MATLAB Implementation of Cubic Spline Interpolation with Algorithm Explanation

This is a MATLAB .m program implementing cubic spline interpolation. The input is a 2D array A(Nx2) containing discrete data points. Spline interpolation is a numerical analysis method used to generate interpolating functions that approximate values between known data points. The interpolation follows the mathematical formulation: S(x) = A(J) + B(J)*(x - x(J)) + C(J)*(x - x(J))**2 + D(J)*(x - x(J))**3, where J is an integer index, x(J) represents a known data point from the input array, and x is the interpolation point. The program uses A(J) as the interpolation function value to the left of the interpolation point, and A(J+1) to the right. The smoothness of the interpolating function is determined by coefficients B(J), C(J), and D(J), which are calculated by applying constraints on the derivatives of the spline function. The implementation typically involves solving a tridiagonal system of equations to ensure continuity of first and second derivatives at the knots. Key implementation aspects include: - Handling of natural or clamped boundary conditions - Efficient computation of spline coefficients using matrix operations - Interval search algorithms to locate the correct segment J for each interpolation point - Vectorized operations for multiple interpolation points This method is widely applicable in mathematics, engineering, and scientific fields where smooth approximation of discrete data points is required, such as in curve fitting, numerical analysis, and data interpolation tasks. The code structure follows MATLAB best practices with proper input validation and error handling for robust performance.