SRC/dggglm.f(3)

💖 NAME

SRC/dggglm.f is the name section, bestie — yes, the whole thing. This is the Library Functions Manual page for SRC/dggglm.f(3), and the star of the show is the DGGGLM subroutine. Periodt. 💅

🦋 SYNOPSIS

🧚 Functions/Subroutines

subroutine dggglm (n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info)
    DGGGLM

🌈 Function/Subroutine Documentation

🫶 subroutine dggglm

subroutine dggglm (integer n, integer m, integer p, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) d, double precision, dimension( * ) x, double precision, dimension( * ) y, double precision, dimension( * ) work, integer lwork, integer info)
    DGGGLM

DGGGLM is literally the main character here. ✨

✨ Purpose ✨

Okay so DGGGLM solves a general Gauss-Markov linear model (GLM) problem, and the vibe is constrained minimization:

        minimize || y ||_2   subject to   d = A*x + B*y
            x

Here A is an N-by-M matrix, B is an N-by-P matrix, and d is a given N-vector. The assumptions are 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. They’re 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.

⚠️ bestie, warning: Callers of this subroutine should note that the singularity/rank-deficiency checks implemented in this subroutine are rudimentary. The DTRTRS subroutine called by this subroutine only signals a failure due to singularity if the problem is exactly singular.

😭 also important: It is conceivable for one (or more) of the factors involved in the generalized QR factorization of the pair (A, B) 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.

🎀 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N). On entry, D is the left hand side of the GLM equation. On exit, D is destroyed. 💀
• X — X is DOUBLE PRECISION array, dimension (M).
• Y — Y is DOUBLE PRECISION array, dimension (P). On exit, X and Y are the solutions of the GLM problem.
• WORK — WORK is DOUBLE PRECISION 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 DGEQRF, SGERQF, DORMQR and SORMRQ.
  • 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 A in the generalized QR factorization of the pair (A, B) is exactly 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 exactly singular, so that rank( A B ) < N; 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 193 of file dggglm.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

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That’s the page, bestie! 🫶 It’s from LAPACK, Version 3.12.0, and the source/page is SRC/dggglm.f(3). The footer didn’t print a date, so I’m not inventing one — just giving you the exact version and source vibes. ✨💜