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. J., Du Croz J., Greenbaum A., Hammerling S., McKenney A., Sorensen D. C. (1999). LAPACK Users' Guide, series Software, Environments, and Tools, Third edition. Society for Industrial and Applied Mathematics, Philadelphia, PA. ISBN 9780898714470. doi:10.1137/1.9780898719604. 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]