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

`chol(x, …)`# S3 method for default
chol(x, pivot = FALSE, LINPACK = FALSE, tol = -1, …)

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 ignored)?

tol

A numeric tolerance for use with `pivot = TRUE`

.

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.

`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.

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.

`chol2inv`

for its *inverse* (without pivoting),
`backsolve`

for solving linear systems with upper
triangular left sides.

# 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 # }