SLARRV 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

          N is INTEGER


          The order of the matrix.  N >= 0.

VL

          VL is REAL


          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 REAL


          Upper bound of the interval that contains the desired


          eigenvalues. VL < VU. 


          Note: VU is currently not used by this implementation of SLARRV, VU is


          passed to SLARRV 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 SLARRV.

D

          D is REAL array, dimension (N)


          On entry, the N diagonal elements of the diagonal matrix D.


          On exit, D may be overwritten.

L

          L is 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 split.) At the end of each block


          is stored the corresponding shift as given by SLARRE.


          On exit, L is overwritten.

PIVMIN

          PIVMIN is REAL


          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 REAL

RTOL1

          RTOL1 is REAL

RTOL2

          RTOL2 is REAL


           Parameters for bisection.


           An interval [LEFT,RIGHT] has converged if


           RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

W

          W is 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

          WERR is REAL array, dimension (N)


          The first M elements contain the semiwidth of the uncertainty


          interval of the corresponding eigenvalue in W

WGAP

          WGAP is REAL 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 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

          Z is REAL 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 REAL array, dimension (12*N)

IWORK

          IWORK is INTEGER array, dimension (7*N)

INFO

          INFO is INTEGER


          = 0:  successful exit


          > 0:  A problem occurred in SLARRV.


          < 0:  One of the called subroutines signaled an internal problem.


                Needs inspection of the corresponding parameter IINFO


                for further information.


          =-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/MINRGP, the user should be aware that


                he might be trading in precision 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 converge to


                full accuracy in MAXITR steps.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2016

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