dlarrv.f - Man Page

subroutine dlarrv (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)
DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.  

Purpose:

 DLARRV 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.

Parameters

N

          N is INTEGER
          The order of the matrix.  N >= 0.

VL

          VL is DOUBLE PRECISION
          Lower bound 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.

VU

          VU is DOUBLE PRECISION
          Upper bound of the interval that contains the desired
          eigenvalues. VL < VU. 
          Note: VU is currently not used by this implementation of DLARRV, VU is
          passed to DLARRV because it could be used compute gaps on the right end
          of the extremal eigenvalues. However, with not much initial accuracy in
          LAMBDA and VU, the formula can lead to an overestimation of the right gap
          and thus to inadequately early RQI 'convergence'. This is currently
          prevented this by forcing a small right gap. And so it turns out that VU
          is currently not used by this implementation of DLARRV.

D

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

L

          L is 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 split.) At the end of each block
          is stored the corresponding shift as given by DLARRE.
          On exit, L is overwritten.

PIVMIN

          PIVMIN is DOUBLE PRECISION
          The minimum pivot allowed in the Sturm sequence.

ISPLIT

          ISPLIT is 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

          M is INTEGER
          The total number of input eigenvalues.  0 <= M <= N.

DOL

          DOL is INTEGER

DOU

          DOU is 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

          MINRGP is DOUBLE PRECISION

RTOL1

          RTOL1 is DOUBLE PRECISION

RTOL2

          RTOL2 is DOUBLE PRECISION
           Parameters for bisection.
           An interval [LEFT,RIGHT] has converged if
           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

W

          W is 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

          WERR is DOUBLE PRECISION array, dimension (N)
          The first M elements contain the semiwidth of the uncertainty
          interval of the corresponding eigenvalue in W

WGAP

          WGAP is DOUBLE PRECISION array, dimension (N)
          The separation from the right neighbor eigenvalue in W.

IBLOCK

          IBLOCK is 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

          INDEXW is 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

          GERS is 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

          Z is DOUBLE PRECISION 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

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

ISUPPZ

          ISUPPZ is 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

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

IWORK

          IWORK is INTEGER array, dimension (7*N)

INFO

          INFO is INTEGER
          = 0:  successful exit

          > 0:  A problem occurred in DLARRV.
          < 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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Definition at line 287 of file dlarrv.f.