Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 2021.1
  • 12/04/2020
  • Public Content
Contents

?sbtrd

Reduces a real symmetric band matrix to tridiagonal form.

Syntax

lapack_int
LAPACKE_ssbtrd
(
int
matrix_layout
,
char
vect
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
float
*
ab
,
lapack_int
ldab
,
float
*
d
,
float
*
e
,
float
*
q
,
lapack_int
ldq
);
lapack_int
LAPACKE_dsbtrd
(
int
matrix_layout
,
char
vect
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
double
*
ab
,
lapack_int
ldab
,
double
*
d
,
double
*
e
,
double
*
q
,
lapack_int
ldq
);
Include Files
  • mkl.h
Description
The routine reduces a real symmetric band matrix
A
 to symmetric tridiagonal form
T
 by an orthogonal similarity transformation:
A
 =
Q*T*Q
T
. The orthogonal matrix
Q
 is determined as a product of Givens rotations.
If required, the routine can also form the matrix
Q
 explicitly.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
vect
Must be
'V'
,
'N'
, or
'U'
.
If
vect
 =
'V'
, the routine returns the explicit matrix
Q
.
If
vect
 =
'N'
, the routine does not return
Q
.
If
vect
 =
'U'
, the routine updates matrix
X
 by forming
X
*
Q
.
uplo
Must be
'U'
 or
'L'
.
If
uplo
 =
'U'
,
ab
 stores the upper triangular part of
A
.
If
uplo
 =
'L'
,
ab
 stores the lower triangular part of
A
.
n
The order of the matrix
A
 (
n
≥
0
).
kd
The number of super- or sub-diagonals in
A
(
kd
≥
0
).
ab
,
q
ab
(size at least max(1,
ldab
*
n
) for column major layout and at least max(1,
ldab
*(
kd
+ 1)) for row major layout)
 is an array containing either upper or lower triangular part of the matrix
A
 (as specified by
uplo
) in band storage format.
q
(size max(1,
ldq
*
n
))
 is an array.
If
vect
 =
'U'
, the
q
 array must contain an
n
-by-
n
 matrix
X
.
If
vect
 =
'N'
 or
'V'
, the
q
 parameter need not be set.
ldab
The leading dimension of
ab
; at least
kd
+1
for column major layout and
n
 for row major layout
 .
ldq
The leading dimension of
q
. Constraints:
ldq
≥
 max(1,
n
)
 if
vect
 =
'V'
 or
'U'
;
ldq
≥
 1
 if
vect
 =
'N'
.
Output Parameters
ab
On exit, the diagonal elements of the array
ab
 are overwritten by the diagonal elements of the tridiagonal matrix
T
. If
kd
 > 0, the elements on the first superdiagonal (if
uplo
 =
'U'
) or the first subdiagonal (if
uplo
 =
'L'
) are ovewritten by the off-diagonal elements of
T
. The rest of
ab
 is overwritten by values generated during the reduction.
d
,
e
,
q
Arrays:
d
 contains the diagonal elements of the matrix
T
.
The size of
d
 must be at least max(1,
n
).
e
 contains the off-diagonal elements of
T
.
The size of
e
 must be at least max(1,
n
-1).
q
 is not referenced if
vect
 =
'N'
.
If
vect
 =
'V'
,
q
 contains the
n
-by-
n
 matrix
Q
.
If
vect
 =
'U'
,
q
 contains the product
X
*
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 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 computed matrix
Q
 differs from an exactly orthogonal matrix by a matrix
E
 such that
||
E
||
2
 =
O
(
ε
)
.
The total number of floating-point operations is approximately
6
n
2
*
kd
 if
vect
 =
'N'
, with
3
n
3
*(
kd
-1)/
kd
 additional operations if
vect
 =
'V'
.
The complex counterpart of this routine is hbtrd.

Product and Performance Information

1

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