Series Expansion Approximation for Function Value Calculation (MATLAB Implementation)

In MATLAB, series expansion (such as Taylor series) is a fundamental numerical method for approximating function values. The core concept involves representing complex functions as infinite series and achieving approximations by truncating to a finite number of terms. This approach proves particularly valuable when analytical solutions are difficult to obtain or when computational costs are high.

For a function ( f(x) ), the Taylor series expansion about point ( a ) is expressed as: [ f(x) = f(a) + f'(a)(x-a) + frac{f''(a)}{2!}(x-a)^2 + cdots + frac{f^{(n)}(a)}{n!}(x-a)^n + R_n(x), ] where ( R_n(x) ) represents the remainder term, indicating truncation error. In practical computations, the first ( N ) terms are typically used for approximation.

Implementation in MATLAB generally follows these steps with corresponding code techniques: Expansion Point Selection: Choose point ( a ) where function values and derivatives are easily computable, often implemented through function handles or symbolic variables. Derivative Calculation: When manual derivative derivation becomes cumbersome, utilize MATLAB's Symbolic Math Toolbox with diff() function to automatically generate higher-order derivatives through recursive differentiation. Summation Truncation: Accumulate the first ( N ) terms using either iterative loops or vectorized operations, computing each term ( frac{f^{(k)}(a)}{k!}(x-a)^k ) through factorial calculations and element-wise power operations. Error Analysis: Evaluate approximation accuracy using remainder term formulas or by comparing results with varying term counts, potentially implementing convergence tests with while loops.

This method finds extensive applications in engineering and scientific computing, particularly when built-in functions are unavailable (e.g., special functions) or when computational efficiency requires optimization. Note that series convergence characteristics and expansion point selection significantly impact approximation quality.

Extension Approaches: For periodic functions, Fourier series may provide better approximations using fft() implementations. When functions are non-differentiable at expansion points, consider piecewise expansions or alternative numerical methods like interpolation using interp1 or spline functions.