Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

?sbgvx

Computes selected eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem with banded matrices.

Syntax

lapack_int
LAPACKE_ssbgvx
(
int
matrix_layout
,
char
jobz
,
char
range
,
char
uplo
,
lapack_int
n
,
lapack_int
ka
,
lapack_int
kb
,
float
*
ab
,
lapack_int
ldab
,
float
*
bb
,
lapack_int
ldbb
,
float
*
q
,
lapack_int
ldq
,
float
vl
,
float
vu
,
lapack_int
il
,
lapack_int
iu
,
float
abstol
,
lapack_int
*
m
,
float
*
w
,
float
*
z
,
lapack_int
ldz
,
lapack_int
*
ifail
);
lapack_int
LAPACKE_dsbgvx
(
int
matrix_layout
,
char
jobz
,
char
range
,
char
uplo
,
lapack_int
n
,
lapack_int
ka
,
lapack_int
kb
,
double
*
ab
,
lapack_int
ldab
,
double
*
bb
,
lapack_int
ldbb
,
double
*
q
,
lapack_int
ldq
,
double
vl
,
double
vu
,
lapack_int
il
,
lapack_int
iu
,
double
abstol
,
lapack_int
*
m
,
double
*
w
,
double
*
z
,
lapack_int
ldz
,
lapack_int
*
ifail
);
Include Files
  • mkl.h
Description
The routine computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form
A
*
x
 =
λ
*
B
*
x
. Here
A
 and
B
 are assumed to be symmetric and banded, and
B
 is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
jobz
Must be
'N'
 or
'V'
.
If
jobz
 =
'N'
, then compute eigenvalues only.
If
jobz
 =
'V'
, then compute eigenvalues and eigenvectors.
range
Must be
'A'
 or
'V'
 or
'I'
.
If
range
 =
'A'
, the routine computes all eigenvalues.
If
range
 =
'V'
, the routine computes eigenvalues
w
[
i
]
 in the half-open interval:
vl
<
w
[
i
]
≤
vu
.
If
range
 =
'I'
, the routine computes eigenvalues in range
il
 to
iu
.
uplo
Must be
'U'
 or
'L'
.
If
uplo
 =
'U'
, arrays
ab
 and
bb
 store the upper triangles of
A
 and
B
;
If
uplo
 =
'L'
, arrays
ab
 and
bb
 store the lower triangles of
A
 and
B
.
n
The order of the matrices
A
 and
B
 (
n
≥
 0
).
ka
The number of super- or sub-diagonals in
A
(
ka
≥
 0
).
kb
The number of super- or sub-diagonals in
B
 (
kb
≥
 0).
ab
,
bb
Arrays:
ab
(size at least max(1,
ldab
*
n
) for column major layout and max(1,
ldab
*(
ka
 + 1)) for row major layout)
 is an array containing either upper or lower triangular part of the symmetric matrix
A
 (as specified by
uplo
) in band storage format.
bb
(size at least max(1,
ldbb
*
n
) for column major layout and max(1,
ldbb
*(
kb
 + 1)) for row major layout)
 is an array containing either upper or lower triangular part of the symmetric matrix
B
 (as specified by
uplo
) in band storage format.
ldab
The leading dimension of the array
ab
; must be at least
ka
+1
for column major layout and at least max(1,
n
) for row major layout
.
ldbb
The leading dimension of the array
bb
; must be at least
kb
+1
for column major layout and at least max(1,
n
) for row major layout
.
vl
,
vu
If
range
 =
'V'
, the lower and upper bounds of the interval to be searched for eigenvalues.
Constraint:
vl
<
vu
.
If
range
 =
'A'
 or
'I'
,
vl
 and
vu
 are not referenced.
il
,
iu
If
range
 =
'I'
, the indices in ascending order of the smallest and largest eigenvalues to be returned.
Constraint:
1
≤
il
≤
iu
≤
n
, if
n
 > 0
;
il
=1
 and
iu
=0
if
n
 = 0
.
If
range
 =
'A'
 or
'V'
,
il
 and
iu
 are not referenced.
abstol
The absolute error tolerance for the eigenvalues.
See
Application Notes
 for more information.
ldz
The leading dimension of the output array
z
;
ldz
≥
 1
. If
jobz
 =
'V'
,
ldz
≥
 max(1,
n
)
.
ldq
The leading dimension of the output array
q
;
ldq
 < 1
.
If
jobz
 =
'V'
,
ldq
 < max(1,
n
)
.
Output Parameters
ab
On exit, the contents of
ab
 are overwritten.
bb
On exit, contains the factor
S
 from the split Cholesky factorization
B
 =
S
T
*S
, as returned by pbstf/pbstf.
m
The total number of eigenvalues found,
0
≤
m
≤
n
. If
range
 =
'A'
,
m
 =
n
, and if
range
 =
'I'
,
m
 =
iu
-
il
+1
.
w
,
z
,
q
Arrays:
w
, size at least max(1,
n
) .
If
info
 = 0
, contains the eigenvalues in ascending order.
z
(size max(1,
ldz
*
m
) for column major layout and max(1,
ldz
*
n
) for row major layout)
 .
If
jobz
 =
'V'
, then if
info
 = 0
,
z
 contains the matrix
Z
 of eigenvectors, with the
i
-th column of
z
 holding the eigenvector associated with
w
(
i
). The eigenvectors are normalized so that
Z
T
*B
*
Z
 = I
.
If
jobz
 =
'N'
, then
z
 is not referenced.
q
(size max(1,
ldq
*
n
))
 .
If
jobz
 =
'V'
, then
q
 contains the
n
-by-
n
 matrix used in the reduction of
A
*
x
 =
lambda
*
B
*
x
 to standard form, that is,
C
*
x
=
lambda
*
x
 and consequently
C
 to tridiagonal form.
If
jobz
 =
'N'
, then
q
 is not referenced.
ifail
Array,
size
m
.
If
jobz
 =
'V'
, then if
info
 = 0
, the first
m
 elements of
ifail
 are zero; if
info
 > 0
, the
ifail
 contains the indices of the eigenvectors that failed to converge.
If
jobz
 =
'N'
, then
ifail
 is not referenced.
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
 > 0
, and
if
i
≤
n
, the algorithm failed to converge, and
i
 off-diagonal elements of an intermediate tridiagonal did not converge to zero;
if
info
 =
n
 +
i
, for
1
≤
i
≤
n
, then pbstf/pbstf returned
info
 =
i
 and
B
 is not positive-definite. The factorization of
B
 could not be completed and no eigenvalues or eigenvectors were computed.
Application Notes
An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to
abstol
+
ε
*max(|a|,|b|)
, where
ε
 is the machine precision.
If
abstol
 is less than or equal to zero, then
ε
*||
T
||
1
 is used as tolerance, where
T
 is the tridiagonal matrix obtained by reducing
A
 to tridiagonal form. Eigenvalues will be computed most accurately when
abstol
 is set to twice the underflow threshold 2*
?lamch
('S'), not zero.
If this routine returns with
info
 > 0
, indicating that some eigenvectors did not converge, try setting
abstol
 to 2*
?lamch
('S').

Product and Performance Information

1

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