Man page - larrv(3)

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Manual

larrv

NAME
SYNOPSIS
Functions
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, realrtol2, 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)
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 precisionrtol1, 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, doubleprecision, dimension( * ) work, integer, dimension( * ) iwork, integerinfo)
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, realrtol2, 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)
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 precisionrtol1, 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)
Author

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, realrtol2, 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 = ā€™Vā€™, 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 childā€™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 userā€™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 precisionrtol1, 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, doubleprecision, dimension( * ) work, integer, dimension( * ) iwork, integerinfo)

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 ā€™convergenceā€™. 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 = ā€™Vā€™, 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 childā€™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 userā€™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, realrtol2, 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 ā€™convergenceā€™. 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 = ā€™Vā€™, 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 childā€™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 userā€™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 precisionrtol1, 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 = ā€™Vā€™, 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 childā€™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 userā€™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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