inverse method

Matrix inverse({ double conditionThreshold = 1e-3, })

Calculates the inverse of a square matrix, falling back to SVD-based inversion or generalized pseudo-inversion if necessary.

This method first attempts regular matrix inversion using LU decomposition. If the matrix is near-singular or ill-conditioned (as determined by the condition number), it falls back to an inversion based on Singular Value Decomposition (SVD). If all else fails, it uses the pseudo-inverse as a generalized solution.

• conditionThreshold: A threshold for the condition number, above which the matrix is considered ill-conditioned. Default is 1e-3.

Throws:

• Exception if the matrix is not square.
• Exception if all inversion methods fail.

Returns:

• A Matrix representing the inverse of the current matrix.

Example:

var matrix = Matrix([[1, 2], [3, 4]]);
var result = matrix.inverse();
print(result);
// Output:
// Matrix: 2x2
// ┌ -2.0  1.0 ┐
// └  1.5 -0.5 ┘

Implementation

Matrix inverse({double conditionThreshold = 1e-3}) {
  // if (rowCount != columnCount) {
  //   throw Exception('Matrix must be square to calculate inverse');
  // }

  // Compute SVD decomposition.
  var svd = decomposition.singularValueDecomposition();

  print('Condition number is: ${svd.conditionNumber}');

  if (svd.conditionNumber > Complex(conditionThreshold)) {
    print(
        'Matrix condition number exceeds threshold $conditionThreshold. Likely singular.');
  } else {
    try {
      // Attempt regular inversion using LU solve.
      var lu = decomposition.luDecompositionDoolittle();
      var identity = Matrix.eye(rowCount);
      return lu
          .solve(identity); // Solve A * X = I for X (generalized inverse).
    } catch (e) {
      print('Failed regular inversion: $e. Falling back to SVD inversion...');
    }
  }

  // SVD Inversion.
  try {
    print('Attempting SVD inversion...');
    // Identity matrix of the same size.
    var identity = Matrix.eye(rowCount);
    return svd.solve(identity);
  } catch (e) {
    print('SVD inversion failed: $e');
  }

  // Generalized Inverse Fallback.
  print('Falling back to generalized inverse...');
  return pseudoInverse();
}