Developer Reference

  • 0.9
  • 09/09/2020
  • Public Content
Contents

?gebrd

Reduces a general matrix to bidiagonal form.

Syntax

lapack_int LAPACKE_sgebrd
(
int
matrix_layout
,
lapack_int
m
,
lapack_int
n
,
float*
a
,
lapack_int
lda
,
float*
d
,
float*
e
,
float*
tauq
,
float*
taup
);
lapack_int LAPACKE_dgebrd
(
int
matrix_layout
,
lapack_int
m
,
lapack_int
n
,
double*
a
,
lapack_int
lda
,
double*
d
,
double*
e
,
double*
tauq
,
double*
taup
);
lapack_int LAPACKE_cgebrd
(
int
matrix_layout
,
lapack_int
m
,
lapack_int
n
,
lapack_complex_float*
a
,
lapack_int
lda
,
float*
d
,
float*
e
,
lapack_complex_float*
tauq
,
lapack_complex_float*
taup
);
lapack_int LAPACKE_zgebrd
(
int
matrix_layout
,
lapack_int
m
,
lapack_int
n
,
lapack_complex_double*
a
,
lapack_int
lda
,
double*
d
,
double*
e
,
lapack_complex_double*
tauq
,
lapack_complex_double*
taup
);
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
The leading dimension of
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

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