Man page - larre(3)

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Manual

larre

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine dlarre (character range, integer n, double precision vl, doubleprecision vu, integer il, integer iu, double precision, dimension( * )d, double precision, dimension( * ) e, double precision, dimension( * )e2, double precision rtol1, double precision rtol2, double precisionspltol, integer nsplit, integer, dimension( * ) isplit, integer m,double precision, dimension( * ) w, double precision, dimension( * )werr, double precision, dimension( * ) wgap, integer, dimension( * )iblock, integer, dimension( * ) indexw, double precision, dimension( *) gers, double precision pivmin, double precision, dimension( * ) work,integer, dimension( * ) iwork, integer info)
subroutine slarre (character range, integer n, real vl, real vu, integeril, integer iu, real, dimension( * ) d, real, dimension( * ) e, real,dimension( * ) e2, real rtol1, real rtol2, real spltol, integer nsplit,integer, dimension( * ) isplit, integer m, real, dimension( * ) w,real, dimension( * ) werr, real, dimension( * ) wgap, integer,dimension( * ) iblock, integer, dimension( * ) indexw, real, dimension(* ) gers, real pivmin, real, dimension( * ) work, integer, dimension( *) iwork, integer info)
Author

NAME

larre - larre: step in stemr

SYNOPSIS

Functions

subroutine dlarre (range, n, vl, vu, il, iu, d, e, e2, rtol1, rtol2, spltol, nsplit, isplit, m, w, werr, wgap, iblock, indexw, gers, pivmin, work, iwork, info)
DLARRE given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.
subroutine slarre (range, n, vl, vu, il, iu, d, e, e2, rtol1, rtol2, spltol, nsplit, isplit, m, w, werr, wgap, iblock, indexw, gers, pivmin, work, iwork, info)
SLARRE given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.

Detailed Description

Function Documentation

subroutine dlarre (character range, integer n, double precision vl, doubleprecision vu, integer il, integer iu, double precision, dimension( * )d, double precision, dimension( * ) e, double precision, dimension( * )e2, double precision rtol1, double precision rtol2, double precisionspltol, integer nsplit, integer, dimension( * ) isplit, integer m,double precision, dimension( * ) w, double precision, dimension( * )werr, double precision, dimension( * ) wgap, integer, dimension( * )iblock, integer, dimension( * ) indexw, double precision, dimension( *) gers, double precision pivmin, double precision, dimension( * ) work,integer, dimension( * ) iwork, integer info)

DLARRE given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.

Purpose:

To find the desired eigenvalues of a given real symmetric
tridiagonal matrix T, DLARRE sets any ’small’ off-diagonal
elements to zero, and for each unreduced block T_i, it finds
(a) a suitable shift at one end of the block’s spectrum,
(b) the base representation, T_i - sigma_i I = L_i D_i L_iˆT, and
(c) eigenvalues of each L_i D_i L_iˆT.
The representations and eigenvalues found are then used by
DSTEMR to compute the eigenvectors of T.
The accuracy varies depending on whether bisection is used to
find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to
compute all and then discard any unwanted one.
As an added benefit, DLARRE also outputs the n
Gerschgorin intervals for the matrices L_i D_i L_iˆT.

Parameters

RANGE

RANGE is CHARACTER*1
= ’A’: (’All’) all eigenvalues will be found.
= ’V’: (’Value’) all eigenvalues in the half-open interval
(VL, VU] will be found.
= ’I’: (’Index’) the IL-th through IU-th eigenvalues (of the
entire matrix) will be found.

N

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

VL

VL is DOUBLE PRECISION
If RANGE=’V’, the lower bound for the eigenvalues.
Eigenvalues less than or equal to VL, or greater than VU,
will not be returned. VL < VU.
If RANGE=’I’ or =’A’, DLARRE computes bounds on the desired
part of the spectrum.

VU

VU is DOUBLE PRECISION
If RANGE=’V’, the upper bound for the eigenvalues.
Eigenvalues less than or equal to VL, or greater than VU,
will not be returned. VL < VU.
If RANGE=’I’ or =’A’, DLARRE computes bounds on the desired
part of the spectrum.

IL

IL is INTEGER
If RANGE=’I’, the index of the
smallest eigenvalue to be returned.
1 <= IL <= IU <= N.

