Compute the Norm of a Matrix

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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

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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" or "f"

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.

Source

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

LAPACK is from http://www.netlib.org/lapack.

References

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

See Also

rcond for the (reciprocal) condition number.

Examples

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(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 types of norm:
(nTyp <- eval(formals(base::norm)$type))
sapply(nTyp, norm, x = h9)

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