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 ("O") norm, the infinity ("I") norm, the Frobenius ("F") norm, the maximum modulus ("M") among elements of a matrix, or the “spectral” or "2"-norm, as determined by the value of type.

Usage

norm(x, type = c("O", "I", "F", "M", "2"))

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", "f", "E" or "e"

specifies the Frobenius norm (the Euclidean norm of x treated as if it were a vector);

"M" or "m"

specifies the maximum modulus of all the elements in x; and

"2"

specifies the “spectral” or 2-norm, which is the largest singular value (svd) of 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.

Unsuccessful results from the underlying LAPACK code will result in an error giving a positive error code: these can only be interpreted by detailed study of the FORTRAN code.

Value

The matrix norm, a non-negative number. Zero for a 0-extent (empty) matrix.

Source

Except for norm = "2", the LAPACK routine DLANGE.

LAPACK is from https://netlib.org/lapack/.

References

⁠Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J, Du Croz J, Greenbaum A, Hammerling S, McKenney A, Sorensen D (1999). LAPACK Users' Guide, series Software, Environments, and Tools, Third edition. Society for Industrial and Applied Mathematics, Philadelphia, PA. ISBN 9780898714470. doi:10.1137/1.9781611971811. https://netlib.org/lapack/lug/lapack_lug.html.

See Also

rcond for the (reciprocal) condition number.

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 5 (4 different) types of norm:
(nTyp <- eval(formals(base::norm)$type))
sapply(nTyp, norm, x = h9)
stopifnot(exprs = { # 0-extent matrices:
    sapply(nTyp, norm, x = matrix(, 1,0)) == 0
    sapply(nTyp, norm, x = matrix(, 0,0)) == 0
})
[Package base version 4.6.0 Index]