slanczos {mgcv} | R Documentation |
Uses Lanczos iteration to find the truncated eigen-decomposition of a symmetric matrix.
slanczos(A,k=10,kl=-1)
A |
A symmetric matrix. |
k |
If kl is negative, then the k largest magnitude eigenvalues are found, together with the corresponding eigenvectors. Otherwise the k highest eigenvalues are found. |
kl |
The kl lowest eigenvalues are returned, unless kl is negative. |
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.
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
## 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]))