Table of Contents

Matrix Frobenius Norm Software Manual

Routine Name: frobenius_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

_F\) of any matrix s.t. \(

M

F = \sqrt{\sum{i=1}^m \sum_{j=1}^n

a_{ij}

^2}\).

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 Frobenius 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 << "Frobenius norm: " << frobenius_norm(m) << std::endl;
}

Output from the lines above

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

Frobenius norm: 14.4

explanation of output:

The first lines are the matrix

The second line is the Frobenius norm of that matrix

Implementation/Code: The following is the code for frobenius_norm

This code uses std::for_each to iterate over the entire matrix summing up the elements squared. It then return the square root using std::sqrt of the sum of the elements squared. This is the definition of the Frobenius Norm](https://en.wikipedia.org/wiki/Matrix_norm#Frobenius_norm).

template <typename T>
double frobenius_norm(Matrix<T> m)
{
  // initalize the sum of the elements squared
  double elems_squared = 0.0;

  // for each row
  std::for_each(begin(m), end(m), [&elems_squared](auto const& row) {
    // for each element
    std::for_each(begin(row), end(row), [&elems_squared](auto const& elem) {
      // add the elem^2 to the sum
      elems_squared += elem * elem;
    });
  });

  // return the square root of the sum of the lements squared
  return std::sqrt(elems_squared);
}

Last Modified: October 2018