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.