SLAQP3RK(3)

📦 version 3.12.0, in opensuse-leap 16.0

💖 SLAQP3RK(3) — truncated QR, but make it fashion 💖

🎀 NAME 🎀

SLAQP3RK(3) is the manual name, and it comes from the Library Functions Manual section. The source file it's about is:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/slaqp3rk.f

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

subroutine SLAQP3RK (m, n, nrhs, ioffset, nb, abstol, reltol, kp1,
    maxc2nrm, a, lda, done, kb, maxc2nrmk, relmaxc2nrmk, jpiv, tau, vn1,
    vn2, auxv, f, ldf, iwork, info)

SLAQP3RK does one step of a truncated QR factorization with column pivoting for a real m-by-n matrix A, using Level 3 BLAS, and it overwrites a real m-by-nrhs matrix B with Q**T * B. Slay. 💅

🦋 Function/Subroutine Documentation 🦋

🌈 subroutine SLAQP3RK 🌈

subroutine SLAQP3RK (integer m, integer n, integer nrhs, integer ioffset,
    integer nb, real abstol, real reltol, integer kp1, real maxc2nrm, real,
    dimension( lda, * ) a, integer lda, logical done, integer kb, real
    maxc2nrmk, real relmaxc2nrmk, integer, dimension( * ) jpiv, real,
    dimension( * ) tau, real, dimension( * ) vn1, real, dimension( * ) vn2,
    real, dimension( * ) auxv, real, dimension( ldf, * ) f, integer ldf,
    integer, dimension( * ) iwork, integer info)

Same energy as the synopsis: SLAQP3RK computes a step of truncated QR factorization with column pivoting of a real m-by-n matrix A using Level 3 BLAS, and it overwrites a real m-by-nrhs matrix B with Q**T * B.

🎯 Purpose 🎯

Okay so here's the whole vibe. SLAQP3RK computes a step of truncated QR factorization with column pivoting of a real M-by-N matrix A block A(IOFFSET+1:M,1:N) by using Level 3 BLAS as:

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

The routine tries to factorize NB columns from A starting from the row IOFFSET+1 and updates the residual matrix with BLAS 3 xGEMM. The number of actually factorized columns is returned, and it can be smaller than NB.

Block A(1:IOFFSET,1:N) is accordingly pivoted, but not factorized.

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

Cases when the number of factorized columns KB < NB:

  1. In some cases, due to catastrophic cancellations, it cannot factorize all NB columns and needs to update the residual matrix. Hence, the actual number of factorized columns in the block returned in KB is smaller than NB. The logical DONE is returned as FALSE. The factorization of the whole original matrix A_orig must proceed with the next block.
  2. Whenever the stopping criterion ABSTOL or RELTOL is satisfied, the factorization of the whole original matrix A_orig is stopped, and the logical DONE is returned as TRUE. The number of factorized columns, which is smaller than NB, is returned in KB.
  3. In case both stopping criteria ABSTOL or RELTOL are not used, and when the residual matrix is a zero matrix in some factorization step KB, the factorization of the whole original matrix A_orig is stopped, and the logical DONE is returned as TRUE. The number of factorized columns, which is smaller than NB, is returned in KB.
  4. Whenever NaN is detected in the matrix A or in the array TAU, the factorization of the whole original matrix A_orig is stopped, and the logical DONE is returned as TRUE. The number of factorized columns, which is smaller than NB, is returned in KB. The INFO parameter is set to the column index of the first NaN occurrence.
Reminder, bestie: NaN detection is a whole stop-the-factorization moment, and INFO tells you the first column where it showed up.

