## Developer Reference

• 2020.2
• 07/15/2020
• Public Content
Contents

# ?laed3

Used by
sstedc
/
dstedc
. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.

## Syntax

Include Files
• mkl.fi
Description
The routine
?laed3
finds the roots of the secular equation, as defined by the values in
d
,
w
, and
rho
, between 1 and
k
.
It makes the appropriate calls to
?laed4
and then updates the eigenvectors by multiplying the matrix of eigenvectors of the pair of eigensystems being combined by the matrix of eigenvectors of the
k
-by-
k
system which is solved here.
This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but none are known.
Input Parameters
k
INTEGER
. The number of terms in the rational function to be solved by
?laed4
(
k
0
).
n
INTEGER
. The number of rows and columns in the
q
matrix.
n
k
(deflation may result in
n
>
k
).
n1
INTEGER
. The location of the last eigenvalue in the leading sub-matrix;
min(1,
n
) ≤
n1
n
/
2.
q
REAL
for
slaed3
DOUBLE PRECISION
for
dlaed3
.
Array
q
(
ldq
, *)
. The second dimension of
q
must be at least
max(1,
n
)
.
Initially, the first
k
columns of this array are used as workspace.
ldq
INTEGER
. The leading dimension of the array
q
;
ldq
max(1,
n
)
.
rho
REAL
for
slaed3
DOUBLE PRECISION
for
dlaed3
.
The value of the parameter in the rank one update equation.
rho
0
required.
dlamda
,
q2
,
w
REAL
for
slaed3
DOUBLE PRECISION
for
dlaed3
.
Arrays:
dlamda
(
k
),
q2
(
ldq2
, *),
w
(
k
)
.
The first
k
elements of the array
dlamda
contain the old roots of the deflated updating problem. These are the poles of the secular equation.
The first
k
columns of the array
q2
contain the non-deflated eigenvectors for the split problem. The second dimension of
q2
must be at least
max(1,
n
)
.
The first
k
elements of the array
w
contain the components of the deflation-adjusted updating vector.
indx
INTEGER
. Array, dimension (
n
).
The permutation used to arrange the columns of the deflated
q
matrix into three groups (see
?laed2
).
The rows of the eigenvectors found by
?laed4
must be likewise permuted before the matrix multiply can take place.
ctot
INTEGER
. Array, dimension (4).
A count of the total number of the various types of columns in
q
, as described in
indx
. The fourth column type is any column which has been deflated.
s
REAL
for
slaed3
DOUBLE PRECISION
for
dlaed3
.
Workspace array, dimension (
n1
+1)*
k
.
Will contain the eigenvectors of the repaired matrix which will be multiplied by the previously accumulated eigenvectors to update the system.
Output Parameters
d
REAL
for
slaed3
DOUBLE PRECISION
for
dlaed3
.
Array, dimension at least