Choleski Decomposition of Symmetric Matrix

chol

Description

Returns an upper-triangular matrix, which is the Choleski decomposition of a (hermitian) symmetric, positive definite (or positive semi-definite) matrix. This function is generic and only the default method is implemented now.

Usage

# Generic function
chol(x, ...)
## Default S3 method:
chol(x, pivot = FALSE, LINPACK = pivot, ...)

Arguments

x	a symmetric, positive semi-definite numeric matrix (for example, a correlation matrix or a cross-product matrix). Missing values are not accepted. Complex matrices are not accepted yet.
pivot	a logical value. If TRUE, pivot. The default is FALSE. This argument is useful to see if x is of full rank, or to decompose a matrix that might not be of full rank.
LINPACK	This argument is ignored unless pivot=TRUE, in which case you can choose between the Linpack dchdc and the Lapack dpstf2. The Lapack code fails if x is numerically indefinite, but the Linpack code does not fail. Rather, the Linpack code does what it can, putting the linearly-independent columns at the front of the result and giving you an estimate of the rank (that is, the number of linearly independent columns). The rank estimate is often an overestimate: use qr, svd, or eigen to get a better estimate. When no no pivoting is specified, chol uses the Lapack dpotrf.
...	other arguments to pass to or from function/methods.

Details

If pivot is TRUE, the result contains attributes rank and pivot. These attributes contain, respectively, the index of the last positive computed diagonal element, and the pivoting of the columns.

For y <- chol(x,TRUE), column i of y corresponds to row/column attr(y,"pivot")[i] of x.

Value

returns an upper-triangular matrix, y, such that t(y) %*% y equals x. # if x is numeric or such that t(Conj(y) %*% y equals x if # x is complex.

If pivot=TRUE was given, then the matrix has the extra attributes pivot and rank, giving the permutation of the columns and the estimated rank, respectively.

Background

A positive definite matrix is decomposed into a triangular matrix that is a "square root" of the original. That is, the (conjugate) transpose of the triangular matrix multiplied by the triangular matrix equals the original matrix.

Systems of equations can be solved efficiently using this decomposition.

As a historical note, Andre-Louis Choleski (1875-1918) was attached primarily to the Geodetic Section of the French government and was involved in several large scale-surveys (Maindonald, 1984, pp 346-348).

References

Thisted, R. A. 1988. Elements of Statistical Computing. New York, NY: Chapman and Hall.

Anderson. E. et al. 1999. LAPACK Users' Guide. Third Edition. Philadelphia, PA: SIAM Publications. Available on-line at http://www.netlib.org/lapack/lug/lapack_lug.html.

Dongarra, J. J., et al. 1979. LINPACK Users' Guide. Philadelphia, PA: SIAM Publications.

Golub, G. H. and Van Loan, C. F. 1983. Matrix Computations. Baltimore, MD: Johns Hopkins University Press.

Maindonald, J. H. 1984. Statistical Computation. New York, NY: Wiley.