Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

Contents

?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The routine is deprecated and has been replaced by routine geqp3.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
i
th 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)
n
3
if
m
 =
n
,
(2/3)
n
2
(3
m
-
n
)
if
m
 >
n
,
(2/3)
m
2
(3
n
-
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
 =
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).

Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.