Least Squares Method Based on QR Orthogonal Decomposition

The QR decomposition least squares method is a stable and efficient approach for solving linear least squares problems. This technique avoids potential numerical instability in direct normal equation solutions by employing orthogonal triangularization of matrices.

The core methodology involves three key stages: First, performing QR decomposition on the coefficient matrix to factor it into an orthogonal matrix and an upper triangular matrix; then leveraging the properties of orthogonal matrices to simplify the least squares problem formulation; finally obtaining the optimal solution through back substitution in the triangular system.

Compared to conventional normal equation approaches, the QR decomposition method offers superior numerical stability, particularly beneficial for ill-conditioned or near-singular coefficient matrices. This method eliminates the need to compute the product of coefficient matrices, thereby avoiding the squaring effect of condition numbers and maintaining higher precision in floating-point operations.

In MATLAB implementation, Householder reflections or Givens rotations are typically employed to construct orthogonal matrices - transformations that preserve the Euclidean norm of vectors. Modern numerical linear algebra libraries often utilize blocked algorithms to enhance computational efficiency for large-scale matrix operations. The backslash operator (\) in MATLAB automatically employs QR decomposition when solving overdetermined systems.

This solution method finds widespread applications in curve fitting, system identification, statistical regression, and represents one of the most fundamental algorithms in scientific computing.