SGEQP3RK(3)

📦 version 3.12.0, in opensuse-leap 16.0

✨ SGEQP3RK(3) ✨ — from the Library Functions Manual, bestie! 💅

NAME

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/sgeqp3rk.f

that's the file, no cap. 📁

SYNOPSIS

Functions/Subroutines

subroutine SGEQP3RK (m, n, nrhs, kmax, abstol, reltol, a, lda, k,
    maxc2nrmk, relmaxc2nrmk, jpiv, tau, work, lwork, iwork, info)

so what does it do? SGEQP3RK computes a truncated Householder QR factorization with column pivoting of a real m-by-n matrix A by using Level 3 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B. iconic. ✨

Function/Subroutine Documentation

okay so here's the full signature with all the types and dimensions, periodt:

subroutine SGEQP3RK (integer m, integer n, integer nrhs, integer kmax, real
    abstol, real reltol, real, dimension( lda, * ) a, integer lda, integer
    k, real maxc2nrmk, real relmaxc2nrmk, integer, dimension( * ) jpiv,
    real, dimension( * ) tau, real, dimension( * ) work, integer lwork,
    integer, dimension( * ) iwork, integer info)

and the one-liner is literally: SGEQP3RK computes a truncated Householder QR factorization with column pivoting of a real m-by-n matrix A by using Level 3 BLAS and overwrites a real m-by-nrhs matrix B with Q**T * B. 🫶

Purpose 💖

okay so SGEQP3RK is literally doing two tasks at the same time, multitasking queen 👑

Task 1: The routine computes a truncated (rank K) or full rank Householder QR factorization with column pivoting of a real M-by-N matrix A using Level 3 BLAS. K is the number of columns that were factorized, i.e. the factorization rank of the factor R, and K <= min(M,N).

Here's the math vibe:

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

        = Q(K) * ( R11(K) R12(K) ) = Q(K) * (   R(K)_approx    )
                 ( 0      R22(K) )          ( 0  R(K)_residual ),

where:

  • P(K) is an N-by-N permutation matrix;
  • Q(K) is an M-by-M orthogonal matrix;
  • R(K)_approx = ( R11(K), R12(K) ) is a rank K approximation of the full rank factor R with K-by-K upper-triangular R11(K) and K-by-N rectangular R12(K). The diagonal entries of R11(K) appear in non-increasing order of absolute value, and absolute values of all of them exceed the maximum column 2-norm of R22(K) up to roundoff error.
  • R(K)_residual = R22(K) is the residual of a rank K approximation of the full rank factor R. It is a an (M-K)-by-(N-K) rectangular matrix;
  • 0 is a an (M-K)-by-K zero matrix.

Task 2: At the same time, the routine overwrites a real M-by-NRHS matrix B with Q(K)**T * B using Level 3 BLAS. 💅

=====================================================================

The matrices A and B are stored on input in the array A as the left and right blocks A(1:M,1:N) and A(1:M, N+1:N+NRHS) respectively.

                                  N     NRHS
             array_A   =   M  [ mat_A, mat_B ]

The truncation criteria (i.e. when to stop the factorization) can be any of the following:

  1. The input parameter KMAX, the maximum number of columns KMAX to factorize, i.e. the factorization rank is limited to KMAX. If KMAX >= min(M,N), the criterion is not used.
  2. The input parameter ABSTOL, the absolute tolerance for the maximum column 2-norm of the residual matrix R22(K). This means that the factorization stops if this norm is less or equal to ABSTOL. If ABSTOL < 0.0, the criterion is not used.
  3. The input parameter RELTOL, the tolerance for the maximum column 2-norm matrix of the residual matrix R22(K) divided by the maximum column 2-norm of the original matrix A, which is equal to abs(R(1,1)). This means that the factorization stops when the ratio of the maximum column 2-norm of R22(K) to the maximum column 2-norm of A is less than or equal to RELTOL. If RELTOL < 0.0, the criterion is not used.
  4. In case both stopping criteria ABSTOL or RELTOL are not used, and when the residual matrix R22(K) is a zero matrix in some factorization step K. ( This stopping criterion is implicit. )

The algorithm stops when any of these conditions is first satisfied, otherwise the whole matrix A is factorized.

