Compute the Choleski factorization of a real symmetric positive-definite square matrix.
chol(x, ...) ## S3 method for class 'default': chol((x, pivot = FALSE, LINPACK = FALSE, tol = -1, ...))
- 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.
- Should pivoting be used?
- logical. Should LINPACK be used (which is deprecated)?
- A numeric tolerance for use with
pivot = TRUE, LINPACK = FALSE).
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.
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.
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. Use of
chol(pivot = FALSE, LINPACK = TRUE) was deprecated in R 1.7.0. Pivoting with LAPACK requires LAPACK >= 3.2 and was added in R 2.15.2. The value of
tol is passed to LAPACK, with negative values selecting the default tolerance of (usually)
nrow(x) * .Machine$double.neg.eps * max(diag(x). The algorithm terminates once the pivot is less than
The LINPACK interface is restricted to matrices
x with less than 2^31 elements.
The upper triangular factor of the Choleski decomposition, i.e., the matrix R such that R'R = x (see example).
If pivoting is used, then two additional attributes
"rank" are also returned.
The code does not check for symmetry.
pivot = TRUE and
x is not non-negative definite then there will be a warning message but a meaningless result will occur. So only use
pivot = TRUE when
x is non-negative definite by construction.
Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM.
Available on-line at http://www.netlib.org/lapack/lug/lapack_lug.html.
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
Dongarra, J. J., Bunch, J. R., Moler, C. B. and Stewart, G. W. (1978) LINPACK Users Guide. Philadelphia: SIAM Publications.
LINPACK = TRUE, pivot = FALSE (for compatibility with R < 1.7.0) was formally deprecated in R 2.15.2.
( m <- matrix(c(5,1,1,3),2,2) ) ( cm <- chol(m) ) t(cm) %*% cm #-- = 'm' crossprod(cm) #-- = 'm' # now for something positive semi-definite x <- matrix(c(1:5, (1:5)^2), 5, 2) x <- cbind(x, x[, 1] + 3*x[, 2]) colnames(x) <- letters[20:22] m <- crossprod(x) qr(m)$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
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