# slanczos: Compute truncated eigen decomposition of a symmetric matrix In mgcv: Mixed GAM Computation Vehicle with Automatic Smoothness Estimation

## Description

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

## Usage

 `1` ```slanczos(A,k=10,kl=-1,tol=.Machine\$double.eps^.5,nt=1) ```

## 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!). `nt` number of threads to use for leading order iterative multiplication of A by vector. May show no speed improvement on two processor machine.

## 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

`cyclic.p.spline`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ``` 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])) ```