chol {base} R Documentation

## The Cholesky Decomposition

### Description

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

### Usage

chol(x, ...)

## Default S3 method:
chol(x, pivot = FALSE,  LINPACK = FALSE, tol = -1, ...)


### Arguments

 x an object for which a method exists. The default method applies to numeric (or logical) symmetric, positive-definite matrices. ... arguments to be based to or from methods. pivot Should pivoting be used? LINPACK logical. Should LINPACK be used (now an error)? tol A numeric tolerance for use with pivot = TRUE.

### Details

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

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 Cholesky 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.

The value of tol is passed to LAPACK, with negative values selecting the default tolerance of (usually) nrow(x) * .Machine$double.neg.eps * max(diag(x)). The algorithm terminates once the pivot is less than tol. Unsuccessful results from the underlying LAPACK code will result in an error giving a positive error code: these can only be interpreted by detailed study of the FORTRAN code. ### Value The upper triangular factor of the Cholesky decomposition, i.e., the matrix R such that R'R = x (see example). If pivoting is used, then two additional attributes "pivot" and "rank" are also returned. ### Warning The code does not check for symmetry. If pivot = TRUE and x is not non-negative definite then there will be a warning message but a meaningless result will occur. So only use pivot = TRUE when x is non-negative definite by construction. ### Source This is an interface to the LAPACK routines DPOTRF and DPSTRF, LAPACK is from https://netlib.org/lapack/ and its guide is listed in the references. ### References Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM. Available on-line at https://netlib.org/lapack/lug/lapack_lug.html. Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole. ### See Also chol2inv for its inverse (without pivoting), backsolve for solving linear systems with upper triangular left sides. qr, svd for related matrix factorizations. ### Examples ( m <- matrix(c(5,1,1,3),2,2) ) ( cm <- chol(m) ) t(cm) %*% cm #-- = 'm' crossprod(cm) #-- = 'm' # now for something positive semi-definite x <- matrix(c(1:5, (1:5)^2), 5, 2) x <- cbind(x, x[, 1] + 3*x[, 2]) colnames(x) <- letters[20:22] m <- crossprod(x) qr(m)$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))
## 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
(Q <- chol(m, pivot = TRUE)) # warning
crossprod(Q)  # not equal to m


[Package base version 4.3.0 Index]