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.