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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: computes the eigenvectors of the tridiagonal matrix T = L D L^T given L, D and APPROXIMATIONS to the eigenvalues of L D

```NAME
LAPACK-3  - 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

SYNOPSIS
SUBROUTINE ZLARRV( N, VL, VU,  D,  L,  PIVMIN,  ISPLIT,  M,  DOL,  DOU,
MINRGP, RTOL1, RTOL2, W, WERR, WGAP, IBLOCK, INDEXW,
GERS, Z, LDZ, ISUPPZ, WORK, IWORK, INFO )

INTEGER        DOL, DOU, INFO, LDZ, M, N

DOUBLE         PRECISION MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU

INTEGER        IBLOCK( * ), INDEXW( * ), ISPLIT( * ), ISUPPZ( *  ),
IWORK( * )

DOUBLE         PRECISION D( * ), GERS( * ), L( * ), W( * ), WERR( *
), WGAP( * ), WORK( * )

COMPLEX*16     Z( LDZ, * )

PURPOSE
ZLARRV 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 DLARRE.

ARGUMENTS
N       (input) INTEGER
The order of the matrix.  N >= 0.

VL      (input) DOUBLE PRECISION
VU      (input) DOUBLE PRECISION
Lower  and  upper  bounds  of  the  interval that contains the
desired
eigenvalues. VL < VU. Needed to compute gaps on  the  left  or
right
end of the extremal eigenvalues in the desired RANGE.

D       (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the N diagonal elements of the diagonal matrix D.
On exit, D may be overwritten.

L       (input/output) DOUBLE PRECISION 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 DLARRE.
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 second 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) DOUBLE PRECISION

RTOL1   (input) DOUBLE PRECISION
RTOL2   (input) DOUBLE PRECISION
Parameters for bisection.
RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

W       (input/output) DOUBLE PRECISION 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 DLARRE 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) DOUBLE PRECISION array, dimension (N)
The first M elements contain the semiwidth of the uncertainty
interval of the corresponding eigenvalue in W

WGAP    (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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*16       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 eigenvalues, with the i-th
column of Z holding the eigenvector associated 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
indicating the nonzero elements in Z. The I-th eigenvector
is nonzero only in elements ISUPPZ( 2*I-1 ) through
ISUPPZ( 2*I ).

WORK    (workspace) DOUBLE PRECISION array, dimension (12*N)

IWORK   (workspace) INTEGER array, dimension (7*N)

INFO    (output) INTEGER
= 0:  successful exit
> 0:  A problem occured in ZLARRV.
<  0:   One  of  the  called  subroutines signaled an internal
problem.
Needs inspection of the corresponding parameter IINFO
for further information.

=-1:  Problem in DLARRB when refining a child's eigenvalues.
=-2:  Problem in DLARRF 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/MINRGP, the user should be aware that
he might be trading in precision when he decreases MINRGP.
=-3:  Problem in DLARRB when refining a single eigenvalue
after the Rayleigh correction was rejected.
= 5:  The Rayleigh Quotient Iteration failed to converge to
full accuracy in MAXITR steps.

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

LAPACK auxiliary routine (versionMarch22011                       LAPACK-3(3)
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