Piecewise Lagrange Interpolation Polynomial with n Nodes

This text introduces a piecewise Lagrange interpolation polynomial method with the following implementation steps:

1. First, partition the original function domain using n interpolation nodes. The algorithm divides the dataset into segments between consecutive nodes, applying local interpolation to each segment.

2. Next, perform computation using the function call format y = lagrange(x0, y0, x, k). The function handles vectorized operations to efficiently compute multiple interpolation points simultaneously.

3. Input parameter x0 represents an n-dimensional vector containing interpolation node coordinates, while y0 is an n-dimensional vector storing corresponding function values. These vectors define the known data points for constructing the interpolant.

4. Parameter x is an m-dimensional vector specifying evaluation points where interpolation values are needed. Parameter k controls the degree of piecewise polynomials (maximum degree 3, default k=1 for linear interpolation). The implementation uses k+1 neighboring points for each local polynomial construction.

5. Finally, the output y returns interpolated function values at points specified in x. The algorithm employs Lagrange basis polynomial calculations for each segment with proper boundary handling.

Beyond these core steps, different function approximations can be obtained by adjusting interpolation points and polynomial degrees. This method is particularly useful for data fitting and prediction tasks, as well as for numerical data processing and scientific computing applications where piecewise approximation provides better stability than global polynomial interpolation.