# chol: The Choleski Decomposition

 chol R Documentation

## 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 = 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 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.

### 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.

### Source

This is an interface to the LAPACK routines `DPOTRF` and `DPSTRF`,

LAPACK is from https://www.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://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.

`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
```