Legendre Polynomial Fitting Algorithm

The algorithm program implements Legendre polynomial fitting to determine the weight coefficients for an N-th order Legendre polynomial. This method employs least squares approximation to fit data by minimizing the sum of squared residuals between the polynomial model and the given dataset. The implementation typically involves: 1. Generating the basis Legendre polynomials up to order N using recurrence relations: P₀(x)=1, P₁(x)=x, and (n+1)Pₙ₊₁(x) = (2n+1)xPₙ(x) - nPₙ₋₁(x) 2. Constructing the design matrix where each column represents Legendre polynomial values at data points 3. Solving the normal equations or using QR decomposition for numerical stability Legendre polynomial fitting serves as a powerful tool with applications across various domains including physics, mathematics, and computer science. Compared to standard polynomial fitting methods, Legendre polynomial fitting offers superior approximation performance and enhanced stability for high-order fits due to the orthogonal nature of Legendre polynomials on the interval [-1,1]. This orthogonality prevents the ill-conditioning issues commonly encountered in monomial basis fits. For accurate data fitting requirements or high-precision computations, the Legendre polynomial fitting algorithm is recommended, particularly when working with normalized data ranges. The implementation can be optimized using orthogonal polynomial properties to avoid numerical instability in higher-order approximations.