Contents

# ?pteqr

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

## Syntax

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
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
.
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.
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.