Contents

# ?geqrf

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

## Syntax

Include Files
• mkl.h
Description
The routine forms the
QR
factorization of a general
m
-by-
n
matrix
A
. 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 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.
tau
Array, size at least max (1, min(
m
,
n
)). Contains scalars that define elementary reflectors for the matrix
Q
in its decomposition in a product of elementary reflectors.
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:
?geqrf
(this routine)
to factorize
A
=
QR
;
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 least squares solution vectors
x
.)
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