Matrix (version 1.7-1)

norm-methods: Matrix Norms

Description

Computes a matrix norm of x, using Lapack for dense matrices. The norm can be the one ("O", or "1") norm, the infinity ("I") norm, the Frobenius ("F") norm, the maximum modulus ("M") among elements of a matrix, or the spectral norm or 2-norm ("2"), as determined by the value of type.

Usage

norm(x, type, ...)

Value

A numeric value of class "norm", representing the quantity chosen according to type.

Arguments

x

a real or complex matrix.

type

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 norm” aka “2-norm”, which is the largest singular value (svd) of x.

The default is "O". Only the first character of type[1] is used.

...

further arguments passed to or from other methods.

Details

For dense matrices, the methods eventually call the Lapack functions dlange, dlansy, dlantr, zlange, zlansy, and zlantr.

References

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

See Also

onenormest(), an approximate randomized estimate of the 1-norm condition number, efficient for large sparse matrices.

The norm() function from R's base package.

Examples

x <- Hilbert(9)
norm(x)# = "O" = "1"
stopifnot(identical(norm(x), norm(x, "1")))
norm(x, "I")# the same, because 'x' is symmetric

allnorms <- function(x) {
    ## norm(NA, "2") did not work until R 4.0.0
    do2 <- getRversion() >= "4.0.0" || !anyNA(x)
    vapply(c("1", "I", "F", "M", if(do2) "2"), norm, 0, x = x)
}
allnorms(x)
allnorms(Hilbert(10))

i <- c(1,3:8); j <- c(2,9,6:10); x <- 7 * (1:7)
A <- sparseMatrix(i, j, x = x)                      ##  8 x 10 "dgCMatrix"
(sA <- sparseMatrix(i, j, x = x, symmetric = TRUE)) ## 10 x 10 "dsCMatrix"
(tA <- sparseMatrix(i, j, x = x, triangular= TRUE)) ## 10 x 10 "dtCMatrix"
(allnorms(A) -> nA)
allnorms(sA)
allnorms(tA)
stopifnot(all.equal(nA, allnorms(as(A, "matrix"))),
	  all.equal(nA, allnorms(tA))) # because tA == rbind(A, 0, 0)
A. <- A; A.[1,3] <- NA
stopifnot(is.na(allnorms(A.))) # gave error