The Choleski Decomposition

Description

Compute the Choleski factorization of a real symmetric positive-definite square matrix.

Usage

chol(x, ...)

## Default S3 method:
chol(x, pivot = FALSE,  LINPACK = pivot, ...)

Arguments

Details

chol is generic: the description here applies to the default method.

This is an interface to the LAPACK routine DPOTRF and the LINPACK routines DPOFA and DCHDC.

Note that only the upper triangular part of x is used, so that R'R = x when x is symmetric.

If pivot = FALSE and x is not non-negative definite an error occurs. If x is positive semi-definite (i.e., some zero eigenvalues) an error will also occur, as a numerical tolerance is used.

If pivot = TRUE, then the Choleski decomposition of a positive semi-definite x can be computed. The rank of x is returned as attr(Q, "rank"), subject to numerical errors. The pivot is returned as attr(Q, "pivot"). It is no longer the case that t(Q) %*% Q equals x. However, setting pivot <- attr(Q, "pivot") and oo <- order(pivot), it is true that t(Q[, oo]) %*% Q[, oo] equals x, or, alternatively, t(Q) %*% Q equals x[pivot, pivot]. See the examples.

Value

The upper triangular factor of the Choleski decomposition, i.e., the matrix R such that R'R = x (see example).

If pivoting is used, then two additional attributes "pivot" and "rank" are also returned.

Warning

The code does not check for symmetry.

If pivot = TRUE and x is not non-negative definite then there will be a warning message but a meaningless result will occur. So only use pivot = TRUE when x is non-negative definite by construction.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Dongarra, J. J., Bunch, J. R., Moler, C. B. and Stewart, G. W. (1978) LINPACK Users Guide. Philadelphia: SIAM Publications.

Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM.
Available on-line at http://www.netlib.org/lapack/lug/lapack_lug.html.

See Also

chol2inv for its inverse (without pivoting), backsolve for solving linear systems with upper triangular left sides.

qr, svd for related matrix factorizations.

Examples

( m <- matrix(c(5,1,1,3),2,2) )
( cm <- chol(m) )
t(cm) %*% cm  #-- = 'm'
crossprod(cm)  #-- = 'm'

# now for something positive semi-definite
x <- matrix(c(1:5, (1:5)^2), 5, 2)
x <- cbind(x, x[, 1] + 3*x[, 2])
m <- crossprod(x)
qr(m)$rank # is 2, as it should be

# chol() may fail, depending on numerical rounding:
# chol() unlike qr() does not use a tolerance.
try(chol(m))

(Q <- chol(m, pivot = TRUE)) # NB wrong rank here - see Warning section.
## we can use this by
pivot <- attr(Q, "pivot")
crossprod(Q[, order(pivot)]) # recover m

## now for a non-positive-definite matrix
( m <- matrix(c(5,-5,-5,3),2,2) )
try(chol(m))  # fails
try(chol(m, LINPACK=TRUE))   # fails
(Q <- chol(m, pivot = TRUE)) # warning
crossprod(Q)  # not equal to m