Contents

# p?stebz

Computes the eigenvalues of a symmetric tridiagonal matrix by bisection.

## Syntax

Include Files
• mkl_scalapack.h
Description
The
p?stebz
function
computes the eigenvalues of a symmetric tridiagonal matrix in parallel. These may be all eigenvalues, all eigenvalues in the interval
[
vl
vu
]
, or the eigenvalues
il
through
iu
. A static partitioning of work is done at the beginning of
p?stebz
which results in all processes finding an (almost) equal number of eigenvalues.
Product and Performance Information
Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.
Notice revision #20201201
Input Parameters
ictxt
(global) The
BLACS
context handle.
range
(global) Must be
'A'
or
'V'
or
'I'
.
If
range
=
'A'
, the
function
computes all eigenvalues.
If
range
=
'V'
, the
function
computes eigenvalues in the interval
[
vl
,
vu
]
.
If
range
=
'I'
, the
function
computes eigenvalues
il
through
iu
.
order
(global) Must be
'B'
or
'E'
.
If
order
=
'B'
, the eigenvalues are to be ordered from smallest to largest within each split-off block.
If
order
=
'E'
, the eigenvalues for the entire matrix are to be ordered from smallest to largest.
n
(global) The order of the tridiagonal matrix
T
(
n
0)
.
vl
,
vu
(global)
If
range
=
'V'
, the
function
computes the lower and the upper bounds for the eigenvalues on the interval
[
1
,
vu
]
.
If
range
=
'A'
or
'I'
,
vl
and
vu
are not referenced.
il
,
iu
(global)
Constraint:
1≤
il
iu
n
.
If
range
=
'I'
, the index of the smallest eigenvalue is returned for
il
and of the largest eigenvalue for
iu
(assuming that the eigenvalues are in ascending order) must be returned.
If
range
=
'A'
or
'V'
,
il
and
iu
are not referenced.
abstol
(global)
The absolute tolerance to which each eigenvalue is required. An eigenvalue (or cluster) is considered to have converged if it lies in an interval of width
abstol
. If
abstol
≤0
, then the tolerance is taken as
ulp
||
T
||, where
ulp
is the machine precision, and ||
T
|| means the 1-norm of
T
Eigenvalues will be computed most accurately when
abstol
is set to the underflow threshold
slamch
(
'U'
), not 0. Note that if eigenvectors are desired later by inverse iteration (
p?stein
),
abstol
should be set to
2*
p?lamch
('S')
.
d
(global)
Array of size
n
.
Contains
n
diagonal elements of the tridiagonal matrix
T
. To avoid overflow, the matrix must be scaled so that its largest entry is no greater than the
overflow
(1/2)
*
underflow
(1/4)
in absolute value, and for greatest accuracy, it should not be much smaller than that.
e
(global)
Array of size
n
- 1
.
Contains
(
n
-1)
off-diagonal elements of the tridiagonal matrix
T
. To avoid overflow, the matrix must be scaled so that its largest entry is no greater than
overflow
(1/2)
* underflow
(1/4)
in absolute value, and for greatest accuracy, it should not be much smaller than that.
work
(local)
Array of size
max(5
n
, 7)
. This is a workspace array.
lwork
(local) The size of the
work
array must be
max
(5
n
, 7)
.
If
lwork
= -1
, then
lwork
is global input and a workspace query is assumed; the
function
only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by
pxerbla
.
iwork
(local) Array of size
max(4
n
, 14)
. This is a workspace array.
liwork
(local) The size of the
iwork
array must ≥
max
(4
n
, 14,
NPROCS
)
.
If
liwork
= -1
, then
liwork
is global input and a workspace query is assumed; the
function
only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by
pxerbla
.
Output Parameters
m
(global) The actual number of eigenvalues found.
0≤
m
n
nsplit
(global) The number of diagonal blocks detected in
T
.
1≤
nsplit
n
w
(global)
Array of size
n
. On exit, the first
m
elements of
w
contain the eigenvalues on all processes.
iblock
(global)
Array of size
n
. At each row/column
j
where
e
[
j
-1]
is zero or small, the matrix
T
is considered to split into a block diagonal matrix. On exit
iblock
[
i
]
specifies which block (from 1 to the number of blocks) the eigenvalue
w
[
i
]
belongs to.
In the (theoretically impossible) event that bisection does not converge for some or all eigenvalues,
info
is set to 1 and the ones for which it did not are identified by a negative block number.
isplit
(global)
Array of size
n
.
Contains the splitting points, at which
T
breaks up into submatrices. The first submatrix consists of rows/columns 1 to
isplit
[0]
, the second of rows/columns
isplit
[0]
+1
through
isplit
[1]
, and so on, and the
nsplit
-th submatrix consists of rows/columns
isplit
[
nsplit
-2]
+1
through
isplit
[
nsplit
-1]
=
n
. (Only the first
nsplit
elements are used, but since the
nsplit
values are not known,
n
words must be reserved for
isplit
.)
info
(global)
If
info
= 0
, the execution is successful.
If
info
< 0
, if
info
= -
i
, the
i
-th argument has an illegal value.
If
info
>
0
, some or all of the eigenvalues fail to converge or are not computed.
If
info
= 1
, bisection fails to converge for some eigenvalues; these eigenvalues are flagged by a negative block number. The effect is that the eigenvalues may not be as accurate as the absolute and relative tolerances.
If
info
= 2
, mismatch between the number of eigenvalues output and the number desired.
If
info
= 3
:
range
='
I'
, and the Gershgorin interval initially used is incorrect. No eigenvalues are computed. Probable cause: the machine has a sloppy floating-point arithmetic. Increase the
fudge
parameter, recompile, and try again.

#### Product and Performance Information

1

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