LANEIG  Compute a few eigenvalues and eigenvectors.
   LANEIG solves the eigenvalue problem A*v=lambda*v, when A is 
   real and symmetric using the Lanczos algorithm with partial 
   reorthogonalization (PRO). 

   [V,D] = LANEIG(A) 
   [V,D] = LANEIG('Afun',N) 

   The first input argument is either a real symmetric matrix, or a 
   string containing the name of an M-file which applies a linear 
   operator to the columns of a given matrix.  In the latter case,
   the second input argument must be N, the order of the problem.

   The full calling sequence is

   [V,D,ERR] = LANEIG(A,K,SIGMA,OPTIONS)
   [V,D,ERR] = LANEIG('Afun',N,K,SIGMA,OPTIONS)

   On exit ERR contains the computed error bounds.  K is the number of
   eigenvalues desired and SIGMA is numerical shift or a two letter string
   which specifies which part of the spectrum should be computed:

   SIGMA            Specified eigenvalues

   'AL'            Algebraically Largest 
   'AS'            Algebraically Smallest
   'LM'            Largest Magnitude   (default)
   'SM'            Smallest Magnitude  (does not work when A is an m-file)
   'BE'            Both Ends.  Computes k/2 eigenvalues
                   from each end of the spectrum (one more
                   from the high end if k is odd.) 

   The OPTIONS structure specifies certain parameters in the algorithm.

    Field name      Parameter                              Default
   
    OPTIONS.tol     Convergence tolerance                  16*eps
    OPTIONS.lanmax  Dimension of the Lanczos basis.
    OPTIONS.v0      Starting vector for the Lanczos        rand(n,1)-0.5
                    iteration.
    OPTIONS.delta   Level of orthogonality among the       sqrt(eps/K)
                    Lanczos vectors.
    OPTIONS.eta     Level of orthogonality after           10*eps^(3/4)
                    reorthogonalization. 
    OPTIONS.cgs     reorthogonalization method used        0
                    '0' : iterated modified Gram-Schmidt 
                    '1' : iterated classical Gram-Schmidt
    OPTIONS.elr     If equal to 1 then extended local      1
                    reorthogonalization is enforced. 

   See also LANPRO, EIGS, EIG.

 References: 
 R.M. Larsen, Ph.D. Thesis, Aarhus University, 1998.

 B. N. Parlett, ``The Symmetric Eigenvalue Problem'', 
 Prentice-Hall, Englewood Cliffs, NJ, 1980.

 H. D. Simon, ``The Lanczos algorithm with partial reorthogonalization'',
 Math. Comp. 42 (1984), no. 165, 115--142.

 Rasmus Munk Larsen, Stanford University, 2000.