Least Squares Polynomial Fitting Using Orthogonal Polynomials Algorithm

This algorithm employs orthogonal polynomials for computing least squares polynomial fits. The core methodology involves representing data points using an orthogonal polynomial basis and applying least squares approximation to derive the optimal fitting polynomial function. In implementation, the algorithm typically constructs orthogonal polynomials (such as Legendre or Chebyshev polynomials) recursively using recurrence relations, then computes coefficients through inner product operations between the orthogonal basis and target values. This approach avoids the numerical instability often encountered when solving normal equations directly with standard polynomial bases. Widely applied in mathematics and statistics, particularly in data analysis and curve fitting applications, this method enhances data interpretation accuracy and provides more reliable predictive modeling. Key computational steps include: initializing orthogonal polynomials, computing orthogonal coefficients via projection operations, and reconstructing the final polynomial using the orthogonal basis expansion. The algorithm's advantage lies in its numerical robustness when handling high-degree polynomial fits, making it particularly suitable for scientific computing applications requiring precise approximations.