mgcv (version 1.9-0)

bandchol: Choleski decomposition of a band diagonal matrix

Description

Computes Choleski decomposition of a (symmetric positive definite) band-diagonal matrix, A.

Usage

bandchol(B)

Value

Let R be the factor such that t(R)%*%R = A. R is upper triangular and if the rows of B contained the diagonals of A on entry, then what is returned is an n by k matrix containing the diagonals of R, packed as B was packed on entry. If B was square on entry, then R is returned directly. See examples.

Arguments

B

An n by k matrix containing the diagonals of the matrix A to be decomposed. First row is leading diagonal, next is first sub-diagonal, etc. sub-diagonals are zero padded at the end. Alternatively gives A directly, i.e. a square matrix with 2k-1 non zero diagonals (those from the lower triangle are not accessed).

Author

Simon N. Wood

Details

Calls dpbtrf from LAPACK. The point of this is that it has \(O(k^2n)\) computational cost, rather than the \(O(n^3)\) required by dense matrix methods.

References

Anderson, E., Bai, Z., Bischof, C., Blackford, S., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A. and Sorensen, D., 1999. LAPACK Users' guide (Vol. 9). Siam.

Examples

require(mgcv)
## simulate a banded diagonal matrix
n <- 7;set.seed(8)
A <- matrix(0,n,n)
sdiag(A) <- runif(n);sdiag(A,1) <- runif(n-1)
sdiag(A,2) <- runif(n-2)
A <- crossprod(A) 

## full matrix form...
bandchol(A)
chol(A) ## for comparison

## compact storage form...
B <- matrix(0,3,n)
B[1,] <- sdiag(A);B[2,1:(n-1)] <- sdiag(A,1)
B[3,1:(n-2)] <- sdiag(A,2)
bandchol(B)