Compute the Norm of a Matrix

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


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 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.


The matrix norm, a non-negative number.


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

See Also

rcond for the (reciprocal) condition number.

  • norm
library(base) # NOT RUN { (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) # }
Documentation reproduced from package base, version 3.6.2, License: Part of R 3.6.2

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