R: Compute the Norm of a Matrix

norm {base}

R Documentation

Compute the Norm of a Matrix

Description

Computes a matrix norm of x using Lapack. The norm can be the one norm, the infinity norm, the Frobenius norm, or the maximum modulus among elements of a matrix, as determined by the value of type.

Usage

norm(x, type = c("O", "I", "F", "M"))

Arguments

x

numeric matrix; note that packages such as Matrix define more norm() methods.

type

character string, specifying the type of matrix norm to be computed. A character indicating the type of norm desired.

"O", "o" or "1": specifies the one norm, (maximum absolute column sum);
"I" or "i": specifies the infinity norm (maximum absolute row sum);
"F" or "f": specifies the Frobenius norm (the Euclidean norm of x treated as if it were a vector); and
"M" or "m": specifies the maximum modulus of all the elements in x.

The default is "O". Only the first character of type[1] is used.

Details

The base method of norm() calls the Lapack function dlange.

Note that the 1-, Inf- and "M" norm is faster to calculate than the Frobenius one.

Value

The matrix norm, a non-negative number.

References

Anderson, E., et al. (1994). LAPACK User's Guide, 2nd edition, SIAM, Philadelphia.

Examples

(x1 <- cbind(1,1:10))
norm(x1)
norm(x1, "I")
norm(x1, "M")
stopifnot(all.equal(norm(x1, "F"),
                    sqrt(sum(x1^2))))

hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+") }
h9 <- hilbert(9)
## all 4 types of norm:
(nTyp <- eval(formals(base::norm)$type))
sapply(nTyp, norm, x=h9)

[Package base version 2.14.1 ]