lp_norm function

lp norm of a vector

Computes the $\ell^p$ norm of an n-dimensional (real/complex) vector $\mathbf{x} \in \mathbf{C}^n$

[REMOVE_ME]$\left|\left| \mathbf{x} \right|\right|_p = \left( \sum_{i=1}^n\left| x_i \right|^p \right)^{1/p}, p \in [0, \infty], [REMOVE_ME_2]$

where $\left| x_i \right|$ is the absolute value of $x_i$. For $p=2$ this is Euclidean norm; for $p=1$ it is Manhattan norm. For $p=0$ it is defined as the number of non-zero elements in $\mathbf{x}$; for $p = \infty$ it is the maximum of the absolute values of $\mathbf{x}$.

The norm of $\mathbf{x}$ equals $0$ if and only if c("$\\mathbf{x} =\n$", "$\\mathbf{0}$").

lp_norm(x, p = 2)

Arguments

• x: n-dimensional vector (possibly complex values)
• p: which norm? Allowed values $p \geq 0$ including Inf. Default: 2 (Euclidean norm).

Returns

Non-negative float, the norm of $\mathbf{x}$.

Description

Computes the $\ell^p$ norm of an n-dimensional (real/complex) vector $\mathbf{x} \in \mathbf{C}^n$

where $\left| x_i \right|$ is the absolute value of $x_i$. For $p=2$ this is Euclidean norm; for $p=1$ it is Manhattan norm. For $p=0$ it is defined as the number of non-zero elements in $\mathbf{x}$; for $p = \infty$ it is the maximum of the absolute values of $\mathbf{x}$.

The norm of $\mathbf{x}$ equals $0$ if and only if c("$\\mathbf{x} =\n$", "$\\mathbf{0}$").

Examples

kRealVec <- c(3, 4)
# Pythagoras
lp_norm(kRealVec)
# did not know Manhattan,
lp_norm(kRealVec, p = 1)

# so he just imagined running in circles.
kComplexVec <- exp(1i * runif(20, -pi, pi))
plot(kComplexVec)
sapply(kComplexVec, lp_norm)