Contents

# ?gebrd

Reduces a general matrix to bidiagonal form.

## Syntax

Include Files
• mkl.h
Description
The routine reduces a general
m
-by-
n
matrix
A
to a bidiagonal matrix
B
by an orthogonal (unitary) transformation.
If
m
n
, the reduction is given by where
B
1
is an
n
-by-
n
upper diagonal matrix,
Q
and
P
are orthogonal or, for a complex
A
, unitary matrices;
Q
1
consists of the first
n
columns of
Q
.
If
m
<
n
, the reduction is given by
A
=
Q*
B*
P
H
=
Q
*(
B
1
0)*
P
H
=
Q
1
*B
1
*P
1
H
,
where
B
1
is an
m
-by-
m
lower diagonal matrix,
Q
and
P
are orthogonal or, for a complex
A
, unitary matrices;
P
1
consists of the first
m
columns of
P
.
The routine does not form the matrices
Q
and
P
explicitly, but represents them as products of elementary reflectors. Routines are provided to work with the matrices
Q
and
P
in this representation:
If the matrix
A
is real,
• to compute
Q
and
P
explicitly, call orgbr.
• to multiply a general matrix by
Q
or
P
, call ormbr.
If the matrix
A
is complex,
• to compute
Q
and
P
explicitly, call ungbr.
• to multiply a general matrix by
Q
or
P
, call unmbr.
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
Arrays:
a
(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
If
m
n
, the diagonal and first super-diagonal of
a
are overwritten by the upper bidiagonal matrix
B
. The elements below the diagonal, with the array
tauq
, represent the orthogonal matrix
Q
as a product of elementary reflectors, and the elements above the first superdiagonal, with the array
taup
, represent the orthogonal matrix
P
as a product of elementary reflectors.
If
m
<
n
, the diagonal and first sub-diagonal of
a
are overwritten by the lower bidiagonal matrix
B
. The elements below the first subdiagonal, with the array
tauq
, represent the orthogonal matrix
Q
as a product of elementary reflectors, and the elements above the diagonal, with the array
taup
, represent the orthogonal matrix
P
as a product of elementary reflectors.
d
Array, size at least
max(1, min(
m
,
n
))
.
Contains the diagonal elements of
B
.
e
Array, size at least
max(1, min(
m
,
n
) - 1)
. Contains the off-diagonal elements of
B
.
tauq
,
taup
Arrays, size at least
max (1, min(
m
,
n
))
. The scalar factors of the elementary reflectors which represent the orthogonal or unitary matrices
P
and
Q
.
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 matrices
Q
,
B
, and
P
satisfy
QBP
H
=
A
+
E
, where
||
E
||
2
=
c
(
n
)
ε
||
A
||
2
,
c
(
n
)
is a modestly increasing function of
n
, and
ε
is the machine precision.
The approximate number of floating-point operations for real flavors is
(4/3)*
n
2
*(3*
m
-
n
) for
m
n
,
(4/3)*
m
2
*(3*
n
-
m
) for
m
<
n
.
The number of operations for complex flavors is four times greater.
If
n
is much less than
m
, it can be more efficient to first form the
QR
factorization of
A
by calling geqrf and then reduce the factor
R
to bidiagonal form. This requires approximately
2*
n
2
*(
m
+
n
)
floating-point operations.
If
m
is much less than
n
, it can be more efficient to first form the
LQ
factorization of
A
by calling gelqf and then reduce the factor
L
to bidiagonal form. This requires approximately
2*
m
2
*(
m
+
n
)
floating-point operations.

#### Product and Performance Information

1

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Notice revision #20110804