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dlarrd.f(3)				      LAPACK				      dlarrd.f(3)

NAME
       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.

Version 3.4.2				 Tue Sep 25 2012			      dlarrd.f(3)
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