Matrix Norms

Routine Name: Matrix Norm

Author: Kyle Hovey

Language: C++

Description/Purpose:

This method computes the matrix \(1\)-norm and \(\infty\)-norm. The former is equivalent to the maximum column sum, and the latter is equivalent to the maximum row sum.

Input:

A square matrix.

Output:

The \(1\)-norm or \(\infty\)-norm of that matrix.

Usage/Example:

#include "../../matrix/src/matrix/matrix.h"
#include <iostream>
#include <limits>

using Mtx = Matrix::Matrix<double>;

int main() {
  Mtx A({
      {1, 5, 1, 0},
      {2, 0, 0, 1},
      {3, 8, 5, 2},
      {4, 9, 2, 1}
  });

  auto oneNorm = Mtx::mNorm(A, 1);
  auto infNorm = Mtx::mNorm(A, std::numeric_limits<uint>::max());

  std::cout << "1-norm of matrix:" << std::endl;
  std::cout << oneNorm << std::endl;
  std::cout << "Infinity-norm of matrix:" << std::endl;
  std::cout << infNorm << std::endl;

  return EXIT_SUCCESS;
}

Output:

1-norm of matrix:
22
Infinity-norm of matrix:
18

Implementation/Code:

template <typename T>
T Matrix<T>::mNorm(const Matrix<T>& A, const uint& n) {
  const auto [ M, N ] = A.getSize();
  T max = -std::numeric_limits<uint>::max();

  if (n == std::numeric_limits<uint>::max()) {
    // Max row-sum
    for (uint row = 0; row < M; ++row) {
      T sum = 0;

      for (uint col = 0; col < N; ++col) {
        sum += A.getVal(row, col);
      }

      max = sum > max ? sum : max;
    }

    return max;
  } else if (n == 1) {
    // Max col-sum
    for (uint col = 0; col < N; ++col) {
      T sum = 0;

      for (uint row = 0; row < N; ++row) {
        sum += A.getVal(row, col);
      }

      max = sum > max ? sum : max;
    }

    return max;
  } else {
    throw std::domain_error("Matrix norm order not implemented.");
  }
}

Numerical Solutions (Math 5620)

Matrix Norms