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dstevr(3P)		    Sun Performance Library		    dstevr(3P)

NAME
       dstevr  - compute selected eigenvalues and, optionally, eigenvectors of
       a real symmetric tridiagonal matrix T

SYNOPSIS
       SUBROUTINE DSTEVR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W,
	     Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)

       CHARACTER * 1 JOBZ, RANGE
       INTEGER N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
       INTEGER ISUPPZ(*), IWORK(*)
       DOUBLE PRECISION VL, VU, ABSTOL
       DOUBLE PRECISION D(*), E(*), W(*), Z(LDZ,*), WORK(*)

       SUBROUTINE DSTEVR_64(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M,
	     W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)

       CHARACTER * 1 JOBZ, RANGE
       INTEGER*8 N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
       INTEGER*8 ISUPPZ(*), IWORK(*)
       DOUBLE PRECISION VL, VU, ABSTOL
       DOUBLE PRECISION D(*), E(*), W(*), Z(LDZ,*), WORK(*)

   F95 INTERFACE
       SUBROUTINE STEVR(JOBZ, RANGE, [N], D, E, VL, VU, IL, IU, ABSTOL, M,
	      W, Z, [LDZ], ISUPPZ, [WORK], [LWORK], [IWORK], [LIWORK], [INFO])

       CHARACTER(LEN=1) :: JOBZ, RANGE
       INTEGER :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
       INTEGER, DIMENSION(:) :: ISUPPZ, IWORK
       REAL(8) :: VL, VU, ABSTOL
       REAL(8), DIMENSION(:) :: D, E, W, WORK
       REAL(8), DIMENSION(:,:) :: Z

       SUBROUTINE STEVR_64(JOBZ, RANGE, [N], D, E, VL, VU, IL, IU, ABSTOL,
	      M, W, Z, [LDZ], ISUPPZ, [WORK], [LWORK], [IWORK], [LIWORK], [INFO])

       CHARACTER(LEN=1) :: JOBZ, RANGE
       INTEGER(8) :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
       INTEGER(8), DIMENSION(:) :: ISUPPZ, IWORK
       REAL(8) :: VL, VU, ABSTOL
       REAL(8), DIMENSION(:) :: D, E, W, WORK
       REAL(8), DIMENSION(:,:) :: Z

   C INTERFACE
       #include <sunperf.h>

       void dstevr(char jobz, char range, int n, double *d, double *e,	double
		 vl,  double vu, int il, int iu, double abstol, int *m, double
		 *w, double *z, int ldz, int *isuppz, int *info);

       void dstevr_64(char jobz, char range, long n,  double  *d,  double  *e,
		 double	 vl,  double vu, long il, long iu, double abstol, long
		 *m, double *w,	 double	 *z,  long  ldz,  long	*isuppz,  long
		 *info);

PURPOSE
       dstevr computes selected eigenvalues and, optionally, eigenvectors of a
       real symmetric tridiagonal matrix T.  Eigenvalues and eigenvectors  can
       be  selected  by	 specifying  either  a	range  of values or a range of
       indices for the desired eigenvalues.

       Whenever possible, DSTEVR calls DSTEGR to compute the
       eigenspectrum using Relatively Robust Representations.  DSTEGR computes
       eigenvalues  by	the  dqds algorithm, while orthogonal eigenvectors are
       computed from various "good" L D L^T  representations  (also  known  as
       Relatively  Robust  Representations). Gram-Schmidt orthogonalization is
       avoided as far as possible. More specifically, the various steps of the
       algorithm are as follows. For the i-th unreduced block of T,
	  (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
	       is a relatively robust representation,
	  (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
	      relative accuracy by the dqds algorithm,
	  (c) If there is a cluster of close eigenvalues, "choose" sigma_i
	      close to the cluster, and go to step (a),
	  (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
	      compute the corresponding eigenvector by forming a
	      rank-revealing twisted factorization.
       The desired accuracy of the output can be specified by the input param‐
       eter ABSTOL.

       For more details, see "A new O(n^2) algorithm for the symmetric	tridi‐
       agonal  eigenvalue/eigenvector  problem", by Inderjit Dhillon, Computer
       Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May
       1997.

       Note  1	:  DSTEVR  calls DSTEGR when the full spectrum is requested on
       machines which conform to the ieee-754 floating point standard.	DSTEVR
       calls DSTEBZ and DSTEIN on non-ieee machines and
       when partial spectrum requests are made.

       Normal execution of DSTEGR may create NaNs and infinities and hence may
       abort due to a floating point exception in environments	which  do  not
       handle NaNs and infinities in the ieee standard default manner.

ARGUMENTS
       JOBZ (input)
		 = 'N':	 Compute eigenvalues only;
		 = 'V':	 Compute eigenvalues and eigenvectors.

