Developer Reference for Intel® oneAPI Math Kernel Library for C
?orgqr
Generates the real orthogonal matrix Q of the QR factorization formed by ?geqrf .
Syntax
lapack_intLAPACKE_sorgqr ( intmatrix_layout , lapack_intm , lapack_intn , lapack_intk , float*a , lapack_intlda , constfloat*tau );
lapack_intLAPACKE_dorgqr ( intmatrix_layout , lapack_intm , lapack_intn , lapack_intk , double*a , lapack_intlda , constdouble*tau );
Include Files
mkl.h
Description
sorgqr dorgqr orgqr
The routine generates the whole or part of m -by- m orthogonal matrix Q of the QR factorization formed by the routine ?geqrf or geqpf . Use this routine after a call to sgeqrf / dgeqrf or sgeqpf / dgeqpf .
Usually Q is determined from the QR factorization of an m by p matrix A with m≥p . To compute the whole matrix Q , use:
call LAPACKE_?orgqr ( matrix_layout,m , m , p , a , lda , tau, work, lwork, info )
To compute the leading p columns of Q (which form an orthonormal basis in the space spanned by the columns of A ):
call LAPACKE_?orgqr ( matrix_layout,m , p , p , a , lda, work, lwork, info )
To compute the matrix Q:code:`k` of the QR factorization of leading k columns of the matrix A :
call LAPACKE_?orgqr ( matrix_layout,m , m , k , a , lda , tau, work, lwork, info )
To compute the leading k columns of Q:code:`k` (which form an orthonormal basis in the space spanned by leading k columns of the matrix A ):
call LAPACKE_?orgqr ( matrix_layout,m , k , k , a , lda , tau, work, lwork, info )
Input Parameters
- matrix_layout
-
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
m
The order of the orthogonal matrix Q ( m≥ 0 ).
n
The number of columns of Q to be computed
(0 ≤ n≤m ).
k
The number of elementary reflectors whose product defines the matrix Q (0 ≤ k ≤ n ).
- a , tau , work
-
REAL for sorgqr DOUBLE PRECISION for dorgqr Arrays:
a and tau are the arrays returned by sgeqrf / dgeqrf or sgeqpf / dgeqpf .
The size of a is max(1, lda * n ) for column major layout and max(1, lda * m ) for row major layout . The second dimension of a must be at least max(1, n ). The size of tau must be at least max(1, k ). 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 ( lwork≥n ).
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 for the suggested value of lwork .
Output Parameters
- a
-
Overwritten by n leading columns of the m -by- m orthogonal matrix Q .
- work(1)
-
If info = 0 , on exit work(1) contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.
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
The computed \(Q\) differs from an exactly orthogonal matrix by a matrix \(E\) such that
\(||E||_{2} = O(\varepsilon) ||A||_{2}\) where \(\varepsilon\) is the machine precision.
The total number of floating-point operations is approximately \(4 m n k - 2 (m + n) k^{2} + (4/3) k^{3}\) .
If n = k , the number is approximately \((2/3) n^{2} (3m - n)\) .
The complex counterpart of this routine is ungqr .