## Developer Reference

• 0.10
• 10/21/2020
• Public Content
Contents

# ?geqpf

Computes the QR factorization of a general m-by-n matrix with pivoting.

## Syntax

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The routine is deprecated and has been replaced by routine geqp3.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
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
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
,
jpvt
, ...,
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

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804