To factorize the whole matrix A, use the values KMAX >= min(M,N), ABSTOL < 0.0 and RELTOL < 0.0. 🎯

The routine returns:

  • a) Q(K), R(K)_approx = ( R11(K), R12(K) ), R(K)_residual = R22(K), P(K), i.e. the resulting matrices of the factorization; P(K) is represented by JPIV, ( if K = min(M,N), R(K)_approx is the full factor R, and there is no residual matrix R(K)_residual);
  • b) K, the number of columns that were factorized, i.e. factorization rank;
  • c) MAXC2NRMK, the maximum column 2-norm of the residual matrix R(K)_residual = R22(K), ( if K = min(M,N), MAXC2NRMK = 0.0 );
  • d) RELMAXC2NRMK equals MAXC2NRMK divided by MAXC2NRM, the maximum column 2-norm of the original matrix A, which is equal to abs(R(1,1)), ( if K = min(M,N), RELMAXC2NRMK = 0.0 );
  • e) Q(K)**T * B, the matrix B with the orthogonal transformation Q(K)**T applied on the left.

The N-by-N permutation matrix P(K) is stored in a compact form in the integer array JPIV. For 1 <= j <= N, column j of the matrix A was interchanged with column JPIV(j). 🔄

The M-by-M orthogonal matrix Q is represented as a product of elementary Householder reflectors

     Q(K) = H(1) *  H(2) * . . . * H(K),

where K is the number of columns that were factorized.

Each H(j) has the form

     H(j) = I - tau * v * v**T,

where 1 <= j <= K and:

  • I is an M-by-M identity matrix,
  • tau is a real scalar,
  • v is a real vector with v(1:j-1) = 0 and v(j) = 1.

v(j+1:M) is stored on exit in A(j+1:M,j) and tau in TAU(j).

See the Further Details section for more information. 👀

Parameters

M

M is INTEGER. The number of rows of the matrix A. M >= 0. no negative rows allowed, bestie 💀

N

N is INTEGER. The number of columns of the matrix A. N >= 0. periodt 🌸

NRHS

NRHS is INTEGER. The number of right hand sides, i.e. the number of columns of the matrix B. NRHS >= 0. 🫶

KMAX

KMAX is INTEGER.

The first factorization stopping criterion. KMAX >= 0.

The maximum number of columns of the matrix A to factorize, i.e. the maximum factorization rank.

  • a) If KMAX >= min(M,N), then this stopping criterion is not used, the routine factorizes 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 are not modified, and the matrix A is itself the residual. 😭

ABSTOL

ABSTOL is REAL.

The second factorization stopping criterion, cannot be NaN.

The absolute tolerance (stopping threshold) for maximum column 2-norm of the residual matrix R22(K). The algorithm converges (stops the factorization) when the maximum column 2-norm of the residual matrix R22(K) is less than or equal to ABSTOL. Let SAFMIN = SLAMCH('S').

  • a) If ABSTOL is NaN, then no computation is performed and an error message ( INFO = -5 ) is issued by XERBLA.
  • b) 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.
  • c) If 0.0 <= ABSTOL < 2*SAFMIN, then ABSTOL = 2*SAFMIN is used. This includes the case ABSTOL = -0.0.
  • d) If 2*SAFMIN <= ABSTOL then the input value of ABSTOL is used.

Let MAXC2NRM be the maximum column 2-norm of the whole original matrix A. If ABSTOL chosen above is >= MAXC2NRM, then this stopping criterion is satisfied on input and routine exits immediately after MAXC2NRM is computed. The routine returns MAXC2NRM in MAXC2NORMK, and 1.0 in RELMAXC2NORMK. This includes the case ABSTOL = +Inf. This means that the factorization is not performed, the matrices A and B are not modified, and the matrix A is itself the residual. 💅

RELTOL

(and that's where part 1 cuts off — RELTOL details are coming in part 2, stay tuned bestie! 💜)

  • RELTOL is REAL — the third factorization stopping criterion, and bestie, it literally cannot be NaN. It's the tolerance (aka the stopping threshold) for the ratio abs(R(K+1,K+1))/abs(R(1,1)) — that's the max column 2-norm of the residual matrix R22(K) over the max column 2-norm of the original matrix A. The algorithm converges (i.e. stops the factorization) when abs(R(K+1,K+1))/abs(R(1,1)) of A is less than or equal to RELTOL. And let EPS = SLAMCH('E'), periodt. ✨

Here's the vibe check on what RELTOL can be, fr:

  • a) if RELTOL is NaN, then literally no computation happens and an error message (INFO = -6) gets issued by XERBLA. Not the moment. 💀
  • b) if RELTOL < 0.0, this stopping criterion is just… not used, and the routine factorizes columns depending on KMAX and ABSTOL. This includes the case RELTOL = -Inf.
  • c) if 0.0 <= RELTOL < EPS, then RELTOL = EPS is used instead. This includes the case RELTOL = -0.0 (yes, negative zero counts, she's included 🫶).
  • d) if EPS <= RELTOL, then the input value of RELTOL is used, as-is. Iconic.

