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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 dgeqp3rk | ( | integer | m, |
| integer | n, | ||
| integer | nrhs, | ||
| integer | kmax, | ||
| double precision | abstol, | ||
| double precision | reltol, | ||
| 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( * ) | work, | ||
| integer | lwork, | ||
| integer, dimension( * ) | iwork, | ||
| integer | info | ||
| ) |
DGEQP3RK 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.
Download DGEQP3RK + dependencies [TGZ] [ZIP] [TXT]
DGEQP3RK performs two tasks simultaneously:
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. factorization rank of the
factor R, K <= min(M,N).
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. | [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] | 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. |
| [in] | ABSTOL | ABSTOL is DOUBLE PRECISION
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 = DLAMCH('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. |
| [in] | RELTOL | RELTOL is DOUBLE PRECISION
The third factorization stopping criterion, cannot be NaN.
The tolerance (stopping threshold) for the ratio
abs(R(K+1,K+1))/abs(R(1,1)) of the maximum column 2-norm of
the residual matrix R22(K) to the maximum column 2-norm of
the original matrix A. The algorithm converges (stops the
factorization), when abs(R(K+1,K+1))/abs(R(1,1)) A is less
than or equal to RELTOL. Let EPS = DLAMCH('E').
a) If RELTOL is NaN, then no computation is performed
and an error message ( INFO = -6 ) is issued
by XERBLA.
b) 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.
c) If 0.0 <= RELTOL < EPS, then RELTOL = EPS is used.
This includes the case RELTOL = -0.0.
d) If EPS <= RELTOL then the input value of RELTOL
is used.
Let MAXC2NRM be the maximum column 2-norm of the
whole original matrix A.
If RELTOL chosen above is >= 1.0, 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 RELTOL = +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.
NOTE: We recommend that RELTOL satisfy
min( max(M,N)*EPS, sqrt(EPS) ) <= RELTOL |
| [in,out] | A | A is DOUBLE PRECISION 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.
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:
1) If K = 0, A(1:M,1:N) contains the original matrix A.
2) If K > 0, 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.
2. The elements on and above the diagonal of
the subarray A(1:K,1:N) contain K-by-N
upper-trapezoidal matrix
R(K)_approx = ( R11(K), R12(K) ).
NOTE: If K=min(M,N), i.e. 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.
3. The subarray A(K+1:M,K+1:N) contains (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. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
This is the leading dimension for both matrices, A and B. |
| [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,KMAX,N).
K also represents the number of non-zero Householder
vectors.
NOTE: If K = 0, 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. |
| [out] | MAXC2NRMK | MAXC2NRMK is DOUBLE PRECISION
The maximum column 2-norm of the residual matrix R22(K),
when the factorization stopped at rank K. 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.
NOTE: MAXC2NRMK in the factorization step K would equal
R(K+1,K+1) in the next factorization step K+1. |
| [out] | RELMAXC2NRMK | RELMAXC2NRMK is DOUBLE PRECISION
The ratio MAXC2NRMK / MAXC2NRM of the maximum column
2-norm of the residual matrix R22(K) (when the factorization
stopped at rank K) to the maximum column 2-norm of the
whole original matrix A. 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. |
| [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).
The elements of the array JPIV(1:N) are always set
by the routine, for example, even when no columns
were factorized, i.e. when K = 0, the elements are
set as JPIV(j) = j for j = 1:N. |
| [out] | TAU | TAU is DOUBLE PRECISION 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 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 therefore
undefined.
( If K = 0, all elements of TAU are set to zero, if
the matrix A does not contain NaN. ) |
| [out] | WORK | WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
| [in] | LWORK | LWORK is INTEGER
The dimension of the array WORK.
For optimal performance LWORK >= (2*N + NB*( N+NRHS+1 )),
where NB is the optimal block size for DGEQP3RK 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 availbale workspace WORK, but also also on the
matrix A dimension N via crossover point NX returned
by ILAENV. (For N less than NX, unblocked code should be
used.)
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. |
| [out] | IWORK | IWORK is INTEGER array, dimension (N-1).
Is a work array. ( IWORK is used to store indices
of "bad" columns for norm downdating in the residual
matrix in the blocked step auxiliary subroutine DLAQP3RK ). |
| [out] | INFO | INFO is INTEGER
1) INFO = 0: successful exit.
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.
4) 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 ). |
DGEQP3RK is based on the same BLAS3 Householder QR factorization
algorithm with column pivoting as in DGEQP3 routine which uses
DLARFG 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 ) [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 582 of file dgeqp3rk.f.
1.9.7