       RANGE (input)
		 = 'A': all eigenvalues will be found.
		 = 'V': all eigenvalues in the half-open interval (VL,VU] will
		 be found.  = 'I': the IL-th through IU-th eigenvalues will be
		 found.

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

       D (input/output)
		 On  entry,  the n diagonal elements of the tridiagonal matrix
		 A.  On exit, D may be multiplied by a constant factor	chosen
		 to avoid over/underflow in computing the eigenvalues.

       E (input/output)
		 On  entry,  the (n-1) subdiagonal elements of the tridiagonal
		 matrix A in elements 1 to N-1 of E; E(N) need not be set.  On
		 exit,	E  may	be  multiplied	by a constant factor chosen to
		 avoid over/underflow in computing the eigenvalues.

       VL (input)
		 If RANGE='V', the lower and upper bounds of the  interval  to
		 be  searched  for  eigenvalues.  VL  < VU.  Not referenced if
		 RANGE = 'A' or 'I'.

       VU (input)
		 See the description of VL.

       IL (input)
		 If RANGE='I', the indices (in ascending order) of the	small‐
		 est 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'.

       IU (input)
		 See the description of IL.

       ABSTOL (input)
		 The absolute error tolerance for the eigenvalues.  An approx‐
		 imate eigenvalue is accepted as converged when it  is	deter‐
		 mined to lie in an interval [a,b] of width less than or equal
		 to

		 ABSTOL + EPS *	  max( |a|,|b| ) ,

		 where EPS is the machine precision.  If ABSTOL is  less  than
		 or  equal  to zero, then  EPS*|T|  will be used in its place,
		 where |T| is the 1-norm of the tridiagonal matrix obtained by
		 reducing A to tridiagonal form.

		 See  "Computing  Small Singular Values of Bidiagonal Matrices
		 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
		 LAPACK Working Note #3.

		 If high relative accuracy is important, set ABSTOL to DLAMCH(
		 'Safe minimum' ).  Doing so will guarantee  that  eigenvalues
		 are  computed	to  high  relative  accuracy  when possible in
		 future releases.  The current code does not make any  guaran‐
		 tees  about high relative accuracy, but future releases will.
		 See J. Barlow and J. Demmel, "Computing Accurate Eigensystems
		 of  Scaled Diagonally Dominant Matrices", LAPACK Working Note
		 #7, for a discussion of which matrices define their eigenval‐
		 ues to high relative accuracy.

       M (output)
		 The  total  number  of	 eigenvalues  found.  0 <= M <= N.  If
		 RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

       W (output)
		 The first M elements  contain	the  selected  eigenvalues  in
		 ascending order.

       Z (output)
		 If  JOBZ  =  'V',  then if INFO = 0, the first M columns of Z
		 contain the orthonormal eigenvectors of the matrix  A	corre‐
		 sponding to the selected eigenvalues, with the i-th column of
		 Z holding the eigenvector associated with  W(i).   Note:  the
		 user  must ensure that at least max(1,M) columns are supplied
		 in the array Z; if RANGE = 'V', the exact value of M  is  not
		 known in advance and an upper bound must be used.

       LDZ (input)
		 The  leading dimension of the array Z.	 LDZ >= 1, and if JOBZ
		 = 'V', LDZ >= max(1,N).

       ISUPPZ (output) INTEGER array, dimension (2*max(1,M))
		 The support of the eigenvectors in Z, i.e., the indices indi‐
		 cating	 the  nonzero  elements	 in Z. The i-th eigenvector is
		 nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ(  2*i
		 ).

       WORK (workspace)
		 On  exit, if INFO = 0, WORK(1) returns the optimal (and mini‐
		 mal) LWORK.

       LWORK (input)
		 The dimension of the array WORK.  LWORK >= 20*N.

		 If LWORK = -1, then a workspace query is assumed; the routine
		 only  calculates  the optimal size of the WORK array, returns
		 this value as the first entry of the WORK array, and no error
		 message related to LWORK is issued by XERBLA.

       IWORK (workspace/output)
		 On exit, if INFO = 0, IWORK(1) returns the optimal (and mini‐
		 mal) LIWORK.

       LIWORK (input)
		 The dimension of the array IWORK.  LIWORK >= 10*N.

		 If LIWORK = -1, then a workspace query is assumed;  the  rou‐
		 tine  only  calculates	 the  optimal size of the IWORK array,
		 returns this value as the first entry of the IWORK array, and
		 no error message related to LIWORK is issued by XERBLA.

       INFO (output)
		 = 0:  successful exit
		 < 0:  if INFO = -i, the i-th argument had an illegal value
		 > 0:  Internal error

FURTHER DETAILS
       Based on contributions by
	  Inderjit Dhillon, IBM Almaden, USA
	  Osni Marques, LBNL/NERSC, USA
	  Ken Stanley, Computer Science Division, University of
	    California at Berkeley, USA

				  6 Mar 2009			    dstevr(3P)
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