Contents

# ?larre2a

Given a tridiagonal matrix, sets small off-diagonal elements to zero and for each unreduced block, finds base representations and eigenvalues.

## Syntax

Include Files
• mkl_scalapack.h
Description
To find the desired eigenvalues of a given real symmetric tridiagonal matrix
T
,
?larre2a
sets any "small" off-diagonal elements to zero, and for each unreduced block
T
i
, it finds
• a suitable shift at one end of the block's spectrum,
• the base representation,
T
i
-
σ
i
I
=
L
i
D
i
L
i
T
, and
• eigenvalues of each
L
i
D
i
L
i
T
.
The algorithm obtains a crude picture of all the wanted eigenvalues (as selected by
range
). However, to reduce work and improve scalability, only the eigenvalues
dol
to
dou
are refined. Furthermore, if the matrix splits into blocks, RRRs for blocks that do not contain eigenvalues from
dol
to
dou
are skipped. The DQDS algorithm (
function
?lasq2
) is not used, unlike in the sequential case. Instead, eigenvalues are computed in parallel to some figures using bisection.
Product and Performance Information
Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.
Notice revision #20201201
Input Parameters
range
= 'A': ("All") all eigenvalues will be found.
= 'V': ("Value") all eigenvalues in the half-open interval (
vl
,
vu
] will be found.
= 'I': ("Index")
eigenvalues of the entire matrix with the indices in a given range
will be found.
n
The order of the matrix.
n
> 0.
vl
,
vu
If
range
='V', the lower and upper bounds for the eigenvalues. Eigenvalues less than or equal to
vl
, or greater than
vu
, will not be returned.
vl
<
vu
.
If
range
='I' or ='A',
?larre2a
computes bounds on the desired part of the spectrum.
il
,
iu
If
range
='I', the indices (in ascending order) of the
smallest eigenvalue, to be returned in
w
[
il
-1], and largest eigenvalue, to be returned in
w
[
iu
-1]
.
1
il
iu
n
.
d
Array of size
n
On entry, the
n
diagonal elements of the tridiagonal matrix
T
.
e
Array of size
n
The first (
n
-1) entries contain the subdiagonal elements of the tridiagonal matrix
T
;
e
[
n
-1]
need not be set.
e2
Array of size
n
The first (
n
-1) entries contain the squares of the subdiagonal elements of the tridiagonal matrix
T
;
e2
[
n
-1]
need not be set.
rtol1
,
rtol2
Parameters for bisection.
An interval [
left
,
right
] has converged if
right
-
left
< max(
rtol1
*
gap
,
rtol2
*max(|
left
|,|
right
|) )
spltol
The threshold for splitting.
dol
,
dou
If the user wants to work on only a selected part of the representation tree, he can specify an index range
dol
:
dou
.
Otherwise, the setting
dol
=1,
dou
=
n
should be applied.
Note that
dol
and
dou
refer to the order in which the eigenvalues are stored in
w
.
work
Workspace array of size 6*
n
iwork
Workspace array of size 5*
n
minrgp
The minimum relative gap threshold to decide whether an eigenvalue or a cluster boundary is reached.
OUTPUT Parameters
vl
,
vu
If
range
='V', the lower and upper bounds for the eigenvalues. Eigenvalues less than or equal to
vl
, or greater than
vu
, are not returned.
vl
<
vu
.
If
range
='I' or
range
='A',
?larre2a
computes bounds on the desired part of the spectrum.
d
The
n
diagonal elements of the diagonal matrices
D
i
.
e
e
contains the subdiagonal elements of the unit bidiagonal matrices
L
i
. The entries
e
[
isplit
[
i
]], 0
i
<
nsplit
, contain the base points
σ
i+1
on output
.
e2
The entries
e2
[
isplit
[
i
]], 0
i
<
nsplit
have been set to zero.
nsplit
The number of blocks
T
splits into. 1
nsplit
n
.
isplit
Array of size
n
The splitting points, at which
T
breaks up into blocks.
The first block consists of rows/columns 1 to
isplit
, the second of rows/columns
isplit
+1 through
isplit
, etc., and the
nsplit
-th block consists of rows/columns
isplit
[
nsplit
-2]+1 through
isplit
[
nsplit-1
]=
n
.
m
The total number of eigenvalues (of all
L
i
D
i
L
i
T
) found.
needil
,
neediu
The indices of the leftmost and rightmost eigenvalues of the root node RRR which are needed to accurately compute the relevant part of the representation tree.
w
Array of size
n
The first
m
elements contain the eigenvalues. The eigenvalues of each of the blocks,
L
i
D
i
L
i
T
, are sorted in ascending order (
?larre2a
may use the remaining
n
-
m
elements as workspace).
Note that immediately after exiting this
function
, only the eigenvalues in
w
with indices in range
dol
-1:
dou
-1
rely on this processor because the eigenvalue computation is done in parallel.
werr
Array of size
n
The error bound on the corresponding eigenvalue in
w
.
Note that immediately after exiting this
function
, only the uncertainties in
werr
with indices in range
dol
-1:
dou
-1
are reliable on this processor because the eigenvalue computation is done in parallel.
wgap
Array of size
n
The separation from the right neighbor eigenvalue in
w
. The gap is only with respect to the eigenvalues of the same block as each block has its own representation tree.
Exception: at the right end of a block we store the left gap
Note that immediately after exiting this
function
, only the gaps in
wgap
with indices in range
dol
-1:
dou
-1
are reliable on this processor because the eigenvalue computation is done in parallel.
iblock
Array of size
n
The indices of the blocks (submatrices) associated with the corresponding eigenvalues in
w
;
iblock
[
i
]=1 if eigenvalue
w
[
i
] belongs to the first block from the top,
iblock
[
i
]=2 if
w
[
i
] belongs to the second block, and so on
.
indexw
Array of size
n
The indices of the eigenvalues within each block (submatrix); for example,
indexw
[
i
]= 10 and
iblock
[
i
]=2 imply that the (
i
+1)-th eigenvalue
w
[
i
] is the 10th eigenvalue in block 2
.
gers
Array of size 2*
n
The
n
Gerschgorin intervals (the
i
-th Gerschgorin interval is
(
gers
[2*
i
-2],
gers
[2*
i
-1])
).
pivmin
The minimum pivot in the sturm sequence for
T
.
info
= 0: successful exit
> 0: A problem occurred in
?larre2a
.
< 0: One of the called
function
s signaled an internal problem. Needs inspection of the corresponding parameter
info
for further information.
=-1: Problem in
?larrd2
.
=-2: Not enough internal iterations to find base representation.
=-3: Problem in
?larrb2
when computing the refined root representation.
=-4: Problem in
?larrb2
when preforming bisection on the desired part of the spectrum.
= -9 Problem:
m
<
dou
-
dol
+1, that is the code found fewer eigenvalues than it was supposed to.

#### Product and Performance Information

1

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