Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?geqp3
Computes the QR factorization of a general m-by-n matrix with column pivoting using level 3 BLAS.
Syntax
call sgeqp3 ( m , n , a , lda , jpvt , tau , work , lwork , info )
call dgeqp3 ( m , n , a , lda , jpvt , tau , work , lwork , info )
call cgeqp3 ( m , n , a , lda , jpvt , tau , work , lwork , rwork , info )
call zgeqp3 ( m , n , a , lda , jpvt , tau , work , lwork , rwork , info )
call geqp3 ( a , jpvt [ , tau ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
sgeqp3 dgeqp3 cgeqp3 zgeqp3 geqp3
The routine forms the QR factorization of a general m -by- n matrix A with column pivoting: A*P = Q*R (see Orthogonal Factorizations ) using Level 3 BLAS. Here P denotes an n -by- n permutation matrix. Use this routine instead of geqpf for better performance.
The routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min( m , n ) elementary reflectors . Routines are provided to work with Q in this representation.
Input Parameters
m
INTEGER . The number of rows in the matrix A ( m≥ 0 ).
n
INTEGER . The number of columns in A ( n≥ 0 ).
- a , work
-
REAL for sgeqp3 DOUBLE PRECISION for dgeqp3 COMPLEX for cgeqp3 DOUBLE COMPLEX for zgeqp3 . Arrays: a ( lda ,*) contains the matrix A . The second dimension of a must be at least max(1, n ). Array a of size max(1, lda * n ) for column major layout and max(1, lda * m ) for row major layout contains the matrix A . work is a workspace array, its dimension max(1, lwork) .
lda
INTEGER . The leading dimension of a ; at least max(1, m ) .
lwork
INTEGER . The size of the work array; must be at least max(1, 3*n+1) for real flavors, and at least max(1, n+1) for complex flavors.
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 . See Application Notes below for details.
jpvt
INTEGER .
Array, size at least max(1, n) . On entry, if jpvt(i)jpvt[i - 1]≠ 0 , the i -th column of A is moved to the beginning of AP before the computation, and fixed in place during the computation. If jpvt(i)jpvt[i - 1]= 0 , the i -th column of A is a free column (that is, it may be interchanged during the computation with any other free column).
- rwork
-
REAL for cgeqp3 DOUBLE PRECISION for zgeqp3 . A workspace array, size at least max(1, 2* n ). Used in complex flavors only.
Output Parameters
- a
-
Overwritten by the factorization data as follows: The elements on and above the diagonal of the array contain the min( m , n )-by- n upper trapezoidal matrix R ( R is upper triangular if m ≥ n ); the elements below the diagonal, with the array tau , present the orthogonal matrix Q as a product of min( m , n ) elementary reflectors (see Orthogonal Factorizations ).
- tau
-
REAL for sgeqp3 DOUBLE PRECISION for dgeqp3 COMPLEX for cgeqp3 DOUBLE COMPLEX for zgeqp3 . Array, size at least max (1, min( m , n )). Contains scalar factors of the elementary reflectors for the matrix Q .
- jpvt
-
Overwritten by details of the permutation matrix P in the factorization A*P = Q*R . More precisely, the columns of AP are the columns of A in the following order: jpvt(1), jpvt(2), ..., jpvt(n) . jpvt[0], jpvt[1], ..., jpvt[n - 1] .
info
INTEGER .
If info = 0 , the execution is successful.
If info = -i , the i -th parameter had an illegal value.
Return Values
No return value, info is an Output Parameter.
LAPACK 95 Interface Notes
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions .
Specific details for the routine geqp3 interface are the following:
a
Holds the matrix A of size ( m,n ).
jpvt
Holds the vector of length n .
tau
Holds the vector of length min( m,n )
Application Notes
To solve a set of least squares problems minimizing \(||A x - b||_{2}\) for all columns b of a given matrix \(B\) , you can call the following:
?geqp3 (this routine)
to factorize \(A P = Q R\) ;
to compute \(C = Q^{T} B\) (for real matrices);
to compute \(C = Q^{H} B\) (for complex matrices);
trsm (a BLAS routine)
to solve \(R X = C\) .
(The columns of the computed \(X\) are the permuted least squares solution vectors x ; the output array jpvt specifies the permutation order.)
To compute the elements of \(Q\) explicitly, call
(for real matrices)
(for complex matrices).
If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set lwork = -1 .
If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value ( work(1) ) for subsequent runs.
If you set lwork = -1 , the routine returns immediately and provides the recommended workspace in the first element of the corresponding array ( work ). This operation is called a workspace query.
Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.