Table of Contents

l_pNorms Software Manual

Routine Name: l_pNorm

Author: Philip Nelson

Language: C++. The code can be compiled using the GNU C++ compiler (gcc). A make file is included to compile an example program

For example,

make

will produce an executable ./norm.out that can be executed.

Description/Purpose: This is a template function that can be used to calculate the \(l_n\) norm of any vector where \(n \in \mathbb{R}\) s.t. \(|| v ||_p = (|x_1|^p + |x_2|^p + \cdots + |x_n|^p)^\frac{1}{p}\)

Input: The function takes two arguments, a vector and the value of p

Output: The function returns the specified \(l_p\) norm

@tparam T The type of the elements in `a`
@tparam P The type of `p`
@param a  The vector
@param p  The `p` of the l_pNorm

Usage/Example:

int main()
{
  std::vector<double> v{3, 4, 1};
  std::cout << "v : " << v << '\n';
  std::cout << "l_1 norm :   " <<  l_pNorm(v, 1.0) << '\n';
  std::cout << "l_2 norm :   " << l_pNorm(v, 2.0) << '\n';
}

Output from the lines above

v : [          3         4         1 ]

l_1 norm :   8
l_2 norm :   5.1

explanation of output:

The first line is the vector

The second line is the \(l_1\) norm of \(v\)

The third line is the \(l_2\) norm of \(v\)

Implementation/Code: The following is the code for l_pNorm

The implementation for l_pNorm uses std::accumulate. In this case, it accumulates the values of the vector a from the beginning of the vector to the end of the vector starting with an initial sum of 0.0. Before adding each element, it applies the std::pow function, raising the absolute value of the element, with std::abs, to the specified power p. At the end, it uses std::pow again to raise the accumulated sum to 1.0/p. This is the definition of the pNorm.

template <typename T>
inline T l_pNorm(std::vector<T> const& a, P const p)
{
  return std::pow(
      std::accumulate(
        begin(a), end(a), 0.0, [p](T acc, T const e) {
          return acc + std::pow(std::abs(e), p);
        }),
      1.0/p);
}

Last Modified: September 2018