DLAQP2RK(3)

📦 version 3.12.0, in opensuse-leap 16.0

🌸 DLAQP2RK(3) 🌸

Manual name: DLAQP2RK(3) — from the Library Functions Manual, bestie. We're folding it in right here so the header line doesn't have to do all that. 💅

🌟 NAME 🌟

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dlaqp2rk.f — that's the source path for this whole DLAQP2RK(3) situation. ✨

🌈 SYNOPSIS 🌈

🧩 Functions/Subroutines 🧩

       subroutine DLAQP2RK (m, n, nrhs, ioffset, kmax, abstol, reltol, kp1,
           maxc2nrm, a, lda, k, maxc2nrmk, relmaxc2nrmk, jpiv, tau, vn1, vn2,
           work, info)

DLAQP2RK computes truncated QR factorization with column pivoting of a real matrix block using Level 2 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B. Lowkey iconic. 👑

🧚 Function/Subroutine Documentation 🧚

       subroutine DLAQP2RK (integer m, integer n, integer nrhs, integer ioffset,
           integer kmax, double precision abstol, double precision reltol, integer
           kp1, double precision maxc2nrm, double precision, dimension( lda, * ) a,
           integer lda, integer k, double precision maxc2nrmk, double precision
           relmaxc2nrmk, integer, dimension( * ) jpiv, double precision, dimension(
           * ) tau, double precision, dimension( * ) vn1, double precision,
           dimension( * ) vn2, double precision, dimension( * ) work, integer info)

DLAQP2RK computes truncated QR factorization with column pivoting of a real matrix block using Level 2 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B. Same energy, full formal fit. ✨

🎯 Purpose 🎯

Okay so here's the tea: DLAQP2RK computes a truncated (rank K) or full rank Householder QR factorization with column pivoting of a real matrix block A(IOFFSET+1:M,1:N) as

  A * P(K) = Q(K) * R(K).

It uses Level 2 BLAS. The block A(1:IOFFSET,1:N) is accordingly pivoted, but not factorized. No cap.

The routine also overwrites the right-hand-sides matrix block B stored in A(IOFFSET+1:M,N+1:N+NRHS) with Q(K)**T * B. Slay. 💅

🎀 Parameters 🎀

  • M — M is INTEGER. It's the number of rows of matrix A, and M >= 0. Keep it cute and non-negative. 🌸
  • N — N is INTEGER. It's the number of columns of matrix A, with N >= 0. 🌼
  • NRHS — NRHS is INTEGER. The number of right hand sides, i.e. the number of columns of matrix B. Also NRHS >= 0. 💖
  • IOFFSET — IOFFSET is INTEGER. The number of rows of matrix A that must be pivoted but not factorized. IOFFSET >= 0.

IOFFSET also represents the number of columns of the whole original matrix A_orig that have been factorized in the previous steps. So it's giving "we already did some of this" energy.

  • KMAX — KMAX is INTEGER. This is the first factorization stopping criterion, and KMAX >= 0.

It's the maximum number of columns of matrix A to factorize, i.e. the maximum factorization rank.

  • a) If KMAX >= min(M-IOFFSET,N), then this stopping criterion is not used; factorize columns depending on ABSTOL and RELTOL.
  • b) If KMAX = 0, then this stopping criterion is satisfied on input and the routine exits immediately. That means the factorization is not performed, and the matrices A and B and the arrays TAU, IPIV are not modified. Important detail, bestie. 🫶
  • ABSTOL — ABSTOL is DOUBLE PRECISION, cannot be NaN. This is the second factorization stopping criterion.

It's the absolute tolerance (stopping threshold) for maximum column 2-norm of the residual matrix. The algorithm converges (stops the factorization) when the maximum column 2-norm of the residual matrix is less than or equal to ABSTOL.

  • a) If ABSTOL < 0.0, then this stopping criterion is not used; the routine factorizes columns depending on KMAX and RELTOL. This includes the case ABSTOL = -Inf.
  • b) If 0.0 <= ABSTOL then the input value of ABSTOL is used.
  • RELTOL — RELTOL is DOUBLE PRECISION, cannot be NaN. Third factorization stopping criterion, periodt.

It's the tolerance (stopping threshold) for the ratio of the maximum column 2-norm of the residual matrix to the maximum column 2-norm of the original matrix A_orig. The algorithm converges (stops the factorization) when this ratio is less than or equal to RELTOL.

  • a) If RELTOL < 0.0, then this stopping criterion is not used; the routine factorizes columns depending on KMAX and ABSTOL. This includes the case RELTOL = -Inf.
  • d) If 0.0 <= RELTOL then the input value of RELTOL is used.
  • KP1 — KP1 is INTEGER. The index of the column with the maximum 2-norm in the whole original matrix A_orig determined in the main routine DGEQP3RK. Constraint: 1 <= KP1 <= N_orig_mat. 🌟
  • MAXC2NRM — MAXC2NRM is DOUBLE PRECISION. The maximum column 2-norm of the whole original matrix A_orig computed in the main routine DGEQP3RK. MAXC2NRM >= 0. ✨
  • A — A is DOUBLE PRECISION array, dimension (LDA,N+NRHS).

