SRC/zlaqp2rk.f(3)

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version 3.12.1 · 2 versions
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💖 zlaqp2rk 💖

Okay bestie, welcome to the SRC/zlaqp2rk.f(3) page from the Library Functions Manual. This is the whole vibe for zlaqp2rk, and yes, it is a whole lot of factorization detail. Let’s get into it 💅

✨ NAME ✨

SRC/zlaqp2rk.f — the source-file manual page for the zlaqp2rk routine. iconic Fortran business incoming 🦋

🌈 SYNOPSIS 🌈

🛠️ Functions/Subroutines 🛠️

subroutine zlaqp2rk (m, n, nrhs, ioffset, kmax, abstol, reltol, kp1,
    maxc2nrm, a, lda, k, maxc2nrmk, relmaxc2nrmk, jpiv, tau, vn1, vn2,
    work, info)

ZLAQP2RK computes truncated QR factorization with column pivoting of a complex matrix block using Level 2 BLAS and overwrites a complex m-by-nrhs matrix B with Q**H * B. no cap, that’s the whole little summary ✨

💅 Function/Subroutine Documentation 💅

subroutine zlaqp2rk (integer m, integer n, integer nrhs, integer ioffset,
    integer kmax, double precision abstol, double precision reltol, integer
    kp1, double precision maxc2nrm, complex*16, dimension( lda, * ) a,
    integer lda, integer k, double precision maxc2nrmk, double precision
    relmaxc2nrmk, integer, dimension( * ) jpiv, complex*16, dimension( * )
    tau, double precision, dimension( * ) vn1, double precision, dimension(
    * ) vn2, complex*16, dimension( * ) work, integer info)

ZLAQP2RK computes truncated QR factorization with column pivoting of a complex matrix block using Level 2 BLAS and overwrites a complex m-by-nrhs matrix B with Q**H * B. same energy, now with the full signature moment 🎀

🦋 Purpose 🦋

Bestie, here’s the purpose, and it is very factorization-coded:

ZLAQP2RK computes a truncated (rank K) or full rank Householder QR factorization with column pivoting of the complex matrix block A(IOFFSET+1:M,1:N) as

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

The routine uses Level 2 BLAS. The block A(1:IOFFSET,1:N) is accordingly pivoted, but not factorized.

The routine also overwrites the right-hand-sides matrix block B stored in A(IOFFSET+1:M,N+1:N+NRHS) with Q(K)**H * B. that’s the plot twist, and it ate 💖

🎀 Parameters 🎀

  • M — INTEGER. The number of rows of the matrix A. Constraint: M >= 0. 🌸
  • N — INTEGER. The number of columns of the matrix A. Constraint: N >= 0. 🌸
  • NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. Constraint: NRHS >= 0. 🌸
  • IOFFSET — INTEGER. The number of rows of the matrix A that must be pivoted but not factorized. Constraint: IOFFSET >= 0.

IOFFSET also represents the number of columns of the whole original matrix A_orig that have been factorized in the previous steps. iykyk ✨

  • KMAX — INTEGER. The first factorization stopping criterion. Constraint: KMAX >= 0.

KMAX is the maximum number of columns of the 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. This means that the factorization is not performed, the matrices A and B and the arrays TAU, IPIV are not modified. early exit behavior, noted bestie 🫶

  • ABSTOL — DOUBLE PRECISION, cannot be NaN. The second factorization stopping criterion.

ABSTOL is 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 — DOUBLE PRECISION, cannot be NaN. The third factorization stopping criterion.

RELTOL is 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. yes, it says d), and we are preserving that little quirk exactly 😭

  • KP1 — INTEGER. The index of the column with the maximum 2-norm in the whole original matrix A_orig determined in the main routine ZGEQP3RK. Constraint: 1 <= KP1 <= N_orig_mat. 🌟
  • MAXC2NRM — DOUBLE PRECISION. The maximum column 2-norm of the whole original matrix A_orig computed in the main routine ZGEQP3RK. Constraint: MAXC2NRM >= 0. 🌟

A — COMPLEX*16 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 unitary matrix Q(K) as a product of elementary reflectors.
  2. The upper triangular block of the matrix A stored in A(IOFFSET+1:M,1:K) 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,K+1:N+NRHS). The left part A(IOFFSET+1:M,K+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(K)**H. that’s the whole exit state, no details skipped 🫶
  • LDA — INTEGER. The leading dimension of the array A. Constraint: LDA >= max(1,M). 🌈
  • K — 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. Constraint: 0 <= K <= min(M-IOFFSET,KMAX,N).

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

  • MAXC2NRMK — DOUBLE PRECISION. The maximum column 2-norm of the residual matrix, when the factorization stopped at rank K. Constraint: MAXC2NRMK >= 0. ⭐
  • RELMAXC2NRMK — 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. Constraint: 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 — COMPLEX*16 array, dimension (min(M-IOFFSET,N)). The scalar factors of the elementary reflectors. 🦋
  • VN1 — DOUBLE PRECISION array, dimension (N). The vector with the partial column norms. 🌸
  • VN2 — DOUBLE PRECISION array, dimension (N). The vector with the exact column norms. 🌸
  • WORK — COMPLEX*16 array, dimension (N-1). Used in ZLARF1F subroutine to apply an elementary reflector from the left. 💅
  • INFO — INTEGER.
    1. INFO = 0: successful 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 the 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. undefined elements, bestie, do not trust them 💀
  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 K+1 (when K columns have been factorized). it keeps going, and that is the whole note 🔥

👑 Author 👑

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

periodt, that’s the author block exactly as it appears 💜

🌈 References 🌈

okay so the bibliography is giving academic baddie, and every single deet is staying:

  • [1] — A Level 3 BLAS QR factorization algorithm with column pivoting developed in 1996. The squad: G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain; X. Sun, Computer Science Dept., Duke University, USA; and C. H. Bischof, Math. and Comp. Sci. Div., Argonne National Lab, USA. The paper is 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. Credit goes to Z. Drmac and Z. Bujanovic, Dept. of Math., University of Zagreb, Croatia. The title: 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 🫶

The contributor receipts are literally in the source comment, so here they are, untouched:

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

Definition at line 341 of file zlaqp2rk.f.

👑 Author 👑

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okay bestie, that’s the whole page — footer vibes are LAPACK Version 3.12.0 at SRC/zlaqp2rk.f(3), and there’s no date printed in it — periodt 💅✨

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