Contents

# ?ggglm

Solves a general Gauss-Markov linear model problem using a generalized QR factorization.

## Syntax

Include Files
• mkl.h
Description
The routine solves a general Gauss-Markov linear model (GLM) problem:
minimize
x
||
y
||
2
subject to
d
=
A
*
x
+
B
*
y
where
A
is an
n
-by-
m
matrix,
B
is an
n
-by-
p
matrix, and
d
is a given
n
-vector. It is assumed that
m
n
m
+
p
, and
rank(
A
) =
m
and
rank(
A
B
) =
n
.
Under these assumptions, the constrained equation is always consistent, and there is a unique solution
x
and a minimal 2-norm solution
y
, which is obtained using a generalized
QR
factorization of the matrices
(
A
,
B
)
given by In particular, if matrix
B
is square nonsingular, then the problem GLM is equivalent to the following weighted linear least squares problem
minimize
x
||
B
-1
(
d
-
A
*
x
)||
2
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
n
The number of rows of the matrices
A
and
B
(
n
0
).
m
The number of columns in
A
(
m
0
).
p
The number of columns in
B
(
p
n
-
m
).
a
,
b
,
d
Arrays:
a
(size max(1,
lda
*
m
) for column major layout and max(1,
lda
*
n
) for row major layout)
contains the
n
-by-
m
matrix
A
.
b
(size max(1,
ldb
*
p
) for column major layout and max(1,
ldb
*
n
) for row major layout)
contains the
n
-by-
p
matrix
B
.
d
, size at least max(1,
n
), contains the left hand side of the GLM equation.
lda
a
; at least max(1,
n
)
for column major layout and max(1,
m
) for row major layout
.
ldb
b
; at least max(1,
n
)
for column major layout and max(1,
p
) for row major layout
.
Output Parameters
x
,
y
Arrays
x
,
y
. size at least max(1,
m
) for
x
and at least max(1,
p
) for
y
.
On exit,
x
and
y
are the solutions of the GLM problem.
a
On exit, the upper triangular part of the array
a
contains the
m
-by-
m
upper triangular matrix
R
.
b
On exit, if
n
p
, the upper right triangle contains the
n
-by-
n
upper triangular matrix
T
as returned by
?ggrqf
; if
n
>
p
, the elements on and above the
(
n
-
p
)
-th subdiagonal contain the
n
-by-
p
upper trapezoidal matrix
T
.
d
On exit,
d
is destroyed
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
= 1
, the upper triangular factor
R
associated with
A
in the generalized QR factorization of the pair
(
A
,
B
)
is singular, so that
rank(
A
) <
m
; the least squares solution could not be computed.
If
info
= 2
, the bottom
(
n
-
m
)
-by-
(
n
-
m
)
part of the upper trapezoidal factor
T
associated with
B
in the generalized QR factorization of the pair
(
A
,
B
)
is singular, so that
rank(
A
B
) <
n
; the least squares solution could not be computed.

#### Product and Performance Information

1

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