mgcv (version 1.7-4)

slanczos: Compute truncated eigen decomposition of a symmetric matrix

Description

Uses Lanczos iteration to find the truncated eigen-decomposition of a symmetric matrix.

Usage

slanczos(A,k=10,kl=-1)

Arguments

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.

Value

• A list with elements values (array of eigenvalues); vectors (matrix with eigenvectors in its columns); iter (number of iterations required).

Details

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.

References

Demmel, J. (1997) Applied Numerical Linear Algebra. SIAM

See Also

cyclic.p.spline

Examples

## 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]))