--- name: "dlaqp2rk" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/dlaqp2rk.f" --- # dlaqp2rk(3) Part of [dlaqp2rk.f(3)](/dlaqp2rk.f/3.md). ## NAME SRC/dlaqp2rk.f ## 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. ## 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. **Purpose:** ``` !> !> 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). !> !> 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)**T * B. !> ``` **Parameters** *M* ``` !> M is INTEGER !> The number of rows of the matrix A. M >= 0. !> ``` *N* ``` !> N is INTEGER !> The number of columns of the matrix A. N >= 0. !> ``` *NRHS* ``` !> NRHS is INTEGER !> The number of right hand sides, i.e., the number of !> columns of the matrix B. NRHS >= 0. !> ``` *IOFFSET* ``` !> IOFFSET is 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. !> ``` *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-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. !> ``` *ABSTOL* ``` !> ABSTOL is DOUBLE PRECISION, cannot be NaN. !> !> The second factorization stopping criterion. !> !> 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. !> !> The third factorization stopping criterion. !> !> 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. 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 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)**T. !> ``` *LDA* ``` !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !> ``` *K* ``` !> K is 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 <= K <= min(M-IOFFSET,KMAX,N). !> !> K also represents the number of non-zero Householder !> vectors. !> ``` *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 the 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 DLARF1F subroutine to apply an elementary !> reflector from the left. !> ``` *INFO* ``` !> INFO is 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. !> 3) 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. **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 **340** of file **dlaqp2rk.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.