Pnorm-class: The Pnorm class.

Description

This class represents the vector p-norm.

Usage

Pnorm(x, p = 2, axis = NA_real_, max_denom = 1024)
# S4 method for Pnorm
validate_args(object)
# S4 method for Pnorm
name(object)
# S4 method for Pnorm
to_numeric(object, values)
# S4 method for Pnorm
sign_from_args(object)
# S4 method for Pnorm
is_atom_convex(object)
# S4 method for Pnorm
is_atom_concave(object)
# S4 method for Pnorm
is_incr(object, idx)
# S4 method for Pnorm
is_decr(object, idx)
# S4 method for Pnorm
is_pwl(object)
# S4 method for Pnorm
get_data(object)
# S4 method for Pnorm
graph_implementation(object, arg_objs, size,
  data = NA_real_)

Arguments

An '>Expression representing a vector or matrix.

A number greater than or equal to 1, or equal to positive infinity.

axis

(Optional) The dimension across which to apply the function: 1 indicates rows, 2 indicates columns, and NA indicates rows and columns. The default is NA.

max_denom

The maximum denominator considered in forming a rational approximation for $p$.

object

A '>Pnorm object.

values

A list of arguments to the atom.

idx

An index into the atom.

arg_objs

A list of linear expressions for each argument.

size

A vector with two elements representing the size of the resulting expression.

data

A list of additional data required by the atom.

Methods (by generic)

validate_args: Check that the arguments are valid.
name: The name and arguments of the atom.
to_numeric: The p-norm of x.
sign_from_args: The atom is positive.
is_atom_convex: The atom is convex if $p \geq 1$.
is_atom_concave: The atom is concave if $p < 1$.
is_incr: The atom is weakly increasing if $p < 1$ or $p \geq 1$ and x is positive.
is_decr: The atom is weakly decreasing if $p \geq 1$ and x is negative.
is_pwl: The atom is piecewise linear only if x is piecewise linear, and either $p = 1$ or $p = \infty$.
get_data: Returns list(p, axis).
graph_implementation: The graph implementation of the atom.

Slots

x: An '>Expression representing a vector or matrix.

p

A number greater than or equal to 1, or equal to positive infinity.

max_denom

The maximum denominator considered in forming a rational approximation for $p$.

axis

(Optional) The dimension across which to apply the function: 1 indicates rows, 2 indicates columns, and NA indicates rows and columns. The default is NA.

.approx_error

(Internal) The absolute difference between $p$ and its rational approximation.

Details

If given a matrix variable, Pnorm will treat it as a vector and compute the p-norm of the concatenated columns.

For $p \geq 1$, the p-norm is given by $$\|x\|_p = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}$$ with domain $x \in \mathbf{R}^n$. For $p < 1, p\neq 0$, the p-norm is given by $$\|x\|_p = \left(\sum_{i=1}^n x_i^p\right)^{1/p}$$ with domain $x \in \mathbf{R}^n_+$.

Note that the "p-norm" is actually a norm only when $p \geq 1$ or $p = +\infty$. For these cases, it is convex.
The expression is undefined when $p = 0$.
Otherwise, when $p < 1$, the expression is concave, but not a true norm.