dlarrd.f man page

dlarrd.f —

Synopsis

Functions/Subroutines

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.

Function/Subroutine Documentation

subroutine dlarrd (characterRANGE, characterORDER, integerN, double precisionVL, double precisionVU, integerIL, integerIU, double precision, dimension( * )GERS, double precisionRELTOL, double precision, dimension( * )D, double precision, dimension( * )E, double precision, dimension( * )E2, double precisionPIVMIN, integerNSPLIT, integer, dimension( * )ISPLIT, integerM, double precision, dimension( * )W, double precision, dimension( * )WERR, double precisionWL, double precisionWU, integer, dimension( * )IBLOCK, integer, dimension( * )INDEXW, double precision, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)

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

VU

VU is DOUBLE PRECISION
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

IL is INTEGER

IU

IU is 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

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.

Date:

September 2012

Definition at line 319 of file dlarrd.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

dlarrd(3) is an alias of dlarrd.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK