Table of Contents

Matrix One Norm Software Manual

Routine Name: one_norm

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 ./matrixNorms.out that can be executed.

Description/Purpose: This is a template function that can be used to calculate the \(| M |1\) of any matrix s.t. \(| M |_1 = \max{1 \leq j \leq n} \sum_{i=1}^m

a_{ij}

\), ie the max of the absolute column sums.

Input: The routine takes one argument, the matrix

@tparam The type of the elements in the matrix
@param  The matrix to take the norm of

Output: The function returns the one norm of the matrix

Usage/Example:

int main()
{
  std::vector<std::vector<int>> m = {
    {-3, 5, 7},
    {2, 6, 4},
    {0, 2, 8}
  };
  std::cout << m << std::endl;
  std::cout << "one norm: " << one_norm(m) << std::endl;
}

Output from the lines above

|         -3         5         7 |
|          2         6         4 |
|          0         2         8 |

one norm: 19

explanation of output:

The first lines are the matrix

The second line is the one norm of that matrix

Implementation/Code: The following is the code for one_norm

The code finds the sum of the absolute values of the columns using std::abs. It then uses std::max_element to identify the

template <typename T>
T one_norm(Matrix<T> m)
{
  // initalize the array to hold the sums of the columns
  std::vector<T> colSums;
  colSums.reserve(m[0].size());

  // for every column
  for (auto j = 0u; j < m[0].size(); ++j)
  {
    // for every element column wise
    T sum = 0;
    for (auto i = 0u; i < m.size(); ++i)
    {
      // sum the absolute values of the column elements
      sum += std::abs(m[i][j]);
    }
    // push the column sum on the end of the colSum array
    colSums.push_back(sum);
  }

  // find and return the maximum element of the column sums
  return *std::max_element(begin(colSums), end(colSums));
}

Last Modified: October 2018