R: Compute truncated eigen decomposition of a symmetric matrix

slanczos {mgcv} | R Documentation

## 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,tol=.Machine$double.eps^.5)

### Arguments

`A` | A symmetric matrix.
`k` | Must be non-negative. If `kl` is negative, then the `k` largest magnitude eigenvalues are found, together with the corresponding eigenvectors. If `kl` is non-negative then the `k` highest eigenvalues are found together with their eigenvectors and the `kl` lowest eigenvalues with eigenvectors are also returned.
`kl` | If `kl` is non-negative then the `kl` lowest eigenvalues are returned together with their corresponding eigenvectors (in addition to the `k` highest eignevalues + vectors). negative `kl` signals that the `k` largest magnitude eigenvalues should be returned, with eigenvectors.
`tol` | tolerance to use for convergence testing of eigenvalues. Error in eigenvalues will be less than the magnitude of the dominant eigenvalue multiplied by `tol` (or the machine precision!).

### Details

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.

### Value

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

### Author(s)

Simon N. Wood simon.wood@r-project.org

### References

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

### See Also

`cyclic.p.spline`

### Examples

     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]))

[Package mgcv version 1.8-3 Index]