes for
!>          ZGEQRF, ZGERQF, ZUNMQR and ZUNMRQ.
!>
!>          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.
!>

🙈 LWORK = -1 is the little "actually, just tell me how much room you need" move — she won't yell at you via XERBLA for that one. very cutesy of her.

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 A in the
!>                generalized QR factorization of the pair (A, B) is
!>                singular, so that rank(A) < M; the least squares
!>                solution could not be computed.
!>          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
!>                factor T associated with B in the generalized QR
!>                factorization of the pair (A, B) is singular, so that
!>                rank( A B ) < N; the least squares solution could not
!>                be computed.
!>

so like — 0 slays, negative means somebody passed a bad argument, 1 means R is singular so rank(A) < M and the least squares solution could not be computed 😭, and 2 means that bottom (N-M) by (N-M) chunk of T is singular so rank( A B ) < N and again, no solution for you, the least squares solution could not be computed. heartbreaking. 💔

---

👑 Author 👑

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

the whole collective. the avengers of numerical linear algebra fr fr 🌈

Definition at line 183 of file zggglm.f.

---

⭐ Author (again, yes really) ⭐

Generated automatically by Doxygen for LAPACK from the source code. she didn't write herself, bestie, the robots did it 🧚

---

✿ that's the whole page ✿ — LAPACK, Version 3.12.0, straight out of /home/abui...RC/zggglm.f(3). go forth and solve your GLM problems, you gorgeous genius 🎀💖🦋