dlarrd-f(3)

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

 VL is DOUBLE PRECISION

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

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

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

 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 is DOUBLE PRECISION

 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

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