zlarrv(3S)

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

SYNOPSIS
     SUBROUTINE ZLARRV( 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( *
                        )

         COMPLEX*16     Z( LDZ, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     ZLARRV 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.


ARGUMENTS
     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) COMPLEX*16 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 ).

     GERSCH  (workspace) DOUBLE PRECISION array, dimension (2*N)

     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 ZSTEIN

FURTHER DETAILS
     Based on contributions by
        Inderjit Dhillon, IBM Almaden, USA
        Osni Marques, LBNL/NERSC, USA
        Ken Stanley, Computer Science Division, University of
          California at Berkeley, USA


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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