## Developer Reference

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

# ?laed1

Used by
sstedc
/
dstedc
. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.

## Syntax

Include Files
• mkl.fi
Description
The routine
?laed1
computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. This routine is used only for the eigenproblem which requires all eigenvalues and eigenvectors of a tridiagonal matrix.
?laed7
handles the case in which eigenvalues only or eigenvalues and eigenvectors of a full symmetric matrix (which was reduced to tridiagonal form) are desired.
T
=
Q
(in)*(
D
(in)+
rho
*
Z
*
Z
T
)*
Q
T
(in) =
Q
(out)*
D
(out)*
Q
T
(out)
where
Z
=
Q
T
u
,
u
is a vector of length
n
with ones in the
cutpnt
and (
cutpnt
+1
) -th elements and zeros elsewhere. The eigenvectors of the original matrix are stored in
Q
, and the eigenvalues are in
D
. The algorithm consists of three stages:
The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the
z
vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine
?laed2
.
The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine
?laed4
(as called by
?laed3
). This routine also calculates the eigenvectors of the current problem.
The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.
Input Parameters
n
INTEGER
. The dimension of the symmetric tridiagonal matrix (
n
0
).
d
,
q
,
work
REAL
for
slaed1
DOUBLE PRECISION
for
dlaed1
.
Arrays:
d
(*)
contains the eigenvalues of the rank-1-perturbed matrix. The dimension of
d
must be at least
max(1,
n
)
.
q
(
ldq
, *)
contains the eigenvectors of the rank-1-perturbed matrix. The second dimension of
q
must be at least
max(1,
n
)
.
work
(
*
)
is a workspace array, dimension at least (
4
n
+
n
2
).
ldq
INTEGER
. The leading dimension of the array
q
;
ldq
max(1,
n
)
.
indxq
INTEGER
. Array, dimension (
n
).
On entry, the permutation which separately sorts the two subproblems in
d
into ascending order.
rho
REAL
for
slaed1
DOUBLE PRECISION
for
dlaed1
.
The subdiagonal entry used to create the rank-1 modification. This parameter can be modified by
?laed2
, where it is input/output.
cutpnt
INTEGER
.
The location of the last eigenvalue in the leading sub-matrix.
min(1,
n
) ≤
cutpnt
n
/2
.
iwork
INTEGER
.
Workspace array, dimension (4
n
).
Output Parameters
d
On exit, contains the eigenvalues of the repaired matrix.
q
On exit,
q
contains the eigenvectors of the repaired tridiagonal matrix.
indxq
On exit