larrv(3)

NAME
larrv - larrv: eig tridiagonal, step in stemr & stegr

SYNOPSIS

  Functions

subroutine 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)
CLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.
subroutine 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)
DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.
subroutine 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)
SLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.
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)
ZLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

Detailed Description

Function Documentation

  subroutine clarrv (integer n, real vl, real vu, real, dimension( * ) d, real, dimension( * ) l, real pivmin, integer, dimension( * ) isplit, integer m, integer dol, integer dou, real minrgp, real rtol1, real rtol2, real, dimension( * ) w, real, dimension( * ) werr, real, dimension( * ) wgap, integer, dimension( * ) iblock, integer, dimension( * ) indexw, real, dimension( * ) gers, complex, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

CLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

Purpose:

    !>
    !> CLARRV 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 SLARRE.
    !>

Parameters
    N

    !>          N is INTEGER
    !>          The order of the matrix.  N >= 0.
    !>

    VL

    !>          VL is REAL
    !>          Lower bound 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.
    !>

    VU

    !>          VU is REAL
    !>          Upper bound 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

    !>          D is REAL array, dimension (N)
    !>          On entry, the N diagonal elements of the diagonal matrix D.
    !>          On exit, D may be overwritten.
    !>

    L

    !>          L is REAL 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 split.) At the end of each block
    !>          is stored the corresponding shift as given by SLARRE.
    !>          On exit, L is overwritten.
    !>

    PIVMIN

    !>          PIVMIN is REAL
    !>          The minimum pivot allowed in the Sturm sequence.
    !>

    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.
    !>

    M

    !>          M is INTEGER
    !>          The total number of input eigenvalues.  0 <= M <= N.
    !>

    DOL

    !>          DOL is INTEGER
    !>

    DOU

    !>          DOU is 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

    !>          MINRGP is REAL
    !>

    RTOL1

    !>          RTOL1 is REAL
    !>

    RTOL2

    !>          RTOL2 is REAL
    !>           Parameters for bisection.
    !>           An interval [LEFT,RIGHT] has converged if
    !>           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
    !>

    W

    !>          W is REAL 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 SLARRE 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

    !>          WERR is REAL array, dimension (N)
    !>          The first M elements contain the semiwidth of the uncertainty
    !>          interval of the corresponding eigenvalue in W
    !>

    WGAP

    !>          WGAP is REAL array, dimension (N)
    !>          The separation from the right neighbor eigenvalue in W.
    !>

    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 the second block.
    !>

    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)). The Gerschgorin intervals should
    !>          be computed from the original UNshifted matrix.
    !>

    Z

    !>          Z is COMPLEX 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

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z.  LDZ >= 1, and if
    !>          JOBZ = 'V', LDZ >= max(1,N).
    !>

    ISUPPZ

    !>          ISUPPZ is 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

    !>          WORK is REAL array, dimension (12*N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (7*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>
    !>          > 0:  A problem occurred in CLARRV.
    !>          < 0:  One of the called subroutines signaled an internal problem.
    !>                Needs inspection of the corresponding parameter IINFO
    !>                for further information.
    !>
    !>          =-1:  Problem in SLARRB when refining a child's eigenvalues.
    !>          =-2:  Problem in SLARRF 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 SLARRB 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    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

  subroutine dlarrv (integer n, double precision vl, double precision vu, double precision, dimension( * ) d, double precision, dimension( * ) l, double precision pivmin, integer, dimension( * ) isplit, integer m, integer dol, integer dou, double precision minrgp, double precision rtol1, double precision rtol2, double precision, dimension( * ) w, double precision, dimension( * ) werr, double precision, dimension( * ) wgap, integer, dimension( * ) iblock, integer, dimension( * ) indexw, double precision, dimension( * ) gers, double precision, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

Purpose:

    !>
    !> DLARRV 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.
    !>

Parameters
    N

    !>          N is INTEGER
    !>          The order of the matrix.  N >= 0.
    !>

    VL

    !>          VL is DOUBLE PRECISION
    !>          Lower bound 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.
    !>

