Developer Reference for Intel® oneAPI Math Kernel Library for C
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?geqrfp
Computes the QR factorization of a general m-by-n matrix with non-negative diagonal elements.
Syntax
lapack_int LAPACKE_sgeqrfp (int matrix_layout, lapack_int m, lapack_int n, float* a, lapack_int lda, float* tau);
lapack_int LAPACKE_dgeqrfp (int matrix_layout, lapack_int m, lapack_int n, double* a, lapack_int lda, double* tau);
lapack_int LAPACKE_cgeqrfp (int matrix_layout, lapack_int m, lapack_int n, lapack_complex_float* a, lapack_int lda, lapack_complex_float* tau);
lapack_int LAPACKE_zgeqrfp (int matrix_layout, lapack_int m, lapack_int n, lapack_complex_double* a, lapack_int lda, lapack_complex_double* tau);
Include Files
- mkl.h
Description
The routine forms the QR factorization of a general m-by-n matrix A (see Orthogonal Factorizations). No pivoting is performed. The diagonal entries of R are real and nonnegative.
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.
This routine supports the Progress Routine feature. See Progress Function for details.
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
-
Array, size max(1,lda*n) for column major layout and max(1,lda*m) for row major layout, containing the matrix A.
- lda
-
The leading dimension of a; at least max(1, m) for column major layout and at least max(1, n) for row major layout.
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).
The diagonal elements of the matrix R are real and non-negative.
- tau
-
Array, size at least max (1, min(m, n)). Contains scalars that define elementary reflectors for the matrix Qin its decomposition in a product of elementary reflectors (see Orthogonal Factorizations).
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.
Application Notes
The computed factorization is the exact factorization of a matrix A + E, where
||E||2 = O(ε)||A||2.
The approximate number of floating-point operations for real flavors is
(4/3)n3 |
if m = n, |
(2/3)n2(3m-n) |
if m > n, |
(2/3)m2(3n-m) |
if m < n. |
The number of operations for complex flavors is 4 times greater.
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:
?geqrfp (this routine) |
to factorize A = QR; |
to compute C = QT*B (for real matrices); |
|
to compute C = QH*B (for complex matrices); |
|
trsm (a BLAS routine) |
to solve R*X = C. |
(The columns of the computed X are the least squares solution vectors x.)
To compute the elements of Q explicitly, call