Contents

# ?sterf

Computes all eigenvalues of a real symmetric tridiagonal matrix using QR algorithm.

## Syntax

Include Files
• mkl.h
Description
The routine computes all the eigenvalues of a real symmetric tridiagonal matrix
T
(which can be obtained by reducing a symmetric or Hermitian matrix to tridiagonal form). The routine uses a square-root-free variant of the
QR
algorithm.
If you need not only the eigenvalues but also the eigenvectors, call steqr.
Input Parameters
n
The order of the matrix
T
(
n
0
).
d
,
e
Arrays:
d
contains the diagonal elements of
T
.
The dimension of
d
must be at least max(1,
n
).
e
contains the off-diagonal elements of
T
.
The dimension of
e
must be at least max(1,
n
-1).
Output Parameters
d
The
n
eigenvalues in ascending order, unless
info
> 0.
See also
info
.
e
On exit, the array is overwritten; see
info
.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
=
i
, the algorithm failed to find all the eigenvalues after 30
n
iterations:
i
off-diagonal elements have not converged to zero. On exit,
d
and
e
contain, respectively, the diagonal and off-diagonal elements of a tridiagonal matrix orthogonally similar to
T
.
If
info
=
-i
, the
i
-th parameter had an illegal value.
Application Notes
The computed eigenvalues and eigenvectors are exact for a matrix
T
+
E
such that
||
E
||
2
=
O
(
ε
)*||
T
||
2
, where
ε
is the machine precision.
If
λ
i
is an exact eigenvalue, and
m
i
is the corresponding computed value, then
|
μ
i
-
λ
i
|
c
(
n
)*
ε
*||
T
||
2
where
c
(
n
)
is a modestly increasing function of
n
.
The total number of floating-point operations depends on how rapidly the algorithm converges. Typically, it is about
14
n
2
.

#### Product and Performance Information

1

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