CLARRV(1) LAPACK auxiliary routine (version 3.2) CLARRV(1)NAME
CLARRV - computes the eigenvectors of the tridiagonal matrix T = L D
L^T given L, D and APPROXIMATIONS to the eigenvalues of L D L^T
SUBROUTINE CLARRV( N, VL, VU, D, L, PIVMIN, ISPLIT, M, DOL, DOU, MIN‐
RGP, RTOL1, RTOL2, W, WERR, WGAP, IBLOCK, INDEXW,
GERS, Z, LDZ, ISUPPZ, WORK, IWORK, INFO )
INTEGER DOL, DOU, INFO, LDZ, M, N
REAL MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU
INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ), ISUPPZ( * ),
IWORK( * )
REAL D( * ), GERS( * ), L( * ), W( * ), WERR( * ), WGAP(
* ), WORK( * )
COMPLEX Z( LDZ, * )
CLARRV computes the eigenvectors of the tridiagonal matrix T = L D L^T
given L, D and APPROXIMATIONS to the eigenvalues of L D L^T. The input
eigenvalues should have been computed by SLARRE.
N (input) INTEGER
The order of the matrix. N >= 0.
VL (input) REAL
VU (input) REAL Lower and upper bounds of the interval
that contains the desired eigenvalues. VL < VU. Needed to com‐
pute gaps on the left or right end of the extremal eigenvalues
in the desired RANGE.
D (input/output) REAL array, dimension (N)
On entry, the N diagonal elements of the diagonal matrix D. On
exit, D may be overwritten.
L (input/output) REAL array, dimension (N)
On entry, the (N-1) subdiagonal elements of the unit bidiagonal
matrix L are in elements 1 to N-1 of L (if the matrix is not
splitted.) At the end of each block is stored the corresponding
shift as given by SLARRE. On exit, L is overwritten.
PIVMIN (in) DOUBLE PRECISION
The minimum pivot allowed in the Sturm sequence.
ISPLIT (input) INTEGER array, dimension (N)
The splitting points, at which T breaks up into blocks. The
first block consists of rows/columns 1 to ISPLIT( 1 ), the sec‐
ond of rows/columns ISPLIT( 1 )+1 through ISPLIT( 2 ), etc.
M (input) INTEGER
The total number of input eigenvalues. 0 <= M <= N.
DOL (input) INTEGER
DOU (input) INTEGER If the user wants to compute only
selected eigenvectors from all the eigenvalues supplied, he can
specify an index range DOL:DOU. Or else the setting DOL=1,
DOU=M should be applied. Note that DOL and DOU refer to the
order in which the eigenvalues are stored in W. If the user
wants to compute only selected eigenpairs, then the columns
DOL-1 to DOU+1 of the eigenvector space Z contain the computed
eigenvectors. All other columns of Z are set to zero.
MINRGP (input) REAL
RTOL1 (input) REAL
RTOL2 (input) REAL Parameters for bisection. RIGHT-
LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
W (input/output) REAL array, dimension (N)
The first M elements of W contain the APPROXIMATE eigenvalues
for which eigenvectors are to be computed. The eigenvalues
should be grouped by split-off block and ordered from smallest
to largest within the block ( The output array W from SLARRE is
expected here ). Furthermore, they are with respect to the
shift of the corresponding root representation for their block.
On exit, W holds the eigenvalues of the UNshifted matrix.
WERR (input/output) REAL array, dimension (N)
The first M elements contain the semiwidth of the uncertainty
interval of the corresponding eigenvalue in W
WGAP (input/output) REAL array, dimension (N)
The separation from the right neighbor eigenvalue in W.
IBLOCK (input) INTEGER array, dimension (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, =2 if W(i) belongs to
the second block, etc.
INDEXW (input) INTEGER array, dimension (N)
The indices of the eigenvalues within each block (submatrix);
for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the i-th
eigenvalue W(i) is the 10-th eigenvalue in the second block.
GERS (input) REAL array, dimension (2*N)
The N Gerschgorin intervals (the i-th Gerschgorin interval is
(GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should be
computed from the original UNshifted matrix.
Z (output) COMPLEX array, dimension (LDZ, max(1,M) )
If INFO = 0, the first M columns of Z contain the orthonormal
eigenvectors of the matrix T corresponding to the input eigen‐
values, with the i-th column of Z holding the eigenvector asso‐
ciated with W(i). Note: the user must ensure that at least
max(1,M) columns are supplied in the array Z.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
'V', LDZ >= max(1,N).
ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices indi‐
cating the nonzero elements in Z. The I-th eigenvector is
nonzero only in elements ISUPPZ( 2*I-1 ) through ISUPPZ( 2*I ).
WORK (workspace) REAL array, dimension (12*N)
IWORK (workspace) INTEGER array, dimension (7*N)
INFO (output) INTEGER
= 0: successful exit > 0: A problem occured in CLARRV.
< 0: One of the called subroutines signaled an internal prob‐
lem. Needs inspection of the corresponding parameter IINFO for
=-1: Problem in SLARRB when refining a child's eigenvalues.
=-2: Problem in SLARRF when computing the RRR of a child. When
a child is inside a tight cluster, it can be difficult to find an
RRR. A partial remedy from the user's point of view is to make
the parameter MINRGP smaller and recompile. However, as the
orthogonality of the computed vectors is proportional to 1/MIN‐
RGP, the user should be aware that he might be trading in preci‐
sion when he decreases MINRGP. =-3: Problem in SLARRB when
refining a single eigenvalue after the Rayleigh correction was
rejected. = 5: The Rayleigh Quotient Iteration failed to con‐
verge to full accuracy in MAXITR steps.
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
LAPACK auxiliary routine (versioNovember 2008 CLARRV(1)