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DLARRD(1)	    LAPACK auxiliary routine (version 3.2)	     DLARRD(1)

NAME
       DLARRD  -  computes the eigenvalues of a symmetric tridiagonal matrix T
       to suitable accuracy

SYNOPSIS
       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 )

	   CHARACTER	  ORDER, RANGE

	   INTEGER	  IL, INFO, IU, M, N, NSPLIT

	   DOUBLE	  PRECISION PIVMIN, RELTOL, VL, VU, WL, WU

	   INTEGER	  IBLOCK( * ), INDEXW( * ), ISPLIT( * ), IWORK( * )

	   DOUBLE	  PRECISION D( * ), E( * ), E2( * ), GERS( * ),	 W(  *
			  ), WERR( * ), WORK( * )

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	eigen‐
       values.
       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.

ARGUMENTS
       RANGE   (input) CHARACTER
	       = '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   (input) CHARACTER
	       = '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	eigen‐
	       values  for  the entire matrix will be ordered from smallest to
	       largest.

       N       (input) INTEGER
	       The order of the tridiagonal matrix T.  N >= 0.

       VL      (input) DOUBLE PRECISION
	       VU      (input) 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      (input) INTEGER
	       IU      (input) 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    (input) DOUBLE PRECISION array, dimension (2*N)
	       The  N  Gerschgorin intervals (the i-th Gerschgorin interval is
	       (GERS(2*i-1), GERS(2*i)).

       RELTOL  (input) 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.,  con‐
	       verged.	 Note:	this  should  always be at least radix*machine
	       epsilon.

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

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

       E2      (input) DOUBLE PRECISION array, dimension (N-1)
	       The (n-1) squared  off-diagonal	elements  of  the  tridiagonal
	       matrix T.

       PIVMIN  (input) DOUBLE PRECISION
	       The minimum pivot allowed in the Sturm sequence for T.

       NSPLIT  (input) INTEGER
	       The  number of diagonal blocks in the matrix T.	1 <= NSPLIT <=
	       N.

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

       W       (output) 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 corre‐
	       sponding error is bounded by WERR(j) = abs( a_j - b_j)/2

       WERR    (output) DOUBLE PRECISION array, dimension (N)
	       The  error  bound on the corresponding eigenvalue approximation
	       in W.

       WL      (output) DOUBLE PRECISION
	       WU      (output) DOUBLE PRECISION The interval  (WL,  WU]  con‐
	       tains 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 com‐
	       puted by DLAEBZ from the index range specified.

       IBLOCK  (output) 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  (output) 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    (workspace) DOUBLE PRECISION array, dimension (4*N)

       IWORK   (workspace) INTEGER array, dimension (3*N)

       INFO    (output) 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  Ger‐
	       shgorin	interval initially used was too small.	No eigenvalues
	       were computed.  Probable cause: your machine has sloppy	float‐
	       ing-point  arithmetic.	Cure:  Increase the PARAMETER "FUDGE",
	       recompile, and try again.

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	 solu‐
	       tion.   Based on contributions by W. Kahan, University of Cali‐
	       fornia, Berkeley, USA Beresford Parlett, University of Califor‐
	       nia, Berkeley, USA Jim Demmel, University of California, Berke‐
	       ley, USA Inderjit Dhillon, University  of  Texas,  Austin,  USA
	       Osni  Marques,  LBNL/NERSC,  USA Christof Voemel, University of
	       California, Berkeley, USA

 LAPACK auxiliary routine (versioNovember 2008			     DLARRD(1)
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