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.
norm(x, type = c("O", "I", "F", "M", "2"))numeric matrix; note that packages such as Matrix
define more norm() methods.
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.
The matrix norm, a non-negative number.
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.
Anderson, E., et al (1994). LAPACK User's Guide, 2nd edition, SIAM, Philadelphia.
rcond for the (reciprocal) condition number.
(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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