Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 0.9
  • 09/09/2020
  • Public Content
Contents

?lar1va

Computes scaled eigenvector corresponding to given eigenvalue.

Syntax

void slar1va
(
MKL_INT*
n
,
MKL_INT*
b1
,
MKL_INT*
bn
,
float*
lambda
,
float*
d
,
float*
l
,
float*
ld
,
float*
lld
,
float*
pivmin
,
float*
gaptol
,
float*
z
,
MKL_INT*
wantnc
,
MKL_INT*
negcnt
,
float*
ztz
,
float*
mingma
,
MKL_INT*
r
,
MKL_INT*
isuppz
,
float*
nrminv
,
float*
resid
,
float*
rqcorr
,
float*
work
);
void dlar1va
(
MKL_INT*
n
,
MKL_INT*
b1
,
MKL_INT*
bn
,
double*
lambda
,
double*
d
,
double*
l
,
double*
ld
,
double*
lld
,
double*
pivmin
,
double*
gaptol
,
double*
z
,
MKL_INT*
wantnc
,
MKL_INT*
negcnt
,
double*
ztz
,
double*
mingma
,
MKL_INT*
r
,
MKL_INT*
isuppz
,
double*
nrminv
,
double*
resid
,
double*
rqcorr
,
double*
work
);
Include Files
  • mkl_scalapack.h
Description
?slar1va
 computes the (scaled) r-th column of the inverse of the submatrix in rows
b1
 through
bn
 of the tridiagonal matrix
L
D
L
T
 -
λ
I
. When
λ
 is close to an eigenvalue, the computed vector is an accurate eigenvector. Usually,
r
 corresponds to the index where the eigenvector is largest in magnitude. The following steps accomplish this computation :
  1. Stationary qd transform,
    L
    D
    L
    T
     -
    λ
    I
     =
    L
    +
    D
    +
    L
    +
    T
    ,
  2. Progressive qd transform,
    L
    D
    L
    T
     -
    λ
    I
     =
    U
    -
    D
    -
    U
    -
    T
    ,
  3. Computation of the diagonal elements of the inverse of
    L
    D
    L
    T
     -
    λ
    I
     by combining the above transforms, and choosing
    r
     as the index where the diagonal of the inverse is (one of the) largest in magnitude.
  4. Computation of the (scaled)
    r
    -th column of the inverse using the twisted factorization obtained by combining the top part of the stationary and the bottom part of the progressive transform.
Input Parameters
n
The order of the matrix
L
D
L
T
.
b1
First index of the submatrix of
L
D
L
T
.
bn
Last index of the submatrix of
L
D
L
T
.
lambda
The shift
λ
. In order to compute an accurate eigenvector,
lambda
 should be a good approximation to an eigenvalue of
L
D
L
T
.
l
Array of size
n
-1
The (
n
-1) subdiagonal elements of the unit bidiagonal matrix
L
, in elements
0 to
n
-2
.
d
Array of size
n
The
n
 diagonal elements of the diagonal matrix
D
.
ld
Array of size
n
-1
The
n
-1 elements
l
[
i
]*
d
[
i
],
i
=0,...,
n
-2
.
lld
Array of size
n
-1
The
n
-1 elements
l
[
i
]*
l
]
i
]*
d
[
i
],
i
=0,...,
n
-2
.
pivmin
The minimum pivot in the Sturm sequence.
gaptol
Tolerance that indicates when eigenvector entries are negligible with respect to their contribution to the residual.
z
Array of size
n
On input, all entries of
z
 must be set to 0.
wantnc
Specifies whether
negcnt
 has to be computed.
r
The twist index for the twisted factorization used to compute
z
.
On input, 0
≤
r
≤
n
. If
r
 is input as 0,
r
 is set to the index where (
L
D
L
T
 -
σ
I
)
-1
 is largest in magnitude. If 1
≤
r
≤
n
,
r
 is unchanged.
Ideally,
r
 designates the position of the maximum entry in the eigenvector.
work
(Workspace) array of size 4*
n
OUTPUT Parameters
z
On output,
z
 contains the (scaled)
r
-th column of the inverse. The scaling is such that
z
[
r
-1]
 equals 1.
negcnt
If
wantnc
is non-zero
 then
negcnt
 = the number of pivots <
pivmin
 in the matrix factorization
L
D
L
T
, and
negcnt
 = -1 otherwise.
ztz
The square of the 2-norm of
z
.
mingma
The reciprocal of the largest (in magnitude) diagonal element of the inverse of
L
D
L
T
 -
σ
I
.
r
On output,
r
 contains the twist index used to compute
z
.
isuppz
array of size 2
The support of the vector in
z
, i.e., the vector
z
 is non-zero only in elements
isuppz
[0] and
isuppz
[1]
.
nrminv
nrminv
 = 1/
SQRT
(
ztz
 )
resid
The residual of the FP vector.
resid
 =
ABS
(
mingma
 )/
SQRT
(
ztz
 )
rqcorr
The Rayleigh Quotient correction to
lambda
.

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