R: The Choleski Decomposition

chol {base} | 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 = pivot, ...)

### Arguments

`x` | an object for which a method exists. The default method applies to real symmetric, positive-definite matrices.
`...` | arguments to be based to or from methods.
`pivot` | Should pivoting be used?
`LINPACK` | logical. Should LINPACK be used in the non-pivoting case (for compatibility with R < 1.7.0)?

### Details

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

This is an interface to the LAPACK routine DPOTRF and the LINPACK routines DPOFA and DCHDC.

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.

### 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.

### References

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.

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.

### 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])
    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)) # NB wrong rank here - see Warning section.
    ## 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
    try(chol(m, LINPACK=TRUE))   # fails
    (Q <- chol(m, pivot = TRUE)) # warning
    crossprod(Q)  # not equal to m

[Package base version 2.15.1 Index]