Developer Reference for Intel® oneAPI Math Kernel Library for C
?geqp3
Computes the QR factorization of a general m-by-n matrix with column pivoting using level 3 BLAS.
Syntax
lapack_intLAPACKE_sgeqp3 ( intmatrix_layout , lapack_intm , lapack_intn , float*a , lapack_intlda , lapack_int*jpvt , float*tau );
lapack_intLAPACKE_dgeqp3 ( intmatrix_layout , lapack_intm , lapack_intn , double*a , lapack_intlda , lapack_int*jpvt , double*tau );
lapack_intLAPACKE_cgeqp3 ( intmatrix_layout , lapack_intm , lapack_intn , lapack_complex_float*a , lapack_intlda , lapack_int*jpvt , lapack_complex_float*tau );
lapack_intLAPACKE_zgeqp3 ( intmatrix_layout , lapack_intm , lapack_intn , lapack_complex_double*a , lapack_intlda , lapack_int*jpvt , lapack_complex_double*tau );
Include Files
mkl.h
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
- matrix_layout
-
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
m
The number of rows in the matrix A ( m≥ 0 ).
n
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
The leading dimension of a ; at least max(1, m ) for column major layout and max(1, n ) for row major layout .
lwork
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
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] .
Return Values
This function returns a value info .
If info=0 , the execution is successful.
If info = -i , the i -th parameter had an illegal value.
LAPACK 95 Interface Notes
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.
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).