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larrd

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine dlarrd (character range, character order, integer n, doubleprecision vl, double precision vu, integer il, integer iu, doubleprecision, dimension( * ) gers, double precision reltol, doubleprecision, dimension( * ) d, double precision, dimension( * ) e, doubleprecision, dimension( * ) e2, double precision pivmin, integer nsplit,integer, dimension( * ) isplit, integer m, double precision, dimension(* ) w, double precision, dimension( * ) werr, double precision wl,double precision wu, integer, dimension( * ) iblock, integer,dimension( * ) indexw, double precision, dimension( * ) work, integer,dimension( * ) iwork, integer info)
subroutine slarrd (character range, character order, integer n, real vl,real vu, integer il, integer iu, real, dimension( * ) gers, realreltol, real, dimension( * ) d, real, dimension( * ) e, real,dimension( * ) e2, real pivmin, integer nsplit, integer, dimension( * )isplit, integer m, real, dimension( * ) w, real, dimension( * ) werr,real wl, real wu, integer, dimension( * ) iblock, integer, dimension( *) indexw, real, dimension( * ) work, integer, dimension( * ) iwork,integer info)
Author

NAME

larrd - larrd: step in stemr, tridiag eig

SYNOPSIS

Functions

subroutine dlarrd (range, order, n, vl, vu, il, iu, gers, reltol, d, e, e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw, work, iwork, info)
DLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.
subroutine slarrd (range, order, n, vl, vu, il, iu, gers, reltol, d, e, e2, pivmin, nsplit, isplit, m, w, werr, wl, wu, iblock, indexw, work, iwork, info)
SLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.

Detailed Description

Function Documentation

subroutine dlarrd (character range, character order, integer n, doubleprecision vl, double precision vu, integer il, integer iu, doubleprecision, dimension( * ) gers, double precision reltol, doubleprecision, dimension( * ) d, double precision, dimension( * ) e, doubleprecision, dimension( * ) e2, double precision pivmin, integer nsplit,integer, dimension( * ) isplit, integer m, double precision, dimension(* ) w, double precision, dimension( * ) werr, double precision wl,double precision wu, integer, dimension( * ) iblock, integer,dimension( * ) indexw, double precision, dimension( * ) work, integer,dimension( * ) iwork, integer info)

DLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.

Purpose:

DLARRD computes the eigenvalues of a symmetric tridiagonal
matrix T to suitable accuracy. This is an auxiliary code to be
called from DSTEMR.
The user may ask for all eigenvalues, all eigenvalues
in the half-open interval (VL, VU], or the IL-th through IU-th
eigenvalues.

To avoid overflow, the matrix must be scaled so that its
largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
accuracy, it should not be much smaller than that.

See W. Kahan ’Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix’, Report CS41, Computer Science Dept., Stanford
University, July 21, 1966.

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.

ORDER

ORDER is CHARACTER*1
= ’B’: (’By Block’) the eigenvalues will be grouped by
split-off block (see IBLOCK, ISPLIT) and
ordered from smallest to largest within
the block.
= ’E’: (’Entire matrix’)
the eigenvalues for the entire matrix
will be ordered from smallest to
largest.

N

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

VL

VL is DOUBLE PRECISION
If RANGE=’V’, the lower bound of the interval to
be searched for eigenvalues. Eigenvalues less than or equal
to VL, or greater than VU, will not be returned. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

VU

VU is DOUBLE PRECISION
If RANGE=’V’, the upper bound of the interval to
be searched for eigenvalues. Eigenvalues less than or equal
to VL, or greater than VU, will not be returned. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

IL

IL is INTEGER
If RANGE=’I’, the index of the
smallest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = ’A’ or ’V’.

IU

IU is INTEGER
If RANGE=’I’, the index of the
largest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = ’A’ or ’V’.

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)).

RELTOL

RELTOL is DOUBLE PRECISION
The minimum relative width of an interval. When an interval
is narrower than RELTOL times the larger (in
magnitude) endpoint, then it is considered to be
sufficiently small, i.e., converged. Note: this should
always be at least radix*machine epsilon.

D

D is DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.

E

E is DOUBLE PRECISION array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix T.

E2

E2 is DOUBLE PRECISION array, dimension (N-1)
The (n-1) squared off-diagonal elements of the tridiagonal matrix T.

PIVMIN

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

NSPLIT

NSPLIT is INTEGER
The number of diagonal blocks in the matrix T.
1 <= NSPLIT <= N.

