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.
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

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804