# ?larrv

Computes the eigenvectors of the tridiagonal matrix T = L*D* LT given L, D and the eigenvalues of L*D* LT.

## Syntax

call slarrv( 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 )

call 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 )

call clarrv( 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 )

call 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 )

## Description

The routine ?larrv computes the eigenvectors of the tridiagonal matrix T = L*D* LT given L, D and approximations to the eigenvalues of L*D* LT.

The input eigenvalues should have been computed by slarre for real flavors (slarrv/clarrv) and by dlarre for double precision flavors (dlarre/zlarre).

## Input Parameters

n

  INTEGER. The order of the matrix. `n ≥ 0`.

vl, vu

  REAL for slarrv/clarrv

  DOUBLE PRECISION for dlarrv/zlarrv

  Lower and upper bounds respectively 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

  REAL for slarrv/clarrv

  DOUBLE PRECISION for dlarrv/zlarrv

  Array, DIMENSION (n). On entry, the n diagonal elements of the diagonal matrix D.

l

  REAL for slarrv/clarrv

  DOUBLE PRECISION for dlarrv/zlarrv

  Array, DIMENSION (n).

  On entry, the (n-1) subdiagonal elements of the unit bidiagonal matrix L are contained in elements 1 to n-1 of L if the matrix is not splitted. At the end of each block the corresponding shift is stored as given by slarre for real flavors and by dlarre for double precision flavors.

pivmin

  REAL for slarrv/clarrv

  DOUBLE PRECISION for dlarrv/zlarrv

  The minimum pivot allowed in the Sturm sequence.

isplit

  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

  INTEGER. The total number of eigenvalues found.

  `0 ≤ m ≤ n`. If `range = 'A'`, `m = n`, and if `range = 'I'`, `m = iu - il +1`.

dol, dou

  INTEGER.

  If you want to compute only selected eigenvectors from all the eigenvalues supplied, specify an index range dol:dou. Or else apply the setting dol=1, dou=m. Note that dol and dou refer to the order in which the eigenvalues are stored in w.

  If you want 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, rtol1, rtol2

  REAL for slarrv/clarrv

  DOUBLE PRECISION for dlarrv/zlarrv

  Parameters for bisection. An interval [LEFT,RIGHT] has converged if `RIGHT-LEFT.LT.MAX( rtol1*gap, rtol2*max(|LEFT|,|RIGHT|) )`.

w

  REAL for slarrv/clarrv

  DOUBLE PRECISION for dlarrv/zlarrv

  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 ?larre is expected here). These eigenvalues are set with respect to the shift of the corresponding root representation for their block.

werr

  REAL for slarrv/clarrv

  DOUBLE PRECISION for dlarrv/zlarrv

  Array, DIMENSION (n). The first m elements contain the semiwidth of the uncertainty interval of the corresponding eigenvalue in w.

wgap

  REAL for slarrv/clarrv

  DOUBLE PRECISION for dlarrv/zlarrv

  Array, DIMENSION (n). The separation from the right neighbor eigenvalue in w.

iblock

  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

  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

  REAL for slarrv/clarrv

  DOUBLE PRECISION for dlarrv/zlarrv

  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.

ldz

  INTEGER. The leading dimension of the output array Z. `ldz ≥ 1`, and if `jobz = 'V'`, `ldz ≥ max(1,n)`.

work

  REAL for slarrv/clarrv

  DOUBLE PRECISION for dlarrv/zlarrv

  Workspace array, DIMENSION (12*n).

iwork

  INTEGER.

  Workspace array, DIMENSION (7*n).

## Output Parameters

d

  On exit, d may be overwritten.

l

  On exit, l is overwritten.

w

  On exit, w holds the eigenvalues of the unshifted matrix.

werr

  On exit, werr contains refined values of its input approximations.

wgap

  On exit, wgap contains refined values of its input approximations. Very small gaps are changed.

z

  REAL for slarrv

  DOUBLE PRECISION for dlarrv

  COMPLEX for clarrv

  COMPLEX*16 for zlarrv

  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.

isuppz

  INTEGER .

  Array, DIMENSION `(2*max(1,m))`. The support of the eigenvectors in z, that is, the indices indicating the nonzero elements in z. The i-th eigenvector is nonzero only in elements isuppz(2i-1) through isuppz(2i).

info

  INTEGER.

  If `info = 0`: successful exit

  If `info > 0`: A problem occured in ?larrv. If `info = 5`, the Rayleigh Quotient Iteration failed to converge to full accuracy.

  If `info < 0`: One of the called subroutines signaled an internal problem. Inspection of the corresponding parameter info for further information is required.

  - If `info = -1`, there is a problem in ?larrb when refining a child eigenvalue;

  - If `info = -2`, there is a problem in ?larrf when computing the relatively robust representation (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, you should be aware that you might be trading in precision when you decrease minrgp.

  - If `info = -3`, there is a problem in ?larrb when refining a single eigenvalue after the Rayleigh correction was rejected.

## See Also

?larrb Provides limited bisection to locate eigenvalues for more accuracy.
?larre Given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.
?larrf Finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.
Parent topic: Auxiliary Routines