1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
|
\name{entrywise.norm}
\alias{entrywise.norm}
\title{ Compute the entrywise norm of a matrix }
\description{
This function returns the \eqn{\left\| {\bf{x}} \right\|_p } norm of the matrix \eqn{{\mathbf{x}}}.
}
\usage{
entrywise.norm(x,p)
}
\arguments{
\item{x}{ a numeric vector or matrix }
\item{p}{ a real value for the power }
}
\details{
Let \eqn{{\bf{x}}} be an \eqn{m \times n} numeric matrix.
The formula used to compute the norm is \eqn{\left\| {\bf{x}} \right\|_p = \left( {\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {\left| {x_{i,j} } \right|^p } } } \right)^{{1 \mathord{\left/
{\vphantom {1 p}} \right.
} p}}}.
}
\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.
}
\note{
If argument x is not numeric, the function displays an error message and terminates.
If argument x is neither a matrix nor a vector, the function displays an error message and terminates.
If argument p is zero, the function displays an error message and terminates.
}
\author{ Frederick Novomestky \email{fnovomes@poly.edu} }
\seealso{
\code{\link{one.norm}},
\code{\link{inf.norm}}
}
\examples{
A <- matrix( c( 3, 5, 7, 2, 6, 4, 0, 2, 8 ), nrow=3, ncol=3, byrow=TRUE )
print( entrywise.norm( A, 2 ) )
}
\keyword{ math }
|
