Developer Reference

Contents

?sptrd

Reduces a real symmetric matrix to tridiagonal form using packed storage.

Syntax

lapack_int
LAPACKE_ssptrd
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
float
*
ap
,
float
*
d
,
float
*
e
,
float
*
tau
);
lapack_int
LAPACKE_dsptrd
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
double
*
ap
,
double
*
d
,
double
*
e
,
double
*
tau
);
Include Files
  • mkl.h
Description
The routine reduces a packed real symmetric matrix
A
to symmetric tridiagonal form
T
by an orthogonal similarity transformation:
A
=
Q*T*Q
T
. The orthogonal matrix
Q
is not formed explicitly but is represented as a product of
n
-1 elementary reflectors. Routines are provided for working with
Q
in this representation. See
Application Notes
below for details.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
uplo
Must be
'U'
or
'L'
.
If
uplo
=
'U'
,
ap
stores the packed upper triangle of
A
.
If
uplo
=
'L'
,
ap
stores the packed lower triangle of
A
.
n
The order of the matrix
A
(
n
0
).
ap
Array, size at least max(1,
n
(
n
+1)/2). Contains either upper or lower triangle of
A
(as specified by
uplo
) in the packed form described in Matrix Storage Schemes.
Output Parameters
ap
Overwritten by the tridiagonal matrix
T
and details of the orthogonal matrix
Q
, as specified by
uplo
.
d
,
e
,
tau
Arrays:
d
contains the diagonal elements of the matrix
T
.
The dimension of
d
must be at least max(1,
n
).
e
contains the off-diagonal elements of
T
.
The dimension of
e
must be at least max(1,
n
-1).
tau
Stores (
n
-1) scalars that define elementary reflectors in decomposition of the matrix
Q
in a product of
n
-1 reflectors.
The dimension of
tau
must be at least max(1,
n
-1).
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 matrix
Q
is represented as a product of
n
-1 elementary reflectors, as follows :
  • If
    uplo
    =
    'U'
    ,
    Q
    =
    H
    (
    n
    -1) ...
    H
    (2)
    H
    (1)
    Each
    H
    (i) has the form
    H
    (i) =
    I
    -
    tau
    *
    v
    *
    v
    T
    where
    tau
    is a real scalar and
    v
    is a real vector with
    v
    (i+1:
    n
    ) = 0
    and
    v
    (i) = 1
    .
    On exit,
    tau
    is stored in
    tau
    [
    i
    - 1]
    , and
    v
    (1:i-1)
    is stored in
    AP
    , overwriting
    A
    (1:i-1, i+1)
    .
  • If
    uplo
    =
    'L'
    ,
    Q
    =
    H
    (1)
    H
    (2)
    ... H
    (
    n
    -1)
    Each
    H
    (i) has the form
    H
    (i) =
    I
    -
    tau
    *
    v
    *
    v
    T
    where
    tau
    is a real scalar and
    v
    is a real vector with
    v
    (1:i) = 0
    and
    v
    (i+1) = 1
    .
    On exit,
    tau
    is stored in
    tau
    [
    i
    - 1]
    , and
    v
    (i+2:n)
    is stored in
    AP
    , overwriting
    A
    (i+2:n, i)
    .
The computed matrix
T
is exactly similar to a matrix
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 is
(4/3)
n
3
.
After calling this routine, you can call the following:
to form the computed matrix
Q
explicitly
to multiply a real matrix by
Q
.
The complex counterpart of this routine is hptrd.

Product and Performance Information

1

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