ISPLIT

ISPLIT is 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., and the NSPLIT-th consists of rows/columns
ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
(Only the first NSPLIT elements will actually be used, but
since the user cannot know a priori what value NSPLIT will
have, N words must be reserved for ISPLIT.)

M

M is INTEGER
The actual number of eigenvalues found. 0 <= M <= N.
(See also the description of INFO=2,3.)

W

W is DOUBLE PRECISION array, dimension (N)
On exit, the first M elements of W will contain the
eigenvalue approximations. DLARRD computes an interval
I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue
approximation is given as the interval midpoint
W(j)= ( a_j + b_j)/2. The corresponding error is bounded by
WERR(j) = abs( a_j - b_j)/2

WERR

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

WL

WL is DOUBLE PRECISION

WU

WU is DOUBLE PRECISION
The interval (WL, WU] contains all the wanted eigenvalues.
If RANGE=’V’, then WL=VL and WU=VU.
If RANGE=’A’, then WL and WU are the global Gerschgorin bounds
on the spectrum.
If RANGE=’I’, then WL and WU are computed by DLAEBZ from the
index range specified.

IBLOCK

IBLOCK is INTEGER array, dimension (N)
At each row/column j where E(j) is zero or small, the
matrix T is considered to split into a block diagonal
matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which
block (from 1 to the number of blocks) the eigenvalue W(i)
belongs. (DLARRD may use the remaining N-M elements as
workspace.)

INDEXW

INDEXW is INTEGER array, dimension (N)
The indices of the eigenvalues within each block (submatrix);
for example, INDEXW(i)= j and IBLOCK(i)=k imply that the
i-th eigenvalue W(i) is the j-th eigenvalue in block k.

WORK

WORK is DOUBLE PRECISION array, dimension (4*N)

IWORK

IWORK is INTEGER array, dimension (3*N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: some or all of the eigenvalues failed to converge or
were not computed:
=1 or 3: Bisection failed to converge for some
eigenvalues; these eigenvalues are flagged by a
negative block number. The effect is that the
eigenvalues may not be as accurate as the
absolute and relative tolerances. This is
generally caused by unexpectedly inaccurate
arithmetic.
=2 or 3: RANGE=’I’ only: Not all of the eigenvalues
IL:IU were found.
Effect: M < IU+1-IL
Cause: non-monotonic arithmetic, causing the
Sturm sequence to be non-monotonic.
Cure: recalculate, using RANGE=’A’, and pick
out eigenvalues IL:IU. In some cases,
increasing the PARAMETER ’FUDGE’ may
make things work.
= 4: RANGE=’I’, and the Gershgorin interval
initially used was too small. No eigenvalues
were computed.
Probable cause: your machine has sloppy
floating-point arithmetic.
Cure: Increase the PARAMETER ’FUDGE’,
recompile, and try again.

Internal Parameters:

FUDGE DOUBLE PRECISION, default = 2
A ’fudge factor’ to widen the Gershgorin intervals. Ideally,
a value of 1 should work, but on machines with sloppy
arithmetic, this needs to be larger. The default for
publicly released versions should be large enough to handle
the worst machine around. Note that this has no effect
on accuracy of the solution.

Contributors:

W. Kahan, University of California, Berkeley, USA
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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine slarrd (character range, character order, integer n, real vl,real vu, integer il, integer iu, real, dimension( * ) gers, realreltol, real, dimension( * ) d, real, dimension( * ) e, real,dimension( * ) e2, real pivmin, integer nsplit, integer, dimension( * )isplit, integer m, real, dimension( * ) w, real, dimension( * ) werr,real wl, real wu, integer, dimension( * ) iblock, integer, dimension( *) indexw, real, dimension( * ) work, integer, dimension( * ) iwork,integer info)

SLARRD computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.

Purpose:

SLARRD computes the eigenvalues of a symmetric tridiagonal
matrix T to suitable accuracy. This is an auxiliary code to be
called from SSTEMR.
The user may ask for all eigenvalues, all eigenvalues
in the half-open interval (VL, VU], or the IL-th through IU-th
eigenvalues.

To avoid overflow, the matrix must be scaled so that its
largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
accuracy, it should not be much smaller than that.

See W. Kahan ’Accurate Eigenvalues of a Symmetric Tridiagonal
Matrix’, Report CS41, Computer Science Dept., Stanford
University, July 21, 1966.

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.

ORDER

ORDER is CHARACTER*1
= ’B’: (’By Block’) the eigenvalues will be grouped by
split-off block (see IBLOCK, ISPLIT) and
ordered from smallest to largest within
the block.
= ’E’: (’Entire matrix’)
the eigenvalues for the entire matrix
will be ordered from smallest to
largest.

