## Developer Reference

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

# ?gelqf

Computes the LQ factorization of a general m-by-n matrix.

## Syntax

Include Files
• mkl.h
Description
The routine forms the
LQ
factorization of a general
m
-by-
n
matrix
A
(seeOrthogonal Factorizations). No pivoting is performed.
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.
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
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
.
Output Parameters
a
Overwritten by the factorization data as follows:
The elements on and below the diagonal of the array contain the
m
-by-min(
m
,
n
) lower trapezoidal matrix
L
(
L
is lower triangular if
m
n
); the elements above the diagonal, with the array
tau
, represent the orthogonal matrix
Q
as a product of elementary reflectors.
tau
Array, size at least
max(1, min(
m
,
n
))
.
Contains scalars that define elementary reflectors for the matrix
Q
(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)
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 find the minimum-norm solution of an underdetermined least squares problem minimizing
||
A*x
-
b
||
2
for all columns
b
of a given matrix
B
, you can call the following:
?gelqf
(this routine)
to factorize
A
=
L*Q
;
trsm (a BLAS routine)
to solve
L*Y
=
B
for
Y
;
to compute
X
= (
Q
1
)
T
*Y
(for real matrices);
to compute
X
= (
Q
1
)
H
*Y
(for complex matrices).
(The columns of the computed
X
are the minimum-norm solution vectors
x
. Here
A
is an
m
-by-
n
matrix with
m
<
n
;
Q
1
denotes the first
m
columns of
Q
).
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