DLARRV - compute the eigenvectors of the tridiagonal matrix T = L D L^T given L, D and the eigenvalues of L D L^T

SUBROUTINE DLARRV(

N, D, L, ISPLIT, M, W, IBLOCK, GERSCH, TOL, Z, LDZ, ISUPPZ, WORK, IWORK, INFO )

INTEGER INFO, LDZ, M, N

DOUBLE PRECISION TOL

INTEGER IBLOCK( * ), ISPLIT( * ), ISUPPZ( * ), IWORK( * )

DOUBLE PRECISION D( * ), GERSCH( * ), L( * ), W( * ), WORK( * ), Z( LDZ, * )

DLARRV computes the eigenvectors of the tridiagonal matrix T = L D L^T given L, D and the eigenvalues of L D L^T. The input eigenvalues should have high relative accuracy with respect to the entries of L and D. The desired accuracy of the output can be specified by the input parameter TOL.

N (input) INTEGER

The order of the matrix. N >= 0.

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-1)

On entry, the (n-1) subdiagonal elements of the unit bidiagonal matrix L in elements 1 to N-1 of L. L(N) need not be set. On exit, L is overwritten.

ISPLIT (input) INTEGER array, dimension (N)

The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 through ISPLIT( 2 ), etc.

TOL (input) DOUBLE PRECISION

The absolute error tolerance for the eigenvalues/eigenvectors. Errors in the input eigenvalues must be bounded by TOL. The eigenvectors output have residual norms bounded by TOL, and the dot products between different eigenvectors are bounded by TOL. TOL must be at least N*EPS*|T|, where EPS is the machine precision and |T| is the 1-norm of the tridiagonal matrix.

M (input) INTEGER

The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W (input) DOUBLE PRECISION array, dimension (N)

The first M elements of W contain the 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 ). Errors in W must be bounded by TOL (see above).

IBLOCK (input) INTEGER array, dimension (N)

The submatrix indices associated with the corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to the first submatrix from the top, =2 if W(i) belongs to the second submatrix, etc.

Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )

If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix T corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used.

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 (13*N)

IWORK (workspace) INTEGER array, dimension (6*N)

INFO (output) INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = 1, internal error in DLARRB if INFO = 2, internal error in DSTEIN

Based on contributions by

Inderjit Dhillon, IBM Almaden, USA

Osni Marques, LBNL/NERSC, USA