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Choleski Decomposition of Symmetric Matrix

DESCRIPTION:: Returns an upper-triangular matrix which is the Choleski decomposition of a (hermitian) symmetric, positive definite (or positive semi-definite) matrix.

USAGE:

chol(x, pivot=F)

REQUIRED ARGUMENTS:

x:: (hermitian) symmetric, positive (semi-)definite matrix (e.g., correlation matrix or cross-product matrix). Missing values are not accepted.

OPTIONAL ARGUMENTS:

pivot:: should pivoting be done? Useful to see if x is of full rank, or to decompose a matrix that may not be of full rank.

VALUE:: 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.

DETAILS:: If pivot is TRUE, the result will contain attributes rank and pivot; these contain respectively, the index of the last positive computed diagonal element, and the pivoting of the columns. For y <- chol(x,T), column i of y corresponds to row/column attr(y,"pivot")[i] of x. That is, Conj(t(y[,order(attr(y,"pivot"))])) %*% y[,order(attr(y,"pivot"))] == x, or put another way Conj(t(y)) %*% y ==x[attr(y,"pivot"),attr(y,"pivot")]. If pivot is FALSE, any singularities will result in an error.

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 efficiently solved using this decomposition.
As a historical note, Androleski (1875-1918) was primarily attached to the Geodetic Section of the French government and was involved in several large scale surveys (Maindonald, 1984, pp 346-348).

REFERENCES:

Dongarra, J. J., Bunch, J. R., Moler, C. B., and Stewart, G. W. (1979). LINPACK Users' Guide. SIAM, Philadelphia.

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

Maindonald, J. H. (1984). Statistical Computation. Wiley, New York.

Thisted, R. A. (1988). Elements of Statistical Computing. Chapman and Hall, New York.