Vector Norm

The Norm function calculates several different types of vector norms for x, depending on the argument p.

Norm(x, p = 2)

Arguments

x

Numeric vector; matrices not allowed.

p

Numeric scalar or Inf, -Inf; default is 2

Details

Norm returns a scalar that gives some measure of the magnitude of the elements of x. It is called the \(p\)-norm for values \(-Inf \le p \le Inf\), defining Hilbert spaces on \(R^n\).

Norm(x) is the Euclidean length of a vecor x; same as Norm(x, 2).
Norm(x, p) for finite p is defined as sum(abs(A)^p)^(1/p).
Norm(x, Inf) returns max(abs(x)), while Norm(x, -Inf) returns min(abs(x)).

Value

Numeric scalar (or Inf), or NA if an element of x is NA.

Note

In Matlab/Octave this is called norm; R's norm function norm(x, "F") (`Frobenius Norm') is the same as Norm(x).

See also

norm of a matrix

Examples

Norm(c(3, 4))          #=> 5  Pythagoras triple
#> [1] 5
Norm(c(1, 1, 1), p=2)  #   sqrt(3)
#> [1] 1.732051
Norm(1:10, p = 1)      #   sum(1:10)
#> [1] 55
Norm(1:10, p = 0)      #   Inf
#> [1] Inf
Norm(1:10, p = Inf)    #   max(1:10)
#> [1] 10
Norm(1:10, p = -Inf)   #   min(1:10)
#> [1] 1