dlarre-f(3)

dlarre.f –

Synopsis

Functions/Subroutines

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.

Function/Subroutine Documentation

subroutine dlarre (characterRANGE, integerN, double precisionVL, double precisionVU, integerIL, integerIU, double precision, dimension( * )D, double precision, dimension( * )E, double precision, dimension( * )E2, double precisionRTOL1, double precisionRTOL2, double precisionSPLTOL, integerNSPLIT, integer, dimension( * )ISPLIT, integerM, double precision, dimension( * )W, double precision, dimension( * )WERR, double precision, dimension( * )WGAP, integer, dimension( * )IBLOCK, integer, dimension( * )INDEXW, double precision, dimension( * )GERS, double precisionPIVMIN, double precision, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)

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_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.

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

 VL is DOUBLE PRECISION

 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 is INTEGER

 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 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 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 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.LT.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 is INTEGER
 The total number of eigenvalues (of all L_i D_i L_i^T)
 found.

 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

 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 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.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

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

Definition at line 295 of file dlarre.f.

Author

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