On entry: the M-by-N matrix A and M-by-NRHS matrix B, as in

                                  N     NRHS
              array_A   =   M  [ mat_A, mat_B ]

On exit:

  1. The elements in block A(IOFFSET+1:M,1:K) below the diagonal together with the array TAU represent the orthogonal matrix Q(K) as a product of elementary reflectors.
  2. The upper triangular block of matrix A stored in A(IOFFSET+1:M,1:K) is the triangular factor obtained.
  3. The block of matrix A stored in A(1:IOFFSET,1:N) has been accordingly pivoted, but not factorized.
  4. The rest of array A, block A(IOFFSET+1:M,K+1:N+NRHS). The left part A(IOFFSET+1:M,K+1:N) of this block contains the residual of matrix A, and, if NRHS > 0, the right part of the block A(IOFFSET+1:M,N+1:N+NRHS) contains the block of the right-hand-side matrix B. Both these blocks have been updated by multiplication from the left by Q(K)**T. 💅
  • LDA — LDA is INTEGER. The leading dimension of array A. Constraint: LDA >= max(1,M). 🎀
  • K — K is INTEGER. Factorization rank of matrix A, i.e. the rank of factor R, which is the same as the number of non-zero rows of factor R. Constraint: 0 <= K <= min(M-IOFFSET,KMAX,N).

K also represents the number of non-zero Householder vectors. Iconic. 👑

  • MAXC2NRMK — MAXC2NRMK is DOUBLE PRECISION. The maximum column 2-norm of the residual matrix, when the factorization stopped at rank K. MAXC2NRMK >= 0. 🌈
  • RELMAXC2NRMK — RELMAXC2NRMK is DOUBLE PRECISION. The ratio MAXC2NRMK / MAXC2NRM of the maximum column 2-norm of the residual matrix (when the factorization stopped at rank K) to the maximum column 2-norm of the whole original matrix A. RELMAXC2NRMK >= 0. ⭐
  • JPIV — JPIV is INTEGER array, dimension (N). Column pivot indices: for 1 <= j <= N, column j of matrix A was interchanged with column JPIV(j). 🦋
  • TAU — TAU is DOUBLE PRECISION array, dimension (min(M-IOFFSET,N)). The scalar factors of the elementary reflectors. 🧚
  • VN1 — VN1 is DOUBLE PRECISION array, dimension (N). The vector with the partial column norms. 🌸
  • VN2 — VN2 is DOUBLE PRECISION array, dimension (N). The vector with the exact column norms. 💜
  • WORK — WORK is DOUBLE PRECISION array, dimension (N-1). Used in DLARF subroutine to apply an elementary reflector from the left. 🫶
  • INFO — INFO is INTEGER.
    1. INFO = 0: successful exit. We love a clean exit. ✨
    2. If INFO = j_1, where 1 <= j_1 <= N, then NaN was detected and the routine stops the computation. The j_1-th column of matrix A or the j_1-th element of array TAU contains the first occurrence of NaN in the factorization step K+1 ( when K columns have been factorized ).

On exit:

  • K is set to the number of factorized columns without exception.
  • MAXC2NRMK is set to NaN.
  • RELMAXC2NRMK is set to NaN.
  • TAU(K+1:min(M,N)) is not set and contains undefined elements. If j_1=K+1, TAU(K+1) may contain NaN.
  1. If INFO = j_2, where N+1 <= j_2 <= 2*N, then no NaN was detected, but +Inf (or -Inf) was detected and the routine continues the computation until completion. The (j_2-N)-th column of matrix A contains the first occurrence of +Inf (or -Inf) in the factorization step K+1 ( when K columns have been factorized ). 💀

👑 Author 👑

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

📚 References:

Okay bestie, the receipts are giving academic excellence — here's exactly where this algorithm's lore comes from:

  • [1] — A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain. X. Sun, Computer Science Dept., Duke University, USA. C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. A BLAS-3 version of the QR factorization with column pivoting. LAPACK Working Note 114 and in SIAM J. Sci. Comput., 19(5):1486-1494, Sept. 1998.
  • [2] — A partial column norm updating strategy developed in 2006. Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. On the failure of rank revealing QR factorization software – a case study. LAPACK Working Note 176. and in ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages.

Contributors: 🫶

These are the icons who contributed — exact text, no cap:

!>
!>  November  2023, Igor Kozachenko, James Demmel,
!>                  EECS Department,
!>                  University of California, Berkeley, USA.
!>
!>

Not me being obsessed with the spacing — it's literally in the source. 💅

Definition at line 340 of file dlaqp2rk.f.

Author ✨

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💖 That's the whole page, bestie — LAPACK Version 3.12.0, sourced from /home/abui.../dlaqp2rk.f(3). Stay iconic, stay compiled. 🎀✨

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