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cholesky(x [,pivot:T or force:T , nonposok:T]), x a positive definite
  square REAL matrix with no MISSING values

Keywords: matrix algebra
cholesky(A) returns the Cholesky decomposition of the positive definite
REAL symmetric matrix A.  Its value is the REAL upper triangular matrix
r of the same size as A such that r' %*% r = A.  It is an error if A is
not positive definite.

cholesky(A, nonposok:T) does the same, except that a non positive
definite A is not considered to be an error, but results a value of NULL
being returned.  This makes it possible for a macro to take corrective
action when a matrix is not positive definite.  See topics 'macros' and

cholesky(A,pivot:T [,nonposok:T]) reorders the rows and columns as the
computation proceeds so as to obtain the most stable computation.  It
returns a structure with components 'r', a REAL upper triangular matrix,
and 'pivot', a REAL vector of integers describing the reordering.  After
result <- cholesky(A,pivot:T), result$r' %*% result$r should equal
A[result$pivot, result$pivot] except for rounding error.

cholesky(A,force:Vec [,nonposok:T]), where Vec is a REAL vector whose
length is nrows(A), enables pivoting, but allows some control on
reordering.  The elements of Vec should be 1, -1, or 0, since only the
signs are used.  Before factoring, rows and columns of A, if any, with
index j such that Vec[j] > 0 are moved to rows and columns 1, 2, ...,
(initial columns) but are not further moved.  All rows and columns with
Vec[j] < 0 are moved to rows and columns nrows(A), nrows(A) - 1, ...,
(final columns) but are not further moved.  Rows and columns, if any,
with Vec[j] == 0 (pivoted columns), are free to be reordered, but will
follow the initial columns and precede the final columns.  Again the
result is a structure with components 'r' and 'pivot'.

See also qr().

Gary Oehlert 2003-01-15