    VU

    !>          VU is DOUBLE PRECISION
    !>          Upper bound of the interval that contains the desired
    !>          eigenvalues. VL < VU.
    !>          Note: VU is currently not used by this implementation of DLARRV, VU is
    !>          passed to DLARRV because it could be used compute gaps on the right end
    !>          of the extremal eigenvalues. However, with not much initial accuracy in
    !>          LAMBDA and VU, the formula can lead to an overestimation of the right gap
    !>          and thus to inadequately early RQI 'convergence'. This is currently
    !>          prevented this by forcing a small right gap. And so it turns out that VU
    !>          is currently not used by this implementation of DLARRV.
    !>

    D

    !>          D is DOUBLE PRECISION array, dimension (N)
    !>          On entry, the N diagonal elements of the diagonal matrix D.
    !>          On exit, D may be overwritten.
    !>

    L

    !>          L is 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 split.) At the end of each block
    !>          is stored the corresponding shift as given by DLARRE.
    !>          On exit, L is overwritten.
    !>

    PIVMIN

    !>          PIVMIN is DOUBLE PRECISION
    !>          The minimum pivot allowed in the Sturm sequence.
    !>

    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.
    !>

    M

    !>          M is INTEGER
    !>          The total number of input eigenvalues.  0 <= M <= N.
    !>

    DOL

    !>          DOL is INTEGER
    !>

    DOU

    !>          DOU is 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

    !>          MINRGP is DOUBLE PRECISION
    !>

    RTOL1

    !>          RTOL1 is DOUBLE PRECISION
    !>

    RTOL2

    !>          RTOL2 is DOUBLE PRECISION
    !>           Parameters for bisection.
    !>           An interval [LEFT,RIGHT] has converged if
    !>           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
    !>

    W

    !>          W is 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

    !>          WERR is DOUBLE PRECISION array, dimension (N)
    !>          The first M elements contain the semiwidth of the uncertainty
    !>          interval of the corresponding eigenvalue in W
    !>

    WGAP

    !>          WGAP is DOUBLE PRECISION array, dimension (N)
    !>          The separation from the right neighbor eigenvalue in W.
    !>

    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 the second block.
    !>

    GERS

    !>          GERS is 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

    !>          Z is DOUBLE PRECISION 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

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z.  LDZ >= 1, and if
    !>          JOBZ = 'V', LDZ >= max(1,N).
    !>

    ISUPPZ

    !>          ISUPPZ is 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

    !>          WORK is DOUBLE PRECISION array, dimension (12*N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (7*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>
    !>          > 0:  A problem occurred in DLARRV.
    !>          < 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    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

  subroutine slarrv (integer n, real vl, real vu, real, dimension( * ) d, real, dimension( * ) l, real pivmin, integer, dimension( * ) isplit, integer m, integer dol, integer dou, real minrgp, real rtol1, real rtol2, real, dimension( * ) w, real, dimension( * ) werr, real, dimension( * ) wgap, integer, dimension( * ) iblock, integer, dimension( * ) indexw, real, dimension( * ) gers, real, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

Purpose:

    !>
    !> SLARRV 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 SLARRE.
    !>

Parameters
    N

    !>          N is INTEGER
    !>          The order of the matrix.  N >= 0.
    !>

    VL

    !>          VL is REAL
    !>          Lower bound 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.
    !>

    VU

    !>          VU is REAL
    !>          Upper bound of the interval that contains the desired
    !>          eigenvalues. VL < VU.
    !>          Note: VU is currently not used by this implementation of SLARRV, VU is
    !>          passed to SLARRV because it could be used compute gaps on the right end
    !>          of the extremal eigenvalues. However, with not much initial accuracy in
    !>          LAMBDA and VU, the formula can lead to an overestimation of the right gap
    !>          and thus to inadequately early RQI 'convergence'. This is currently
    !>          prevented this by forcing a small right gap. And so it turns out that VU
    !>          is currently not used by this implementation of SLARRV.
    !>

    D

    !>          D is REAL array, dimension (N)
    !>          On entry, the N diagonal elements of the diagonal matrix D.
    !>          On exit, D may be overwritten.
    !>

    L

    !>          L is REAL 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 split.) At the end of each block
    !>          is stored the corresponding shift as given by SLARRE.
    !>          On exit, L is overwritten.
    !>

    PIVMIN

    !>          PIVMIN is REAL
    !>          The minimum pivot allowed in the Sturm sequence.
    !>

    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.
    !>

    M

    !>          M is INTEGER
    !>          The total number of input eigenvalues.  0 <= M <= N.
    !>

