Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

?pteqr

Computes all eigenvalues and (optionally) all eigenvectors of a real symmetric positive-definite tridiagonal matrix.

Syntax

lapack_int LAPACKE_spteqr
(
int
matrix_layout
,
char
compz
,
lapack_int
n
,
float*
d
,
float*
e
,
float*
z
,
lapack_int
ldz
);
lapack_int LAPACKE_dpteqr
(
int
matrix_layout
,
char
compz
,
lapack_int
n
,
double*
d
,
double*
e
,
double*
z
,
lapack_int
ldz
);
lapack_int LAPACKE_cpteqr
(
int
matrix_layout
,
char
compz
,
lapack_int
n
,
float*
d
,
float*
e
,
lapack_complex_float*
z
,
lapack_int
ldz
);
lapack_int LAPACKE_zpteqr
(
int
matrix_layout
,
char
compz
,
lapack_int
n
,
double*
d
,
double*
e
,
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 real symmetric positive-definite tridiagonal matrix
T
. In other words, the routine can compute the spectral factorization:
T
 =
Z
*
Λ
*
Z
T
.
Here
Λ
 is a diagonal matrix whose diagonal elements are the eigenvalues
λ
i
;
Z
 is an orthogonal matrix whose columns are eigenvectors. Thus,
T
*
z
i
 =
λ
i
*
z
i
 for
i
 = 1, 2, ...,
n
.
(The routine normalizes the eigenvectors so that
||
z
i
||
2
 = 1
.)
You can also use the routine for computing the eigenvalues and eigenvectors of real symmetric (or complex Hermitian) positive-definite matrices
A
 reduced to tridiagonal form
T
:
A
 =
Q*T*Q
H
. In this case, the spectral factorization is as follows:
A
 =
Q*T*Q
H
 = (
QZ
)*
Λ
*(
QZ
)
H
. Before calling
?pteqr
, you must reduce
A
 to tridiagonal form and generate the explicit matrix
Q
 by calling the following routines:
 
for real matrices:
for complex matrices:
full storage
?sytrd
,
?orgtr
?hetrd
,
?ungtr
packed storage
?sptrd
,
?opgtr
?hptrd
,
?upgtr
band storage
?sbtrd
(vect
=
'V'
)
?hbtrd
(vect
=
'V'
)
The routine first factorizes
T
 as
L*D*L
H
 where
L
 is a unit lower bidiagonal matrix, and
D
 is a diagonal matrix. Then it forms the bidiagonal matrix
B
 =
L*D
1/2
 and calls
?bdsqr
 to compute the singular values of
B
, which are the square roots of the eigenvalues of
T
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
compz
Must be
'N'
 or
'I'
 or
'V'
.
If
compz
 =
'N'
, the routine computes eigenvalues only.
If
compz
 =
'I'
, the routine computes the eigenvalues and eigenvectors of the tridiagonal matrix
T
.
If
compz
 =
'V'
, the routine computes the eigenvalues and eigenvectors of
A
 (and the array
z
 must contain the matrix
Q
 on entry).
n
The order of the matrix
T
 (
n
≥
 0
).
d
,
e
Arrays:
d
 contains the diagonal elements of
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).
z
Array, size
max(1,
ldz
*
n
)
If
compz
 =
'N'
 or
'I'
,
z
 need not be set.
If
compz
 =
'V'
,
z
 must contain the orthogonal matrix used in the reduction to tridiagonal form..
ldz
The leading dimension of
z
. Constraints:
ldz
≥
 1 if
compz
 =
'N'
;
ldz
≥
 max(1,
n
) if
compz
 =
'V'
 or
'I'
.
Output Parameters
d
The
n
 eigenvalues in descending order, unless
info
 > 0
.
See also
info
.
e
On exit, the array is overwritten.
z
If
info
 = 0
, contains an
n
-by
n
 matrix the columns of which are orthonormal eigenvectors. (The
i
-th column corresponds to the
i
-th eigenvalue.)
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
 =
i
, the leading minor of order
i
 (and hence
T
 itself) is not positive-definite.
If
info
 =
n
 +
i
, the algorithm for computing singular values failed to converge;
i
 off-diagonal elements have not converged to zero.
If
info
 =
-i
, the
i
-th parameter had an illegal value.
Application Notes
If
λ
i
 is an exact eigenvalue, and
μ
i
 is the corresponding computed value, then
|
μ
i
 -
λ
i
|
≤
c
(
n
)*
ε
*
K
*
λ
i
where
c
(
n
) is a modestly increasing function of
n
,
ε
 is the machine precision, and
K
 = ||
DTD
||
2
 *||(
DTD
)
-1
||
2
,
D
 is diagonal with
d
i
i
 =
t
i
i
-1/2
.
If
z
i
 is the corresponding exact eigenvector, and
w
i
 is the corresponding computed vector, then the angle
θ
(
z
i
,
w
i
) between them is bounded as follows:
θ
(
u
i
,
w
i
)
≤
c
(
n
)
ε
K
 / min
i
≠
j
(|
λ
i
 -
λ
j
|/|
λ
i
 +
λ
j
|)
.
Here
min
i
≠
j
(|
λ
i
 -
λ
j
|/|
λ
i
 +
λ
j
|)
 is the relative gap between
λ
i
 and the other eigenvalues.
The total number of floating-point operations depends on how rapidly the algorithm converges.
Typically, it is about
30
n
2
 if
compz
 =
'N'
;
6
n
3
 (for complex flavors,
12
n
3
) if
compz
 =
'V'
 or
'I'
.

Product and Performance Information

1

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