Matrix Analysis and Applications, Vol.Hadoop, Data Science, Statistics & others Sorensen, D.C., "Implicit Application of Polynomial Filters in a k-StepĪrnoldi Method," SIAM J. Methods, SIAM Publications, Philadelphia, 1998. Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Sorensen, "Deflation Techniques for an Implicitly The plot shows the 20 eigenvalues closest to 4 that were computed by eig, along with the 18 eigenvalues closest to 4 - 1e-6 that were computed by eigs. We must use sigma near but not equal to 4 to find those 18 eigenvalues. As lambda gets closer to 4.0, eigs fails. This involves divisions of the form 1/(lambda - 4.0), where lambda is an estimate of an eigenvalue of A. The call eigs(A,18,4.0) to compute 18 eigenvalues near 4.0 tries to find eigenvalues of A - 4.0*I. However, the repeated eigenvalue at 4 must be handled more carefully. It computes and plots the six largest and smallest magnitude eigenvalues of A successfully with: The eig function computes all 632 eigenvalues. Legend('eigs(west0479,8)','eig(full(west0479))')Įxample 3: A = delsq(numgrid('C',30)) is a symmetric positive definite matrix of size 632 with eigenvalues reasonably well-distributed in the interval (0 8), but with 18 eigenvalues repeated at 4. This plot shows the 8 largest magnitude eigenvalues of west0479 as computed by eig and eigs.
eigs easily picks out the largest magnitude eigenvalues. Then pass dnRk's additional arguments, 'C' and 15, to eigs.ĭ2 = 2: west0479 is a real 479-by-479 sparse matrix with both real and pairs of complex conjugate eigenvalues. If the problem fits into memory, it may be quicker to use eig(full(A)).Įigs provides the reverse communication required by the Fortran library ARPACK, namely the routines DSAUPD, DSEUPD, DNAUPD, DNEUPD, ZNAUPD, and ZNEUPD.Įxample 1: This example shows the use of function handles.Įquivalently, if dnRk is the following one-line function: Provide for additional arguments which are passed to Afun(x,p1,p2.).īut is most appropriate for large sparse matrices. In all the eigs syntaxes, eigs(A.) can be replaced by eigs(Afun,n.). The matrix A, A- sigma *I or A- sigma *B represented by Afun is assumed to be real and nonsymmetric unless specified otherwise by opts.isreal and opts.issym. If sigma is a nonzero scalar (generalized eigenvalue problem) I is an identity matrix of the same size as A. If sigma is a nonzero scalar (standard eigenvalue problem). If sigma is not specified, or is a string other than 'sm' MATLAB 5 options stagtol and cheb are no longer allowed.Īccepts the function Afun instead of the matrix A. Permutation vector permB if sparse B is really chol(B(permB,permB)). Note: p must satisfy k < p <= n for real symmetric, k+1 < p <= n otherwise.ġ if B is really its Cholesky factor chol(B), 0 otherwise. Default values are shown in brackets ( ).ġ if A or A- sigma *B represented by Afun is symmetric, 0 otherwise.ġ if A or A- sigma *B represented by Afun is real, 0 otherwise.Ĭonvergence: Ritz estimate residual = 2k ( p >= 2k+1 real nonsymmetric) advised. The MATLAB 5 value sigma = 'be' is obsolete for nonsymmetric and complex problems.Įigs(A,K, sigma,opts) and eigs(A,B,k, sigma,opts) Note, B need only be symmetric (Hermitian) positive semi-definite.įor real symmetric problems, the following are also options:īoth ends (one more from high end if k is odd)įor nonsymmetric and complex problems, the following are also options: If A is a function, Afun must return Y = A\x.
Note, B need only be symmetric (Hermitian) positive semi-definite. If A is a function, Afun must return Y = (A- sigma *B)\x (i.e., Y = A\x when sigma = 0). Return k eigenvalues based on sigma, which can take any of the following values: Return the k largest magnitude eigenvalues. eigs(A.) indicates the standard eigenvalue problem A*V = V*D. B must be symmetric (or Hermitian) positive definite and the same size as A. Solves the generalized eigenvalue problem A*V = B*V*D. If flag is 0 then all the eigenvalues converged otherwise not all converged. Returns a diagonal matrix D of A's six largest magnitude eigenvalues and a matrix V whose columns are the corresponding eigenvectors.Īlso returns a convergence flag. Returns a vector of A's six largest magnitude eigenvalues. Eigs (MATLAB Functions) MATLAB Function Referenceįind a few eigenvalues and eigenvectors of a square large sparse matrixĭ = eigs(Afun,n,k, sigma,options,p1,p2.)ĭ = eigs(Afun,n,B,k, sigma,options,p1,p2.)
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