MATLAB Source Code for Least Squares Fitting Based on Matrix QR Decomposition

QR decomposition is a fundamental matrix factorization method in linear algebra, extensively applied to solve least squares problems. By decomposing the coefficient matrix into the product of an orthogonal matrix and an upper triangular matrix, this method effectively addresses fitting problems for overdetermined equation systems. The core principle of least squares method involves finding optimal parameters that minimize the sum of squared errors. When employing QR decomposition, the original least squares problem transforms into a more easily solvable triangular system of equations. This approach offers superior numerical stability compared to directly solving normal equations, particularly when dealing with ill-conditioned matrices. In MATLAB implementation, the built-in qr function efficiently performs matrix decomposition. The computational process comprises three critical steps: First, perform QR decomposition on the design matrix using [Q,R] = qr(A); then process the right-hand side vector through Q'*b; finally solve the simplified system using back substitution with R\y. This method avoids condition number deterioration issues associated with explicit computation of matrix products. QR decomposition method is particularly suitable for scenarios like experimental data fitting and regression analysis. Through orthogonal transformations, the algorithm automatically handles linearly dependent or nearly linearly dependent column vectors, providing a reliable mathematical tool for engineering computations. The implementation typically includes error handling for rank-deficient cases and can be extended to weighted least squares problems through appropriate matrix scaling.