Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

?hbevd

Computes all eigenvalues and, optionally, all eigenvectors of a complex Hermitian band matrix using divide and conquer algorithm.

Syntax

lapack_int LAPACKE_chbevd
(
int
matrix_layout
,
char
jobz
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_complex_float*
ab
,
lapack_int
ldab
,
float*
w
,
lapack_complex_float*
z
,
lapack_int
ldz
);
lapack_int LAPACKE_zhbevd
(
int
matrix_layout
,
char
jobz
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_complex_double*
ab
,
lapack_int
ldab
,
double*
w
,
lapack_complex_double*
z
,
lapack_int
ldz
);
Include Files
  • mkl.h
Description
The routine computes all the eigenvalues, and optionally all the eigenvectors, of a complex Hermitian band matrix
A
. In other words, it can compute the spectral factorization of
A
 as:
A
 =
Z*
Λ*
Z
H
.
Here
Λ
 is a real diagonal matrix whose diagonal elements are the eigenvalues
λ
i
, and
Z
 is the (complex) unitary matrix whose columns are the eigenvectors
z
i
. Thus,
A*z
i
 =
λ
i
*z
i
 for
i
 = 1, 2, ...,
n
.
If the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan variant of the
QL
 or
QR
 algorithm.
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 only eigenvalues are computed.
If
jobz
 =
'V'
, then eigenvalues and eigenvectors are computed.
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
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 Hermitian matrix
A
 (as specified by
uplo
) in band storage format.
ldab
The leading dimension of
ab
; must be at least
kd
+1
for column major layout and
n
 for row major layout
.
ldz
The leading dimension of the output array
z
.
Constraints:
if
jobz
 =
'N'
, then
ldz
≥
 1
;
if
jobz
 =
'V'
, then
ldz
≥
 max(1,
n
) .
Output Parameters
w
Array, size at least max(1,
n
).
If
info
 = 0
, contains the eigenvalues of the matrix
A
 in ascending order. See also
info
.
z
Array, size
max(1,
ldz
*
n
 if
job
 =
'V'
 and at least 1 if
job
 =
'N'
.
If
jobz
 =
'V'
, then this array is overwritten by the unitary matrix
Z
 which contains the eigenvectors of
A
. The
i
-th column of
Z
 contains the eigenvector which corresponds to the eigenvalue
w
[
i
 - 1]
.
If
jobz
 =
'N'
, then
z
 is not referenced.
ab
On exit, this array is overwritten by the values generated during the reduction to tridiagonal form.
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 eigenvalues and eigenvectors are exact for a matrix
A
 +
E
 such that
||
E
||
2
 =
O
(
ε
)||
A
||
2
, where
ε
 is the machine precision.
The real analogue of this routine is sbevd.
See also heevd for matrices held in full storage, and hpevd for matrices held in packed storage.

Product and Performance Information

1

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