Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

Contents

?hpgvx

Computes selected eigenvalues and, optionally, eigenvectors of a generalized Hermitian positive-definite eigenproblem with matrices in packed storage.

Syntax

lapack_int LAPACKE_chpgvx
(
int
matrix_layout
,
lapack_int
itype
,
char
jobz
,
char
range
,
char
uplo
,
lapack_int
n
,
lapack_complex_float*
ap
,
lapack_complex_float*
bp
,
float
vl
,
float
vu
,
lapack_int
il
,
lapack_int
iu
,
float
abstol
,
lapack_int*
m
,
float*
w
,
lapack_complex_float*
z
,
lapack_int
ldz
,
lapack_int*
ifail
);
lapack_int LAPACKE_zhpgvx
(
int
matrix_layout
,
lapack_int
itype
,
char
jobz
,
char
range
,
char
uplo
,
lapack_int
n
,
lapack_complex_double*
ap
,
lapack_complex_double*
bp
,
double
vl
,
double
vu
,
lapack_int
il
,
lapack_int
iu
,
double
abstol
,
lapack_int*
m
,
double*
w
,
lapack_complex_double*
z
,
lapack_int
ldz
,
lapack_int*
ifail
);
Include Files
  • mkl.h
Description
The routine computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian positive-definite eigenproblem, of the form
A
*
x
 =
λ
*
B
*
x
,
A
*
B
*
x
 =
λ
*
x
, or
B
*
A
*
x
 =
λ
*
x
.
Here
A
 and
B
 are assumed to be Hermitian, stored in packed format, and
B
 is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either 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
).
itype
Must be 1 or 2 or 3. Specifies the problem type to be solved:
if
itype
 = 1
, the problem type is
A*x
 =
lambda
*
B
*
x
;
if
itype
 = 2
, the problem type is
A*B*x
 =
lambda
*
x
;
if
itype
 = 3
, the problem type is
B
*
A*x
 =
lambda
*
x
.
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 with indices
il
 to
iu
.
uplo
Must be
'U'
 or
'L'
.
If
uplo
 =
'U'
, arrays
ap
 and
bp
 store the upper triangles of
A
 and
B
;
If
uplo
 =
'L'
, arrays
ap
 and
bp
 store the lower triangles of
A
 and
B
.
n
The order of the matrices
A
 and
B
 (
n
≥
 0
).
ap
,
bp
Arrays:
ap
 contains the packed upper or lower triangle of the Hermitian matrix
A
, as specified by
uplo
.
The dimension of
ap
 must be at least max(1,
n
*(
n
+1)/2).
bp
 contains the packed upper or lower triangle of the Hermitian matrix
B
, as specified by
uplo
.
The dimension of
bp
 must be at least max(1,
n
*(
n
+1)/2).
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
)
for column major layout and
ldz
≥
 max(1,
m
) for row major layout
.
Output Parameters
ap
On exit, the contents of
ap
 are overwritten.
bp
On exit, contains the triangular factor
U
 or
L
 from the Cholesky factorization
B
 =
U
H
*
U
 or
B
 =
L
*
L
H
, in the same storage format as
B
.
m
The total number of eigenvalues found,
0
≤
m
≤
n
. If
range
 =
'A'
,
m
 =
n
, and if
range
 =
'I'
,
m
 =
iu
-
il
+1
.
w
Array, size at least max(1,
n
).
If
info
 = 0
, contains the eigenvalues in ascending order.
z
Array
z
(size at least max(1,
ldz
*
m
) for column major layout and max(1,
ldz
*
n
) for row major layout)
.
If
jobz
 =
'V'
, then if
info
 = 0
, the first
m
 columns of
z
 contain the orthonormal eigenvectors of the matrix
A
 corresponding to the selected eigenvalues, with the
i
-th column of
z
 holding the eigenvector associated with
w
(
i
). The eigenvectors are normalized as follows:
if
itype
 = 1
 or
2
,
Z
H
*
B
*
Z
 = I
;
if
itype
 = 3
,
Z
H
*inv(
B
)*
Z
 = I
;
If
jobz
 =
'N'
, then
z
 is not referenced.
If an eigenvector fails to converge, then that column of
z
 contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in
ifail
.
Note: you must ensure that at least max(1,
m
) columns are supplied in the array
z
; if
range
 =
'V'
, the exact value of
m
 is not known in advance and an upper bound must be used.
ifail
Array, size at least max(1,
n
).
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
,
cpptrf
/
zpptrf
 and
chpevx
/
zhpevx
 returned an error code:
If
info
 =
i
≤
n
,
chpevx
/
zhpevx
 failed to converge, and
i
 eigenvectors failed to converge. Their indices are stored in the array
ifail
;
If
info
 =
n
 +
i
, for
1
≤
i
≤
n
, then the leading minor of order
i
 of
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.