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## 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

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## 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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