IU

IU is INTEGER
If RANGE=’I’, the index of the
largest eigenvalue to be returned.
1 <= IL <= IU <= N.

D

D is DOUBLE PRECISION array, dimension (N)
On entry, the N diagonal elements of the tridiagonal
matrix T.
On exit, the N diagonal elements of the diagonal
matrices D_i.

E

E is DOUBLE PRECISION array, dimension (N)
On entry, the first (N-1) entries contain the subdiagonal
elements of the tridiagonal matrix T; E(N) need not be set.
On exit, E contains the subdiagonal elements of the unit
bidiagonal matrices L_i. The entries E( ISPLIT( I ) ),
1 <= I <= NSPLIT, contain the base points sigma_i on output.

E2

E2 is DOUBLE PRECISION array, dimension (N)
On entry, the first (N-1) entries contain the SQUARES of the
subdiagonal elements of the tridiagonal matrix T;
E2(N) need not be set.
On exit, the entries E2( ISPLIT( I ) ),
1 <= I <= NSPLIT, have been set to zero

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|) )

SPLTOL

SPLTOL is DOUBLE PRECISION
The threshold for splitting.

NSPLIT

NSPLIT is INTEGER
The number of blocks T splits into. 1 <= NSPLIT <= N.

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., and the NSPLIT-th consists of rows/columns
ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.

M

M is INTEGER
The total number of eigenvalues (of all L_i D_i L_iˆT)
found.

W

W is DOUBLE PRECISION array, dimension (N)
The first M elements contain the eigenvalues. The
eigenvalues of each of the blocks, L_i D_i L_iˆT, are
sorted in ascending order ( DLARRE may use the
remaining N-M elements as workspace).

WERR

WERR is DOUBLE PRECISION array, dimension (N)
The error bound on the corresponding eigenvalue in W.

WGAP

WGAP is DOUBLE PRECISION array, dimension (N)
The separation from the right neighbor eigenvalue in W.
The gap is only with respect to the eigenvalues of the same block
as each block has its own representation tree.
Exception: at the right end of a block we store the left gap

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 block 2

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)).

PIVMIN

PIVMIN is DOUBLE PRECISION
The minimum pivot in the Sturm sequence for T.

WORK

WORK is DOUBLE PRECISION array, dimension (6*N)
Workspace.

IWORK

IWORK is INTEGER array, dimension (5*N)
Workspace.

INFO

INFO is INTEGER
= 0: successful exit
> 0: A problem occurred in DLARRE.
< 0: One of the called subroutines signaled an internal problem.
Needs inspection of the corresponding parameter IINFO
for further information.

=-1: Problem in DLARRD.
= 2: No base representation could be found in MAXTRY iterations.
Increasing MAXTRY and recompilation might be a remedy.
=-3: Problem in DLARRB when computing the refined root
representation for DLASQ2.
=-4: Problem in DLARRB when preforming bisection on the
desired part of the spectrum.
=-5: Problem in DLASQ2.
=-6: Problem in DLASQ2.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The base representations are required to suffer very little
element growth and consequently define all their eigenvalues to
high relative accuracy.

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 slarre (character range, integer n, real vl, real vu, integeril, integer iu, real, dimension( * ) d, real, dimension( * ) e, real,dimension( * ) e2, real rtol1, real rtol2, real spltol, integer nsplit,integer, dimension( * ) isplit, integer m, real, dimension( * ) w,real, dimension( * ) werr, real, dimension( * ) wgap, integer,dimension( * ) iblock, integer, dimension( * ) indexw, real, dimension(* ) gers, real pivmin, real, dimension( * ) work, integer, dimension( *) iwork, integer info)

SLARRE given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.

Purpose:

To find the desired eigenvalues of a given real symmetric
tridiagonal matrix T, SLARRE sets any ’small’ off-diagonal
elements to zero, and for each unreduced block T_i, it finds
(a) a suitable shift at one end of the block’s spectrum,
(b) the base representation, T_i - sigma_i I = L_i D_i L_iˆT, and
(c) eigenvalues of each L_i D_i L_iˆT.
The representations and eigenvalues found are then used by
SSTEMR to compute the eigenvectors of T.
The accuracy varies depending on whether bisection is used to
find a few eigenvalues or the dqds algorithm (subroutine SLASQ2) to
compute all and then discard any unwanted one.
As an added benefit, SLARRE also outputs the n
Gerschgorin intervals for the matrices L_i D_i L_iˆT.

