SLARRF
June 2010
Purpose
Given the initial representation L D L^T and its cluster of close
eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ...
W( CLEND ), SLARRF finds a new relatively robust representation
L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the
eigenvalues of L(+) D(+) L(+)^T is relatively isolated.
eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ...
W( CLEND ), SLARRF finds a new relatively robust representation
L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the
eigenvalues of L(+) D(+) L(+)^T is relatively isolated.
Arguments
| N |
(input) INTEGER
The order of the matrix (subblock, if the matrix splitted).
|
| D |
(input) REAL array, dimension (N)
The N diagonal elements of the diagonal matrix D.
|
| L |
(input) REAL array, dimension (N-1)
The (N-1) subdiagonal elements of the unit bidiagonal
matrix L. |
| LD |
(input) REAL array, dimension (N-1)
The (N-1) elements L(i)*D(i).
|
| CLSTRT |
(input) INTEGER
The index of the first eigenvalue in the cluster.
|
| CLEND |
(input) INTEGER
The index of the last eigenvalue in the cluster.
|
| W |
(input) REAL array, dimension
dimension is >= (CLEND-CLSTRT+1)
The eigenvalue APPROXIMATIONS of L D L^T in ascending order. W( CLSTRT ) through W( CLEND ) form the cluster of relatively close eigenalues. |
| WGAP |
(input/output) REAL array, dimension
dimension is >= (CLEND-CLSTRT+1)
The separation from the right neighbor eigenvalue in W. |
| WERR |
(input) REAL array, dimension
dimension is >= (CLEND-CLSTRT+1)
WERR contain the semiwidth of the uncertainty interval of the corresponding eigenvalue APPROXIMATION in W |
| SPDIAM |
(input) REAL
estimate of the spectral diameter obtained from the
Gerschgorin intervals |
| CLGAPL |
(input) REAL
|
| CLGAPR |
(input) REAL
absolute gap on each end of the cluster.
Set by the calling routine to protect against shifts too close to eigenvalues outside the cluster. |
| PIVMIN |
(input) REAL
The minimum pivot allowed in the Sturm sequence.
|
| SIGMA |
(output) REAL
The shift used to form L(+) D(+) L(+)^T.
|
| DPLUS |
(output) REAL array, dimension (N)
The N diagonal elements of the diagonal matrix D(+).
|
| LPLUS |
(output) REAL array, dimension (N-1)
The first (N-1) elements of LPLUS contain the subdiagonal
elements of the unit bidiagonal matrix L(+). |
| WORK |
(workspace) REAL array, dimension (2*N)
Workspace.
|
Further Details
Based on contributions by
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA