LAPACK 3.12.0
LAPACK: Linear Algebra PACKage

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 mbynrhs matrix B with Q**H * B.
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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 righthandsides matrix block B stored in A(IOFFSET+1:M,N+1:N+NRHS) with Q(K)**H * B.
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrix A. N >= 0. 
[in]  NRHS  NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. 
[in]  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. 
[in]  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(MIOFFSET,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. 
[in]  ABSTOL  ABSTOL is DOUBLE PRECISION, cannot be NaN. The second factorization stopping criterion. The absolute tolerance (stopping threshold) for maximum column 2norm of the residual matrix. The algorithm converges (stops the factorization) when the maximum column 2norm 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. 
[in]  RELTOL  RELTOL is DOUBLE PRECISION, cannot be NaN. The third factorization stopping criterion. The tolerance (stopping threshold) for the ratio of the maximum column 2norm of the residual matrix to the maximum column 2norm 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. 
[in]  KP1  KP1 is INTEGER The index of the column with the maximum 2norm in the whole original matrix A_orig determined in the main routine ZGEQP3RK. 1 <= KP1 <= N_orig_mat. 
[in]  MAXC2NRM  MAXC2NRM is DOUBLE PRECISION The maximum column 2norm of the whole original matrix A_orig computed in the main routine ZGEQP3RK. MAXC2NRM >= 0. 
[in,out]  A  A is COMPLEX*16 array, dimension (LDA,N+NRHS) On entry: the MbyN matrix A and MbyNRHS 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 righthandside matrix B. Both these blocks have been updated by multiplication from the left by Q(K)**H. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
[out]  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 nonzero rows of the factor R. 0 <= K <= min(MIOFFSET,KMAX,N). K also represents the number of nonzero Householder vectors. 
[out]  MAXC2NRMK  MAXC2NRMK is DOUBLE PRECISION The maximum column 2norm of the residual matrix, when the factorization stopped at rank K. MAXC2NRMK >= 0. 
[out]  RELMAXC2NRMK  RELMAXC2NRMK is DOUBLE PRECISION The ratio MAXC2NRMK / MAXC2NRM of the maximum column 2norm of the residual matrix (when the factorization stopped at rank K) to the maximum column 2norm of the whole original matrix A. RELMAXC2NRMK >= 0. 
[out]  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). 
[out]  TAU  TAU is COMPLEX*16 array, dimension (min(MIOFFSET,N)) The scalar factors of the elementary reflectors. 
[in,out]  VN1  VN1 is DOUBLE PRECISION array, dimension (N) The vector with the partial column norms. 
[in,out]  VN2  VN2 is DOUBLE PRECISION array, dimension (N) The vector with the exact column norms. 
[out]  WORK  WORK is COMPLEX*16 array, dimension (N1) Used in ZLARF subroutine to apply an elementary reflector from the left. 
[out]  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_1th column of the matrix A or the j_1th 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_2N)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 ). 
[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. http://www.netlib.org/lapack/lawnspdf/lawn176.pdf and in ACM Trans. Math. Softw. 35, 2, Article 12 (July 2008), 28 pages. https://doi.org/10.1145/1377612.1377616
November 2023, Igor Kozachenko, James Demmel, EECS Department, University of California, Berkeley, USA.
Definition at line 341 of file zlaqp2rk.f.