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LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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| subroutine slaqp2rk | ( | integer | m, |
| integer | n, | ||
| integer | nrhs, | ||
| integer | ioffset, | ||
| integer | kmax, | ||
| real | abstol, | ||
| real | reltol, | ||
| integer | kp1, | ||
| real | maxc2nrm, | ||
| real, dimension( lda, * ) | a, | ||
| integer | lda, | ||
| integer | k, | ||
| real | maxc2nrmk, | ||
| real | relmaxc2nrmk, | ||
| integer, dimension( * ) | jpiv, | ||
| real, dimension( * ) | tau, | ||
| real, dimension( * ) | vn1, | ||
| real, dimension( * ) | vn2, | ||
| real, dimension( * ) | work, | ||
| integer | info | ||
| ) |
SLAQP2RK 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.
Download SLAQP2RK + dependencies [TGZ] [ZIP] [TXT]
SLAQP2RK 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.
| [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(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. |
| [in] | 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. |
| [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 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. |
| [in] | 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 SGEQP3RK. 1 <= KP1 <= N_orig_mat. |
| [in] | MAXC2NRM | MAXC2NRM is DOUBLE PRECISION
The maximum column 2-norm of the whole original
matrix A_orig computed in the main routine SGEQP3RK.
MAXC2NRM >= 0. |
| [in,out] | A | A is 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: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. |
| [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 non-zero
rows of the factor R. 0 <= K <= min(M-IOFFSET,KMAX,N).
K also represents the number of non-zero Householder
vectors. |
| [out] | MAXC2NRMK | MAXC2NRMK is DOUBLE PRECISION
The maximum column 2-norm 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
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. |
| [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 REAL array, dimension (min(M-IOFFSET,N))
The scalar factors of the elementary reflectors. |
| [in,out] | VN1 | VN1 is REAL array, dimension (N)
The vector with the partial column norms. |
| [in,out] | VN2 | VN2 is REAL array, dimension (N)
The vector with the exact column norms. |
| [out] | WORK | WORK is REAL array, dimension (N-1)
Used in SLARF 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_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 ). |
[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 340 of file slaqp2rk.f.
1.9.7