MATLAB Implementation of Newton Interpolation Algorithm with Code Analysis

Newton interpolation is a classical numerical approximation method used to construct interpolation polynomials through known discrete data points. Compared to Lagrange interpolation, it offers higher computational efficiency and convenient incremental node addition capabilities. Core Algorithm Principles Difference Quotient Calculation: First construct a difference quotient table using recursive computation for various orders of difference quotient values. In MATLAB implementation, this typically involves creating a 2D array where each column represents higher-order difference quotients, calculated using nested loops to avoid redundant computations. Polynomial Construction: Build the Newton interpolation polynomial N(x) based on the difference quotient table. The implementation often utilizes Horner's rule (nested multiplication form) to enhance computational efficiency, reducing the number of arithmetic operations required for polynomial evaluation. Approximation Solution: Input new x-values into the polynomial to obtain corresponding y approximations. The MATLAB code typically implements this as a function that takes the difference quotient table and new x-points as inputs, returning interpolated values. MATLAB Implementation Key Points The difference quotient table is usually stored in a 2D array, with careful attention to avoid repeated calculations during the recursive computation process. Built-in functions like `diff` and `prod` can assist in simplifying difference quotient calculation steps. For polynomial evaluation, implementing Horner's scheme significantly improves computational performance by reducing the operation count from O(n²) to O(n). Algorithm Advantages Time complexity of O(n²) is superior to Lagrange interpolation's O(n³). When adding new nodes, only the new difference quotient terms need to be calculated without reconstructing the entire polynomial. This makes it particularly suitable for applications requiring dynamic addition of sampling points, such as real-time data processing systems. Important Considerations High-degree interpolation may exhibit Runge's phenomenon, where oscillations occur at the edges of the interval. In practical applications, there's a crucial trade-off between interpolation degree and computational accuracy. MATLAB implementations should include checks for appropriate polynomial degree selection based on the input data characteristics.