Compute truncated eigen decomposition of a symmetric matrix
Uses Lanczos iteration to find the truncated eigen-decomposition of a symmetric matrix.
slanczos(A,k=10,kl=-1,tol=.Machine$double.eps^.5,nt=1)
A |
A symmetric matrix. |
k |
Must be non-negative. If |
kl |
If |
tol |
tolerance to use for convergence testing of eigenvalues. Error in eigenvalues will be less
than the magnitude of the dominant eigenvalue multiplied by |
nt |
number of threads to use for leading order iterative multiplication of A by vector. May show no speed improvement on two processor machine. |
If kl is non-negative, returns the highest k and lowest kl eigenvalues,
with their corresponding eigenvectors. If kl is negative, returns the largest magnitude k
eigenvalues, with corresponding eigenvectors.
The routine implements Lanczos iteration with full re-orthogonalization as described in Demmel (1997). Lanczos
iteraction iteratively constructs a tridiagonal matrix, the eigenvalues of which converge to the eigenvalues of A,
as the iteration proceeds (most extreme first). Eigenvectors can also be computed. For small k and kl the
approach is faster than computing the full symmetric eigendecompostion. The tridiagonal eigenproblems are handled using LAPACK.
The implementation is not optimal: in particular the inner triadiagonal problems could be handled more efficiently, and there would be some savings to be made by not always returning eigenvectors.
A list with elements values (array of eigenvalues); vectors (matrix with eigenvectors in its columns);
iter (number of iterations required).
Simon N. Wood simon.wood@r-project.org
Demmel, J. (1997) Applied Numerical Linear Algebra. SIAM
require(mgcv) ## create some x's and knots... set.seed(1); n <- 700;A <- matrix(runif(n*n),n,n);A <- A+t(A) ## compare timings of slanczos and eigen system.time(er <- slanczos(A,10)) system.time(um <- eigen(A,symmetric=TRUE)) ## confirm values are the same... ind <- c(1:6,(n-3):n) range(er$values-um$values[ind]);range(abs(er$vectors)-abs(um$vectors[,ind]))
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