DLARRE(3) LAPACK routine of NEC Numeric Library Collection DLARRE(3)
NAME
DLARRE
SYNOPSIS
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)
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_i^T, and
(c) eigenvalues of each L_i D_i L_i^T.
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
conpute 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_i^T.
ARGUMENTS
RANGE (input)
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 (input)
N is INTEGER
The order of the matrix. N > 0.
VL (input/output)
VL is DOUBLE PRECISION
VU (input/output)
VU is DOUBLE PRECISION
If RANGE='V', the lower and upper bounds 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 (input)
IL is INTEGER
IU (input)
IU is INTEGER
If RANGE='I', the indices (in ascending order) of the
smallest and largest eigenvalues to be returned.
1 <= IL <= IU <= N.
D (input/output)
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 (input/output)
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 (input/output)
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 (input)
RTOL1 is DOUBLE PRECISION
RTOL2 (input)
RTOL2 is DOUBLE PRECISION
Parameters for bisection.
An interval [LEFT,RIGHT] has converged if
RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
SPLTOL (input)
SPLTOL is DOUBLE PRECISION
The threshold for splitting.
NSPLIT (output)
NSPLIT is INTEGER
The number of blocks T splits into. 1 <= NSPLIT <= N.
ISPLIT (output)
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 (output)
M is INTEGER
The total number of eigenvalues (of all L_i D_i L_i^T)
found.
W (output)
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_i^T, are
sorted in ascending order ( DLARRE may use the
remaining N-M elements as workspace).
WERR (output)
WERR is DOUBLE PRECISION array, dimension (N)
The error bound on the corresponding eigenvalue in W.
WGAP (output)
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 (output)
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 (output)
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 (output)
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 (output)
PIVMIN is DOUBLE PRECISION
The minimum pivot in the Sturm sequence for T.
WORK (output)
WORK is DOUBLE PRECISION array, dimension (6*N)
Workspace.
IWORK (output)
IWORK is INTEGER array, dimension (5*N)
Workspace.
INFO (output)
INFO is INTEGER
= 0: successful exit
> 0: A problem occured 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.
FURTHER DETAILS
The base representations are required to suffer very little
element growth and consequently define all their eigenvalues to
high relative accuracy.
LAPACK routine 31 October 2017 DLARRE(3)