Now, let MAXC2NRM be the maximum column 2-norm of the whole original matrix A. If the RELTOL chosen above is >= 1.0, then this stopping criterion is satisfied on input and the routine exits immediately after MAXC2NRM is computed. It returns MAXC2NRM in MAXC2NORMK, and 1.0 in RELMAXC2NORMK. This includes the case RELTOL = +Inf. Which means: the factorization is not performed, matrices A and B are not modified, and matrix A is itself the residual. Zero notes taken, we're leaving early 💅

NOTE: we recommend that RELTOL satisfy min( max(M,N)*EPS, sqrt(EPS) ) <= RELTOL
  • A is REAL array, dimension (LDA,N+NRHS)

On entry:

  • a) the subarray A(1:M,1:N) contains the M-by-N matrix A.
  • b) the subarray A(1:M,N+1:N+NRHS) contains the M-by-NRHS matrix B.

Ok so picture it like this, bestie:

                                N     NRHS
    array_A   =   M  [ mat_A, mat_B ]

On exit:

  • a) the subarray A(1:M,1:N) contains parts of the factors of the matrix A, and the plot thickens:
    1. if K = 0, then A(1:M,1:N) contains the original matrix A. No thoughts, head empty, unchanged.
    2. if K > 0, then A(1:M,1:N) contains parts of the factors:
      1. the elements below the diagonal of the subarray A(1:M,1:K), together with TAU(1:K), represent the orthogonal matrix Q(K) as a product of K Householder elementary reflectors. Slay.
      2. the elements on and above the diagonal of the subarray A(1:K,1:N) contain the K-by-N upper-trapezoidal matrix R(K)_approx = ( R11(K), R12(K) ).
NOTE: if K=min(M,N), i.e. a full rank factorization, then R_approx(K) is the full factor R, which is upper-trapezoidal. If, in addition, M>=N, then R is upper-triangular. 💅
  1. the subarray A(K+1:M,K+1:N) contains the (M-K)-by-(N-K) rectangular matrix R(K)_residual = R22(K).
  • b) if NRHS > 0, the subarray A(1:M,N+1:N+NRHS) contains the M-by-NRHS product Q(K)**T * B.
  • LDA is INTEGER — the leading dimension of the array A, and LDA >= max(1,M). Heads up: this is the leading dimension for both matrices, A and B. One number, two jobs, main character energy 👑
  • K is INTEGER — the 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. And 0 <= K <= min(M,KMAX,N). K also represents the number of non-zero Householder vectors. Two jobs again, we love a multitasker ✨
NOTE: if K = 0, then: - a) the arrays A and B are not modified; - b) the array TAU(1:min(M,N)) is set to ZERO, if the matrix A does not contain NaN — otherwise the elements TAU(1:min(M,N)) are undefined 🙈; - c) the elements of the array JPIV are set as follows: for j = 1:N, JPIV(j) = j.
  • MAXC2NRMK is REAL — the maximum column 2-norm of the residual matrix R22(K), when the factorization stopped at rank K. And MAXC2NRMK >= 0.
    • a) if K = 0, i.e. the factorization was not performed, the matrix A was not modified and is itself a residual matrix, then MAXC2NRMK equals the maximum column 2-norm of the original matrix A.
    • b) if 0 < K < min(M,N), then MAXC2NRMK is returned.
    • c) if K = min(M,N), i.e. the whole matrix A was factorized and there is no residual matrix, then MAXC2NRMK = 0.0. Clean slate, no notes 🎀
NOTE: MAXC2NRMK in the factorization step K would equal R(K+1,K+1) in the next factorization step K+1. Foreshadowing fr.
  • RELMAXC2NRMK is REAL — the ratio MAXC2NRMK / MAXC2NRM, i.e. the maximum column 2-norm of the residual matrix R22(K) (when the factorization stopped at rank K) over the maximum column 2-norm of the whole original matrix A. And RELMAXC2NRMK >= 0.
    • a) if K = 0, i.e. the factorization was not performed, the matrix A was not modified and is itself a residual matrix, then RELMAXC2NRMK = 1.0.
    • b) if 0 < K < min(M,N), then RELMAXC2NRMK = MAXC2NRMK / MAXC2NRM is returned.
    • c) if K = min(M,N), i.e. the whole matrix A was factorized and there is no residual matrix, then RELMAXC2NRMK = 0.0.
NOTE: RELMAXC2NRMK in the factorization step K would equal abs(R(K+1,K+1))/abs(R(1,1)) in the next factorization step K+1. Told you it was foreshadowing 🔥
  • JPIV is INTEGER array, dimension (N) — the column pivot indices. For 1 <= j <= N, column j of the matrix A was interchanged with column JPIV(j).