    DOL

    !>          DOL is INTEGER
    !>

    DOU

    !>          DOU is 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

    !>          MINRGP is REAL
    !>

    RTOL1

    !>          RTOL1 is REAL
    !>

    RTOL2

    !>          RTOL2 is REAL
    !>           Parameters for bisection.
    !>           An interval [LEFT,RIGHT] has converged if
    !>           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
    !>

    W

    !>          W is REAL 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 SLARRE 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

    !>          WERR is REAL array, dimension (N)
    !>          The first M elements contain the semiwidth of the uncertainty
    !>          interval of the corresponding eigenvalue in W
    !>

    WGAP

    !>          WGAP is REAL array, dimension (N)
    !>          The separation from the right neighbor eigenvalue in W.
    !>

    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 the second block.
    !>

    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)). The Gerschgorin intervals should
    !>          be computed from the original UNshifted matrix.
    !>

    Z

    !>          Z is REAL 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

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z.  LDZ >= 1, and if
    !>          JOBZ = 'V', LDZ >= max(1,N).
    !>

    ISUPPZ

    !>          ISUPPZ is 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

    !>          WORK is REAL array, dimension (12*N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (7*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>
    !>          > 0:  A problem occurred in SLARRV.
    !>          < 0:  One of the called subroutines signaled an internal problem.
    !>                Needs inspection of the corresponding parameter IINFO
    !>                for further information.
    !>
    !>          =-1:  Problem in SLARRB when refining a child's eigenvalues.
    !>          =-2:  Problem in SLARRF 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 SLARRB 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    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

  subroutine zlarrv (integer n, double precision vl, double precision vu, double precision, dimension( * ) d, double precision, dimension( * ) l, double precision pivmin, integer, dimension( * ) isplit, integer m, integer dol, integer dou, double precision minrgp, double precision rtol1, double precision rtol2, double precision, dimension( * ) w, double precision, dimension( * ) werr, double precision, dimension( * ) wgap, integer, dimension( * ) iblock, integer, dimension( * ) indexw, double precision, dimension( * ) gers, complex*16, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

ZLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

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.
    !>

Parameters
    N

    !>          N is INTEGER
    !>          The order of the matrix.  N >= 0.
    !>

    VL

    !>          VL is DOUBLE PRECISION
    !>          Lower bound 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.
    !>

    VU

    !>          VU is DOUBLE PRECISION
    !>          Upper bound 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

    !>          D is DOUBLE PRECISION array, dimension (N)
    !>          On entry, the N diagonal elements of the diagonal matrix D.
    !>          On exit, D may be overwritten.
    !>

    L

    !>          L is 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 split.) At the end of each block
    !>          is stored the corresponding shift as given by DLARRE.
    !>          On exit, L is overwritten.
    !>

    PIVMIN

    !>          PIVMIN is DOUBLE PRECISION
    !>          The minimum pivot allowed in the Sturm sequence.
    !>

    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.
    !>

    M

    !>          M is INTEGER
    !>          The total number of input eigenvalues.  0 <= M <= N.
    !>

    DOL

    !>          DOL is INTEGER
    !>

    DOU

    !>          DOU is 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

    !>          MINRGP is DOUBLE PRECISION
    !>

    RTOL1

    !>          RTOL1 is DOUBLE PRECISION
    !>

    RTOL2

    !>          RTOL2 is DOUBLE PRECISION
    !>           Parameters for bisection.
    !>           An interval [LEFT,RIGHT] has converged if
    !>           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
    !>

    W

    !>          W is 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

    !>          WERR is DOUBLE PRECISION array, dimension (N)
    !>          The first M elements contain the semiwidth of the uncertainty
    !>          interval of the corresponding eigenvalue in W
    !>

    WGAP

    !>          WGAP is DOUBLE PRECISION array, dimension (N)
    !>          The separation from the right neighbor eigenvalue in W.
    !>

    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 the second block.
    !>

    GERS

    !>          GERS is 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

    !>          Z is 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

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z.  LDZ >= 1, and if
    !>          JOBZ = 'V', LDZ >= max(1,N).
    !>

    ISUPPZ

    !>          ISUPPZ is 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

    !>          WORK is DOUBLE PRECISION array, dimension (12*N)
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (7*N)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>
    !>          > 0:  A problem occurred 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    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

Author

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