Developer Reference for Intel® oneAPI Math Kernel Library for C
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?geqpf
Computes the QR factorization of a general m-by-n matrix with pivoting.
Syntax
lapack_int LAPACKE_sgeqpf (int matrix_layout, lapack_int m, lapack_int n, float* a, lapack_int lda, lapack_int* jpvt, float* tau);
lapack_int LAPACKE_dgeqpf (int matrix_layout, lapack_int m, lapack_int n, double* a, lapack_int lda, lapack_int* jpvt, double* tau);
lapack_int LAPACKE_cgeqpf (int matrix_layout, lapack_int m, lapack_int n, lapack_complex_float* a, lapack_int lda, lapack_int* jpvt, lapack_complex_float* tau);
lapack_int LAPACKE_zgeqpf (int matrix_layout, lapack_int m, lapack_int n, lapack_complex_double* a, lapack_int lda, lapack_int* jpvt, lapack_complex_double* tau);
Include Files
- mkl.h
Description
The routine is deprecated and has been replaced by routine geqp3.
The routine ?geqpf forms the QR factorization of a general m-by-n matrix A with column pivoting: A*P = Q*R (see Orthogonal Factorizations). Here P denotes an n-by-n permutation matrix.
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
-
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.
- lda
-
The leading dimension of a; at least max(1, m)for column major layout and max(1, n) for row major layout.
- jpvt
-
Array, size at least max(1, n).
On entry, if jpvt[i - 1] > 0, the i-th column of A is moved to the beginning of A*P before the computation, and fixed in place during the computation.
If jpvt[i - 1] = 0, the ith column of A is a free column (that is, it may be interchanged during the computation with any other free column).
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
-
Array, size at least max (1, min(m, n)). Contains additional information on the matrix Q.
- jpvt
-
Overwritten by details of the permutation matrix P in the factorization A*P = Q*R. More precisely, the columns of A*P are the columns of A in the following order:
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.
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:
?geqpf (this routine) |
to factorize A*P = Q*R; |
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 permuted least squares solution vectors x; the output array jpvt specifies the permutation order.)
To compute the elements of Q explicitly, call