💅 Parameters 💅

  • M — INTEGER. The number of rows of the matrix A. M >= 0.
  • N — INTEGER. The number of columns of the matrix A. N >= 0.
  • NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
  • IOFFSET — INTEGER. The number of rows of the 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.
  • NB — INTEGER. Factorization block size, i.e the number of columns to factorize in the matrix A. 0 <= NB. If NB = 0, then the routine exits immediately. This means that the factorization is not performed, and the matrices A and B and the arrays TAU, IPIV are not modified.
  • ABSTOL — REAL, cannot be NaN. 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, and the routine factorizes columns depending on NB and RELTOL. This includes the case ABSTOL = -Inf.
    • b) If 0.0 <= ABSTOL, then the input value of ABSTOL is used.
  • RELTOL — REAL, cannot be NaN. 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, and the routine factorizes columns depending on NB and ABSTOL. This includes the case RELTOL = -Inf.
    • d) If 0.0 <= RELTOL, then the input value of RELTOL is used. (yes, the original says d) here — I'm preserving the whole vibe, no edits 🫶)
  • KP1 — INTEGER. The index of the column with the maximum 2-norm in the whole original matrix A_orig determined in the main routine SGEQP3RK. 1 <= KP1 <= N_orig.
  • MAXC2NRM — REAL. The maximum column 2-norm of the whole original matrix A_orig computed in the main routine SGEQP3RK. MAXC2NRM >= 0.
  • A — REAL 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:KB) below the diagonal together with the array TAU represent the orthogonal matrix Q(KB) as a product of elementary reflectors.
  2. The upper triangular block of the matrix A stored in A(IOFFSET+1:M,1:KB) is the triangular factor obtained.
  3. The block of the matrix A stored in A(1:IOFFSET,1:N) has been accordingly pivoted, but not factorized.
  4. The rest of the array A, block A(IOFFSET+1:M,KB+1:N+NRHS). The left part A(IOFFSET+1:M,KB+1:N) of this block contains the residual of the 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(KB)**T.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • DONE — the original page's raw text for this parameter is a whole formatting moment. Here it is exactly, .RE artifact and all:
            .RE

           verbatim !> DONE is LOGICAL !> TRUE: a) if the factorization
           completed before processing !> all min(M-IOFFSET,NB,N) columns due
           to ABSTOL !> or RELTOL criterion, !> b) if the factorization
           completed before processing !> all min(M-IOFFSET,NB,N) columns due
           to the !> residual matrix being a ZERO matrix. !> c) when NaN was
           detected in the matrix A !> or in the array TAU. !> FALSE:
           otherwise. !>

🌸 Parameters 🌸

Then the page rolls into another Parameters heading, and here's what's in it:

  • KB — INTEGER. Factorization rank of the matrix A, i.e. the rank of the factor R, which is the same as the number of non-zero rows of the factor R. 0 <= KB <= min(M-IOFFSET,NB,N). KB also represents the number of non-zero Householder vectors.
  • MAXC2NRMK — REAL. The maximum column 2-norm of the residual matrix, when the factorization stopped at rank KB. MAXC2NRMK >= 0.
  • RELMAXC2NRMK — REAL. The ratio MAXC2NRMK / MAXC2NRM of the maximum column 2-norm of the residual matrix (when the factorization stopped at rank KB) to the maximum column 2-norm of the original matrix A_orig. RELMAXC2NRMK >= 0.
  • JPIV — INTEGER array, dimension (N). Column pivot indices: for 1 <= j <= N, column j of the matrix A was interchanged with column JPIV(j).
  • TAU — REAL array, dimension (min(M-IOFFSET,N)). The scalar factors of the elementary reflectors.
  • VN1 — REAL array, dimension (N). The vector with the partial column norms.
  • VN2 — REAL array, dimension (N). The vector with the exact column norms.
  • AUXV — REAL array, dimension (NB). Auxiliary vector.
  • F — REAL array, dimension (LDF,NB). Matrix F**T = L*(Y**T)*A.
  • LDF
  • LDF — INTEGER, bestie, don't skip it. It's the leading dimension of the array F, and the law is LDF >= max(1,N+NRHS). Iconic constraint, respect it 💅
  • IWORK — INTEGER array, dimension (N-1). It's a work array 🛠️ — specifically it stores the indices of columns for norm downdating in the residual matrix. Lowkey a little filing cabinet for column indices. Obsessed.
  • INFO — INTEGER, and this one's the drama queen of the whole page, so let's go 😭
    1. INFO = 0: successful exit. Slay, we love a clean finish ✨
    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 the matrix A or the j_1-th element of array TAU contains the first occurrence of NaN in the factorization step KB+1 (when KB columns have been factorized).

On exit:

  • KB — is set to the number of factorized columns without exception.
  • MAXC2NRMK — is set to NaN.
  • RELMAXC2NRMK — is set to NaN.
  • TAU(KB+1:min(M,N)) — is not set and contains undefined elements 🙈. And if j_1=KB+1, then TAU(KB+1) may contain NaN. So like… don't go reading those, fr fr.
  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 the matrix A contains the first occurrence of +Inf (or -Inf) in the factorization step KB+1 (when KB columns have been factorized).

👑 Author

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

📚 References:

  • [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:

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

Definition at line 398 of file slaqp3rk.f.

💻 Author

Generated automatically by Doxygen for LAPACK from the source code.

Okay so that's the whole vibe check complete 🦋 this page came from LAPACK Version 3.12.0, straight out of /home/abui.../slaqp3rk.f(3). Go forth and pivot your columns responsibly, bestie 💖🎀
📦 where this page comes from