Algorithm Implementation for Continuous Function Best Square Approximation

In this course project, you will develop an algorithm program implementing continuous function best square approximation in function form. You need to conduct thorough problem analysis and design appropriate algorithms based on problem characteristics. During programming, focus on code readability and maintainability by implementing modular functions for polynomial basis generation, normal equation solving, and error calculation. To validate algorithm correctness and effectiveness, perform numerical experiments using test functions (e.g., trigonometric, exponential functions) and document experimental results. The project report should detail algorithm design process, implementation specifics including key functions like Legendre polynomial computation and matrix decomposition methods, plus numerical experiment outcomes. Submit complete source code enabling others to review and reproduce your results. For comprehensive project completion, consider adding algorithm optimizations such as iterative refinement techniques, additional numerical tests with varying approximation degrees, and computational complexity analysis.