N

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

VL

VL is REAL
If RANGE=’V’, the lower bound of the interval to
be searched for eigenvalues. Eigenvalues less than or equal
to VL, or greater than VU, will not be returned. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

VU

VU is REAL
If RANGE=’V’, the upper bound of the interval to
be searched for eigenvalues. Eigenvalues less than or equal
to VL, or greater than VU, will not be returned. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

IL

IL is INTEGER
If RANGE=’I’, the index of the
smallest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = ’A’ or ’V’.

IU

IU is INTEGER
If RANGE=’I’, the index of the
largest eigenvalue to be returned.
1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
Not referenced if RANGE = ’A’ or ’V’.

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)).

RELTOL

RELTOL is REAL
The minimum relative width of an interval. When an interval
is narrower than RELTOL times the larger (in
magnitude) endpoint, then it is considered to be
sufficiently small, i.e., converged. Note: this should
always be at least radix*machine epsilon.

D

D is REAL array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.

E

E is REAL array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix T.

E2

E2 is REAL array, dimension (N-1)
The (n-1) squared off-diagonal elements of the tridiagonal matrix T.

PIVMIN

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

NSPLIT

NSPLIT is INTEGER
The number of diagonal blocks in the matrix T.
1 <= NSPLIT <= N.

ISPLIT

ISPLIT is 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., and the NSPLIT-th consists of rows/columns
ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
(Only the first NSPLIT elements will actually be used, but
since the user cannot know a priori what value NSPLIT will
have, N words must be reserved for ISPLIT.)

M

M is INTEGER
The actual number of eigenvalues found. 0 <= M <= N.
(See also the description of INFO=2,3.)

W

W is REAL array, dimension (N)
On exit, the first M elements of W will contain the
eigenvalue approximations. SLARRD computes an interval
I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue
approximation is given as the interval midpoint
W(j)= ( a_j + b_j)/2. The corresponding error is bounded by
WERR(j) = abs( a_j - b_j)/2

WERR

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

WL

WL is REAL

WU

WU is REAL
The interval (WL, WU] contains all the wanted eigenvalues.
If RANGE=’V’, then WL=VL and WU=VU.
If RANGE=’A’, then WL and WU are the global Gerschgorin bounds
on the spectrum.
If RANGE=’I’, then WL and WU are computed by SLAEBZ from the
index range specified.

IBLOCK

IBLOCK is INTEGER array, dimension (N)
At each row/column j where E(j) is zero or small, the
matrix T is considered to split into a block diagonal
matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which
block (from 1 to the number of blocks) the eigenvalue W(i)
belongs. (SLARRD may use the remaining N-M elements as
workspace.)

INDEXW

INDEXW is INTEGER array, dimension (N)
The indices of the eigenvalues within each block (submatrix);
for example, INDEXW(i)= j and IBLOCK(i)=k imply that the
i-th eigenvalue W(i) is the j-th eigenvalue in block k.

WORK

WORK is REAL array, dimension (4*N)

IWORK

IWORK is INTEGER array, dimension (3*N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: some or all of the eigenvalues failed to converge or
were not computed:
=1 or 3: Bisection failed to converge for some
eigenvalues; these eigenvalues are flagged by a
negative block number. The effect is that the
eigenvalues may not be as accurate as the
absolute and relative tolerances. This is
generally caused by unexpectedly inaccurate
arithmetic.
=2 or 3: RANGE=’I’ only: Not all of the eigenvalues
IL:IU were found.
Effect: M < IU+1-IL
Cause: non-monotonic arithmetic, causing the
Sturm sequence to be non-monotonic.
Cure: recalculate, using RANGE=’A’, and pick
out eigenvalues IL:IU. In some cases,
increasing the PARAMETER ’FUDGE’ may
make things work.
= 4: RANGE=’I’, and the Gershgorin interval
initially used was too small. No eigenvalues
were computed.
Probable cause: your machine has sloppy
floating-point arithmetic.
Cure: Increase the PARAMETER ’FUDGE’,
recompile, and try again.

Internal Parameters:

FUDGE REAL, default = 2
A ’fudge factor’ to widen the Gershgorin intervals. Ideally,
a value of 1 should work, but on machines with sloppy
arithmetic, this needs to be larger. The default for
publicly released versions should be large enough to handle
the worst machine around. Note that this has no effect
on accuracy of the solution.

Contributors:

W. Kahan, University of California, Berkeley, USA
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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Author

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