The elements of the array JPIV(1:N) are always set by the routine — like, for example, even when no columns were factorized, i.e. when K = 0, the elements are still set as JPIV(j) = j for j = 1:N. She shows up even when there's nothing to do 🫶

  • TAU is REAL array, dimension (min(M,N)) — the scalar factors of the elementary reflectors.

If 0 < K <= min(M,N), only the elements TAU(1:K) of the array TAU are modified by the factorization. After the factorization is computed, if no NaN was found during the factorization, the remaining elements TAU(K+1:min(M,N)) are set to zero — otherwise the elements TAU(K+1:min(M,N)) are not set and are therefore undefined. (And if K = 0, all elements of TAU are set to zero, if the matrix A does not contain NaN.) 💀 careful bestie

  • WORK is REAL array, dimension (MAX(1,LWORK)) — on exit, if INFO = 0, then WORK(1) returns the optimal LWORK. She's a gossip, she tells you what size you should have used 💅
  • LWORK is INTEGER — the dimension of the array WORK.
    • LWORK >= (3*N + NRHS - 1)
    • for optimal performance, LWORK >= (2*N + NB*( N+NRHS+1 )), where NB is the optimal block size for SGEQP3RK returned by ILAENV. Minimal block size MINNB=2.
NOTE: the decision whether to use unblocked BLAS 2 or blocked BLAS 3 code is based not only on the dimension LWORK of the available workspace WORK, but also on the matrix A dimension N via the crossover point NX returned by ILAENV. (For N less than NX, unblocked code should be used.)

And 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. It's just asking, not committing 🎀

  • IWORK is INTEGER array, dimension (N-1) — it's a work array. (IWORK is used to store indices of columns for norm downdating in the residual matrix in the blocked step auxiliary subroutine SLAQP3RK.) Little backstage helper, never gets credited 😭
  • INFO
  • INFO is INTEGER, bestie, and it's literally the routine's ✨status report✨:
    1. INFO = 0: successful exit. slay.
    2. INFO < 0: if INFO = -i, the i-th argument had an illegal value.
    3. 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.
  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).

👑 Author 👑

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

🔥 Further Details 🔥

SGEQP3RK is based on the same BLAS3 Householder QR factorization algorithm with column pivoting as in SGEQP3 routine which uses SLARFG routine to generate Householder reflectors for QR factorization.

We can also write:

A = A_approx(K) + A_residual(K)

The low rank approximation matrix A(K)_approx from the truncated QR factorization of rank K of the matrix A is:

A(K)_approx = Q(K) * ( R(K)_approx ) * P(K)**T
                     (     0     0 )

            = Q(K) * ( R11(K) R12(K) ) * P(K)**T
                     (      0      0 )

The residual A_residual(K) of the matrix A is:

A_residual(K) = Q(K) * ( 0              0 ) * P(K)**T =
                       ( 0  R(K)_residual )

              = Q(K) * ( 0        0 ) * P(K)**T
                       ( 0   R22(K) )

The truncated (rank K) factorization guarantees that the maximum column 2-norm of A_residual(K) is less than or equal to MAXC2NRMK up to roundoff error.

NOTE: An approximation of the null vectors of A can be easily computed from R11(K) and R12(K):

Null( A(K) )_approx = P * ( inv(R11(K)) * R12(K) )
                            (         -I           )

📚 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 582 of file sgeqp3rk.f.

🧚 Author 🧚

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