l_inf Software Manual

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Routine Name: l_inf

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 ./norms.out that can be executed.

Description/Purpose: This is a template function that can be used to calculate the \(l_\infty\) norm of any a vector of any type s.t. \(|| v ||_\infty = max {|x_1|, |x_2|, \cdots , |x_n|}\)

Input: The function takes on argument, the vector

Output: The \(l_\infty\) norm of \(v\)

Usage/Example:

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

Output from the lines above

v : [          3         4         1 ]

l_inf norm : 4

explanation of output: The first line is the vector

The second line is the \(l_\infty\) norm of \(v\)

Implementation/Code: The following is the code for l_inf

The implementation of l_inf uses std::max_element which takes a lambda function that compares two elements by their absolute values using std::abs. Once the absolute largest element has been identified, it’s absolute value is returned. This is the definition of the infinity norm.

/**
 * Determine the l_pNorm of a vector
 *
 * @tparam T The type of the elements in `a`
 * @param a  The vector
 * @param p  The `p` of the l_pNorm
 */
template <typename T>
inline T l_inf(std::vector<T> const& a)
{
  return std::abs(
      *std::max_element(
        begin(a), end(a), [](T const a, T const b) {
          return std::abs(a) < std::abs(b);
        })
      );
}

Last Modified: September 2018