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\name{eigen}
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\name{eigen}
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\alias{eigen}
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\alias{eigen}
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\alias{La.eigen}
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\title{Spectral Decomposition of a Matrix}
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\title{Spectral Decomposition of a Matrix}
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\usage{
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\usage{
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eigen(x, symmetric, only.values = FALSE, EISPACK = FALSE)
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eigen(x, symmetric, only.values = FALSE, EISPACK = FALSE)
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La.eigen(x, symmetric, only.values = FALSE,
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         method = c("dsyevr", "dsyev"))
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}
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}
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\arguments{
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\arguments{
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  \item{x}{a matrix whose spectral decomposition is to be computed.}
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  \item{x}{a matrix whose spectral decomposition is to be computed.}
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  \item{symmetric}{if \code{TRUE}, the matrix is assumed to be symmetric
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  \item{symmetric}{if \code{TRUE}, the matrix is assumed to be symmetric
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    (or Hermitian if complex) and only its lower triangle is used.
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    (or Hermitian if complex) and only its lower triangle is used.
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  \item{only.values}{if \code{TRUE}, only the eigenvalues are computed
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  \item{only.values}{if \code{TRUE}, only the eigenvalues are computed
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    and returned, otherwise both eigenvalues and eigenvectors are
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    and returned, otherwise both eigenvalues and eigenvectors are
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    returned.}
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    returned.}
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  \item{EISPACK}{logical. Should EISPACK be used (for compatibility with
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  \item{EISPACK}{logical. Should EISPACK be used (for compatibility with
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    \R < 1.7.0)?}
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    \R < 1.7.0)?}
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  \item{method}{The LAPACK routine to use in the real symmetric case.}
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}
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}
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\description{
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\description{
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  Computes eigenvalues and eigenvectors.
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  Computes eigenvalues and eigenvectors.
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}
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}
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\details{
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\details{
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  These functions use the LAPACK routines DSYEV/DSYEVR,
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  By default \code{eigen} uses the LAPACK routines DSYEVR/DSYEV,
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  DGEEV, ZHEEV and ZGEEV, and \code{eigen(EISPACK=TRUE)} provides an
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  DGEEV, ZHEEV and ZGEEV whereas \code{eigen(EISPACK=TRUE)} provides an
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  interface to the EISPACK routines \code{RS}, \code{RG}, \code{CH}
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  interface to the EISPACK routines \code{RS}, \code{RG}, \code{CH}
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  and \code{CG}.
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  and \code{CG}.
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  If \code{symmetric} is unspecified, the code attempts to
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  If \code{symmetric} is unspecified, the code attempts to
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  determine if the matrix is symmetric up to plausible numerical
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  determine if the matrix is symmetric up to plausible numerical
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  inaccuracies.  It is faster and surer to set the value yourself.
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  inaccuracies.  It is faster and surer to set the value yourself.
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  \code{eigen} is preferred to \code{eigen(EISPACK=TRUE)}
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  \code{eigen} is preferred to \code{eigen(EISPACK = TRUE)}
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  for new projects, but its eigenvectors may differ in sign and
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  for new projects, but its eigenvectors may differ in sign and
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  (in the asymmetric case) in normalization. (They may also differ
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  (in the asymmetric case) in normalization. (They may also differ
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  between methods and between platforms.)
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  between methods and between platforms.)
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  The LAPACK routine DSYEVR is usually substantially
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  In the real symmetric case, LAPACK routine DSYEVR is used which
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  faster than DSYEV: see
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  \url{http://www.cs.berkeley.edu/~demmel/DOE2000/Report0100.html}.
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  Most benefits are seen with an optimized BLAS system.
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  Using \code{method="dsyevr"} requires IEEE 754 arithmetic.  Should
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  requires IEEE 754 arithmetic.  Should this not be supported on
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  this not be supported on your platform, \code{method="dsyev"} is
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  used, with a warning.
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  your platform, DSYEV is used, with a warning.
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  Computing the eigenvectors is the slow part for large matrices.
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  Computing the eigenvectors is the slow part for large matrices.
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}
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}
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\value{
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\value{
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  The spectral decomposition of \code{x} is returned as components of a
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  The spectral decomposition of \code{x} is returned as components of a