The Choleski Decomposition

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

algebra, array
chol(x, …)

# S3 method for 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 (now ignored)?


A numeric tolerance for use with pivot = TRUE.


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 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.

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.


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.


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.


Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM. Available on-line at

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.

  • chol
  • chol.default
library(base) # NOT RUN { ( 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 # }
Documentation reproduced from package base, version 3.6.2, License: Part of R 3.6.2

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