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\name{spectral.norm}
\alias{spectral.norm}
\title{ Spectral norm of matrix }
\description{
This function returns the spectral norm of a real matrix.
}
\usage{
spectral.norm(x)
}
\arguments{
\item{x}{ a numeric matrix or vector }
}
\details{
Let \eqn{{\bf{x}}} be an \eqn{m \times n} real matrix. The
function computes the order \eqn{n} square matrixmatrix \eqn{{\bf{A}} = {\bf{x'}}\;{\bf{x}}}.
The R function \code{eigen} is applied to this matrix to obtain the vector
of eigenvalues \eqn{{\bf{\lambda }} = \left\lbrack {\begin{array}{cccc}
{\lambda _1 } & {\lambda _2 } & \cdots & {\lambda _n } \\
\end{array}} \right\rbrack}. By construction the eigenvalues are in descending
order of value so that the largest eigenvalue is \eqn{\lambda _1}. Then
the spectral norm is \eqn{\left\| {\bf{x}} \right\|_2 = \sqrt {\lambda _1 }}.
If \eqn{{\bf{x}}} is a vector, then \eqn{{\bf{L}}_2 = \sqrt {\bf{A}}} is returned.
}
\value{
A numeric value.
}
\references{
Bellman, R. (1987). \emph{Matrix Analysis}, Second edition, Classics in Applied Mathematics,
Society for Industrial and Applied Mathematics.
Golub, G. H. and C. F. Van Loan (1996). \emph{Matrix Computations}, Third Edition, The John
Hopkins University Press.
Horn, R. A. and C. R. Johnson (1985). \emph{Matrix Analysis}, Cambridge University Press.
}
\author{ Frederick Novomestky \email{fnovomes@poly.edu} }
\note{
If the argument x is not numeric, an error message is displayed and the function terminates.
If the argument is neither a matrix nor a vector, an error message is displayed and the
function terminates.
If the product matrix \eqn{{\bf{x'}}\;{\bf{x}}} is negative definite, an error message
displayed and the function terminates.
}
\examples{
x <- matrix( c( 2, 4, 2, 1, 3, 1, 5, 2, 1, 2, 3, 3 ), nrow=3, ncol=4, byrow=TRUE )
spectral.norm( x )
}
\keyword{ math }
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