spectral.norm {matrixcalc} | R Documentation |
Spectral norm of matrix
Description
This function returns the spectral norm of a real matrix.
Usage
spectral.norm(x)
Arguments
x |
a numeric matrix or vector |
Details
Let be an
real matrix. The
function computes the order
square matrixmatrix
.
The R function
eigen
is applied to this matrix to obtain the vector
of eigenvalues . By construction the eigenvalues are in descending
order of value so that the largest eigenvalue is
. Then
the spectral norm is
.
If
is a vector, then
is returned.
Value
A numeric value.
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 is negative definite, an error message
displayed and the function terminates.
Author(s)
Frederick Novomestky fnovomes@poly.edu
References
Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.
Golub, G. H. and C. F. Van Loan (1996). Matrix Computations, Third Edition, The John Hopkins University Press.
Horn, R. A. and C. R. Johnson (1985). Matrix Analysis, Cambridge University Press.
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 )