SLARRD(1) LAPACK auxiliary routine (version 3.2) SLARRD(1)NAME
SLARRD - computes the eigenvalues of a symmetric tridiagonal matrix T
to suitable accuracy
SYNOPSIS
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 )
CHARACTER ORDER, RANGE
INTEGER IL, INFO, IU, M, N, NSPLIT
REAL PIVMIN, RELTOL, VL, VU, WL, WU
INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ), IWORK( * )
REAL D( * ), E( * ), E2( * ), GERS( * ), W( * ), WERR( *
), WORK( * )
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 eigen‐
values.
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.
ARGUMENTS
RANGE (input) CHARACTER
= '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 (input) CHARACTER
= '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 eigen‐
values for the entire matrix will be ordered from smallest to
largest.
N (input) INTEGER
The order of the tridiagonal matrix T. N >= 0.
VL (input) REAL
VU (input) REAL If RANGE='V', the lower and upper bounds
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 (input) INTEGER
IU (input) INTEGER If RANGE='I', the indices (in ascending
order) of the smallest and largest eigenvalues 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 (input) REAL array, dimension (2*N)
The N Gerschgorin intervals (the i-th Gerschgorin interval is
(GERS(2*i-1), GERS(2*i)).
RELTOL (input) 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., con‐
verged. Note: this should always be at least radix*machine
epsilon.
D (input) REAL array, dimension (N)
The n diagonal elements of the tridiagonal matrix T.
E (input) REAL array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix T.
E2 (input) REAL array, dimension (N-1)
The (n-1) squared off-diagonal elements of the tridiagonal
matrix T.
PIVMIN (input) REAL
The minimum pivot allowed in the Sturm sequence for T.
NSPLIT (input) INTEGER
The number of diagonal blocks in the matrix T. 1 <= NSPLIT <=
N.
ISPLIT (input) INTEGER array, dimension (N)
The splitting points, at which T breaks up into submatrices.
The first submatrix consists of rows/columns 1 to ISPLIT(1),
the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc.,
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 (output) INTEGER
The actual number of eigenvalues found. 0 <= M <= N. (See also
the description of INFO=2,3.)
W (output) 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 corre‐
sponding error is bounded by WERR(j) = abs( a_j - b_j)/2
WERR (output) REAL array, dimension (N)
The error bound on the corresponding eigenvalue approximation
in W.
WL (output) REAL
WU (output) 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 (output) 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 (output) 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 (workspace) REAL array, dimension (4*N)
IWORK (workspace) INTEGER array, dimension (3*N)
INFO (output) 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 Ger‐
shgorin interval initially used was too small. No eigenvalues
were computed. Probable cause: your machine has sloppy float‐
ing-point arithmetic. Cure: Increase the PARAMETER "FUDGE",
recompile, and try again.
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 solu‐
tion. Based on contributions by W. Kahan, University of Cali‐
fornia, Berkeley, USA Beresford Parlett, University of Califor‐
nia, Berkeley, USA Jim Demmel, University of California, Berke‐
ley, USA Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA Christof Voemel, University of
California, Berkeley, USA
LAPACK auxiliary routine (versioNovember 2008 SLARRD(1)