Parameters

RANGE

RANGE is CHARACTER*1
= ’A’: (’All’) all eigenvalues will be found.
= ’V’: (’Value’) all eigenvalues in the half-open interval
(VL, VU] will be found.
= ’I’: (’Index’) the IL-th through IU-th eigenvalues (of the
entire matrix) will be found.

N

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

VL

VL is REAL
If RANGE=’V’, the lower bound for the eigenvalues.
Eigenvalues less than or equal to VL, or greater than VU,
will not be returned. VL < VU.
If RANGE=’I’ or =’A’, SLARRE computes bounds on the desired
part of the spectrum.

VU

VU is REAL
If RANGE=’V’, the upper bound for the eigenvalues.
Eigenvalues less than or equal to VL, or greater than VU,
will not be returned. VL < VU.
If RANGE=’I’ or =’A’, SLARRE computes bounds on the desired
part of the spectrum.

IL

IL is INTEGER
If RANGE=’I’, the index of the
smallest eigenvalue to be returned.
1 <= IL <= IU <= N.

IU

IU is INTEGER
If RANGE=’I’, the index of the
largest eigenvalue to be returned.
1 <= IL <= IU <= N.

D

D is REAL array, dimension (N)
On entry, the N diagonal elements of the tridiagonal
matrix T.
On exit, the N diagonal elements of the diagonal
matrices D_i.

E

E is REAL array, dimension (N)
On entry, the first (N-1) entries contain the subdiagonal
elements of the tridiagonal matrix T; E(N) need not be set.
On exit, E contains the subdiagonal elements of the unit
bidiagonal matrices L_i. The entries E( ISPLIT( I ) ),
1 <= I <= NSPLIT, contain the base points sigma_i on output.

E2

E2 is REAL array, dimension (N)
On entry, the first (N-1) entries contain the SQUARES of the
subdiagonal elements of the tridiagonal matrix T;
E2(N) need not be set.
On exit, the entries E2( ISPLIT( I ) ),
1 <= I <= NSPLIT, have been set to zero

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|) )

SPLTOL

SPLTOL is REAL
The threshold for splitting.

NSPLIT

NSPLIT is INTEGER
The number of blocks T splits into. 1 <= NSPLIT <= N.

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., and the NSPLIT-th consists of rows/columns
ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.

M

M is INTEGER
The total number of eigenvalues (of all L_i D_i L_iˆT)
found.

W

W is REAL array, dimension (N)
The first M elements contain the eigenvalues. The
eigenvalues of each of the blocks, L_i D_i L_iˆT, are
sorted in ascending order ( SLARRE may use the
remaining N-M elements as workspace).

WERR

WERR is REAL array, dimension (N)
The error bound on the corresponding eigenvalue in W.

WGAP

WGAP is REAL array, dimension (N)
The separation from the right neighbor eigenvalue in W.
The gap is only with respect to the eigenvalues of the same block
as each block has its own representation tree.
Exception: at the right end of a block we store the left gap

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 block 2

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)).

PIVMIN

PIVMIN is REAL
The minimum pivot in the Sturm sequence for T.

WORK

WORK is REAL array, dimension (6*N)
Workspace.

IWORK

IWORK is INTEGER array, dimension (5*N)
Workspace.

INFO

INFO is INTEGER
= 0: successful exit
> 0: A problem occurred in SLARRE.
< 0: One of the called subroutines signaled an internal problem.
Needs inspection of the corresponding parameter IINFO
for further information.

=-1: Problem in SLARRD.
= 2: No base representation could be found in MAXTRY iterations.
Increasing MAXTRY and recompilation might be a remedy.
=-3: Problem in SLARRB when computing the refined root
representation for SLASQ2.
=-4: Problem in SLARRB when preforming bisection on the
desired part of the spectrum.
=-5: Problem in SLASQ2.
=-6: Problem in SLASQ2.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The base representations are required to suffer very little
element growth and consequently define all their eigenvalues to
high relative accuracy.

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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