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Wed Jul 18 14:46:20 2007 UTC (12 years, 1 month ago) by maechler
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norm() for sparse; systematic checking of norm() and all "Summary" methods
\title{Estimate the Reciprocal Condition Number}
% most methods are documented in <foo>Matrix-class.Rd
rcond(x, type, \dots)
  Estimate the reciprocal of the condition number of a matrix.

  This is a generic function with several methods, as seen by
  \item{x}{an \R object that inherits from the \code{Matrix} class.}
    Character indicating the type of norm to be used in the estimate.
    The default is \code{"O"} for the 1-norm.  The other possible value is
    \code{"I"} for the infinity norm, see also \code{\link{norm}}.
  \item{\dots}{further arguments passed to or from other methods.}
  An estimate of the reciprocal condition number of \code{x}.
  The condition number of a matrix is the product of the
  \code{\link{norm}} of the matrix and the norm of its inverse (or
  The condition number takes on values between 1 and infinity,
  inclusive, and can be viewed as a factor by which errors in solving
  linear systems with this matrix as coefficient matrix could be

  \code{rcond()} computes the \emph{reciprocal} condition number with
  values in \eqn{[0,1]} and can be viewed as a scaled measure of how
  close a matrix is to being rank deficient (aka \dQuote{singular}).

  Condition numbers are usually estimated, since exact computation is
  costly in terms of floating-point operations.  An (over) estimate of
  reciprocal condition number is given, since by doing so overflow is
  avoided.  Matrices are well-conditioned if the reciprocal condition
  number is near 1 and ill-conditioned if it is near zero.
  \code{\link{norm}}, \code{\link[base]{solve}}.
  Golub, G., and Van Loan, C. F. (1989).
  \emph{Matrix Computations,}
  2nd edition, Johns Hopkins, Baltimore.
x <- Matrix(rnorm(9), 3, 3)
rcond(Hilbert(9))  # should be about 9.1e-13
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