Table of Contents

Matrix Infinity Norm Software Manual

Routine Name: inf_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 i \leq m} \sum_{i=1}^n

a_{ij}

\), ie the max of the absolute row 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 infinity norm of the matrix

Usage/Example:

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

Output from the lines above

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

inf norm: 15

explanation of output:

The first lines are the matrix

The second line is the inf norm of that matrix

Implementation/Code: The following is the code for inf_norm

The code finds the sum of the absolute values of the elements of the rows. First it uses std::for_each to iterate over the rows. Then for each row it uses std::accumulate to add the absolute value of the elements of the rows using std::abs. It then uses std::max_element to identify the largest of the rowSums and returns it.

template <typename T>
T inf_norm(Matrix<T> m)
{
  // initialize the array to put the row sums
  std::vector<T> rowSums;
  rowSums.reserve(m[0].size());

  // for each row
  std::for_each(begin(m), end(m), [&colSums](auto const& row) {
    // sum the absolute values of the elements
    rowSums.push_back(std::accumulate(
      begin(row), end(row), 0.0, [](T acc, T e) { return acc + std::abs(e); }));
  });

  return *std::max_element(begin(rowSums), end(rowSums));
}

Last Modified: October 2018