Contents

# ?gels

Uses QR or LQ factorization to solve a overdetermined or underdetermined linear system with full rank matrix.

## Syntax

Include Files
• mkl.h
Description
The routine solves overdetermined or underdetermined real/ complex linear systems involving an
m
-by-
n
matrix
A
, or its transpose/ conjugate-transpose, using a
QR
or
LQ
factorization of
A
. It is assumed that
A
has full rank.
The following options are provided:
1. If
trans
=
'N'
and
m
n
: find the least squares solution of an overdetermined system, that is, solve the least squares problem
minimize ||
b
-
A
*
x
||
2
2. If
trans
=
'N'
and
m
<
n
: find the minimum norm solution of an underdetermined system
A
*
X
=
B
.
3. If
trans
=
'T'
or
'C'
and
m
n
: find the minimum norm solution of an undetermined system
A
H
*X
=
B
.
4. If
trans
=
'T'
or
'C'
and
m
<
n
: find the least squares solution of an overdetermined system, that is, solve the least squares problem
minimize ||
b
-
A
H
*
x
||
2
Several right hand side vectors
b
and solution vectors
x
can be handled in a single call; they are formed by the columns of the right hand side matrix
B
and the solution matrix
X
(when coefficient matrix is
A
,
B
is
m
-by-
nrhs
and
X
is
n
-by-
nrhs
; if the coefficient matrix is
A
T
or
A
H
,
B
is
n
-by-
nrhs
and
X
is
m
-by-
nrhs
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
trans
Must be
'N'
,
'T'
, or
'C'
.
If
trans
=
'N'
, the linear system involves matrix
A
;
If
trans
=
'T'
, the linear system involves the transposed matrix
A
T
(for real flavors only);
If
trans
=
'C'
, the linear system involves the conjugate-transposed matrix
A
H
(for complex flavors only).
m
The number of rows of the matrix
A
(
m
0
).
n
The number of columns of the matrix
A
(
n
0
).
nrhs
The number of right-hand sides; the number of columns in
B
(
nrhs
0
).
a
,
b
Arrays:
a
(size max(1,
lda
*
n
) for column major layout and max(1,
lda
*
m
) for row major layout)
contains the
m
-by-
n
matrix
A
.
b
(size max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*max(
m
,
n
)) for row major layout)
contains the matrix
B
of right hand side vectors.
lda
a
; at least max(1,
m
)
for column major layout and at least max(1,
n
) for row major layout
.
ldb
b
; must be at least max(1,
m
,
n
)
for column major layout if
trans
='N'
and at least
max(1,
n
)
if
trans
='T'
and at least max(1,
nrhs
) for row major layout regardless of the value of
trans
.
Output Parameters
a
On exit, overwritten by the factorization data as follows:
if
m
n
, array
a
contains the details of the
QR
factorization of the matrix
A
as returned by
?geqrf
;
if
m
<
n
, array
a
contains the details of the
LQ
factorization of the matrix
A
as returned by
?gelqf
.
b
If
info
= 0
,
b
overwritten by the solution vectors, stored columnwise:
if
trans
=
'N'
and
m
n
, rows 1 to
n
of
b
contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of modulus of elements
n
+1 to
m
in that column;
if
trans
=
'N'
and
m
<
n
, rows 1 to
n
of
b
contain the minimum norm solution vectors;
if
trans
=
'T'
or
'C'
and
m
n
, rows 1 to
m
of
b
contain the minimum norm solution vectors;
if
trans
=
'T'
or
'C'
and
m
<
n
, rows 1 to
m
of
b
contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of modulus of elements
m
+1 to
n
in that column.
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.
If
info
=
i
, the
i-
th diagonal element of the triangular factor of
A
is zero, so that
A
does not have full rank; the least squares solution could not be computed.

#### 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