chol                  package: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 <URL:
     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")
     oo <- order(pivot)
     t(Q[, oo]) %*% Q[, oo] # 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

