To find the desired eigenvalues of a given real symmetric


 tridiagonal matrix T, SLARRE sets any "small" off-diagonal


 elements to zero, and for each unreduced block T_i, it finds


 (a) a suitable shift at one end of the block's spectrum,


 (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and


 (c) eigenvalues of each L_i D_i L_i^T.


 The representations and eigenvalues found are then used by


 SSTEMR to compute the eigenvectors of T.


 The accuracy varies depending on whether bisection is used to


 find a few eigenvalues or the dqds algorithm (subroutine SLASQ2) to


 conpute all and then discard any unwanted one.


 As an added benefit, SLARRE also outputs the n


 Gerschgorin intervals for the matrices L_i D_i L_i^T.

RANGE

          RANGE is CHARACTER*1


          = 'A': ("All")   all eigenvalues will be found.


          = 'V': ("Value") all eigenvalues in the half-open interval


                           (VL, VU] will be found.


          = 'I': ("Index") the IL-th through IU-th eigenvalues (of the


                           entire matrix) will be found.

N

          N is INTEGER


          The order of the matrix. N > 0.

VL

          VL is REAL


          If RANGE='V', the lower bound for the eigenvalues.


          Eigenvalues less than or equal to VL, or greater than VU,


          will not be returned.  VL < VU.


          If RANGE='I' or ='A', SLARRE computes bounds on the desired


          part of the spectrum.

VU

          VU is REAL


          If RANGE='V', the upper bound for the eigenvalues.


          Eigenvalues less than or equal to VL, or greater than VU,


          will not be returned.  VL < VU.


          If RANGE='I' or ='A', SLARRE computes bounds on the desired


          part of the spectrum.

IL

          IL is INTEGER


          If RANGE='I', the index of the


          smallest eigenvalue to be returned.


          1 <= IL <= IU <= N.

IU

          IU is INTEGER


          If RANGE='I', the index of the


          largest eigenvalue to be returned.


          1 <= IL <= IU <= N.

D

          D is REAL array, dimension (N)


          On entry, the N diagonal elements of the tridiagonal


          matrix T.


          On exit, the N diagonal elements of the diagonal


          matrices D_i.

E

          E is REAL array, dimension (N)


          On entry, the first (N-1) entries contain the subdiagonal


          elements of the tridiagonal matrix T; E(N) need not be set.


          On exit, E contains the subdiagonal elements of the unit


          bidiagonal matrices L_i. The entries E( ISPLIT( I ) ),


          1 <= I <= NSPLIT, contain the base points sigma_i on output.

E2

          E2 is REAL array, dimension (N)


          On entry, the first (N-1) entries contain the SQUARES of the


          subdiagonal elements of the tridiagonal matrix T;


          E2(N) need not be set.


          On exit, the entries E2( ISPLIT( I ) ),


          1 <= I <= NSPLIT, have been set to zero

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|) )

SPLTOL

          SPLTOL is REAL


          The threshold for splitting.

NSPLIT

          NSPLIT is INTEGER


          The number of blocks T splits into. 1 <= NSPLIT <= N.

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., and the NSPLIT-th consists of rows/columns


          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.

M

          M is INTEGER


          The total number of eigenvalues (of all L_i D_i L_i^T)


          found.

W

          W is REAL array, dimension (N)


          The first M elements contain the eigenvalues. The


          eigenvalues of each of the blocks, L_i D_i L_i^T, are


          sorted in ascending order ( SLARRE may use the


          remaining N-M elements as workspace).

WERR

          WERR is REAL array, dimension (N)


          The error bound on the corresponding eigenvalue in W.

WGAP

          WGAP is REAL array, dimension (N)


          The separation from the right neighbor eigenvalue in W.


          The gap is only with respect to the eigenvalues of the same block


          as each block has its own representation tree.


          Exception: at the right end of a block we store the left gap

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 block 2

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)).

PIVMIN

          PIVMIN is REAL


          The minimum pivot in the Sturm sequence for T.

WORK

          WORK is REAL array, dimension (6*N)


          Workspace.

IWORK

          IWORK is INTEGER array, dimension (5*N)


          Workspace.

INFO

          INFO is INTEGER


          = 0:  successful exit


          > 0:  A problem occurred in SLARRE.


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


                Needs inspection of the corresponding parameter IINFO


                for further information.


          =-1:  Problem in SLARRD.


          = 2:  No base representation could be found in MAXTRY iterations.


                Increasing MAXTRY and recompilation might be a remedy.


          =-3:  Problem in SLARRB when computing the refined root


                representation for SLASQ2.


          =-4:  Problem in SLARRB when preforming bisection on the


                desired part of the spectrum.


          =-5:  Problem in SLASQ2.


          =-6:  Problem in SLASQ2.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2016

  The base representations are required to suffer very little


  element growth and consequently define all their eigenvalues to


  high relative accuracy.

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