Splines:

It is a piecewise fit and therefore can be utilized if big amounts of accurate data is available. Spline functions let small subsets of data and fit these along lower order polynomials. It is a significant technique utilized in a broad range of applications. The interpolating polynomial might be linear, cubic, quadratic etc.

The cubic spline functions are the most widely utilized. Letting two arbitrary points x_i and x_{i + 1} at which the function f (x) is to be determined

The cubic that passes through these points is provided as

f_i (x) = a₀ + a₁ x_i + a₂ x²_i + a₃ x³_i ... for x among x _i and x _{i + 1}

By Substituting x = x_i and x_{i + 1}, we obtain, respectively

As there are 4 constants to be find out, other 2 conditions are attained by using the continuity of first and second derivatives at x_i & x_{i + 1} (for smooth transition).

The cubic f_i (x) over x_i = x = x_{i +1} is get as

Here, Δ x_i = x_{i + 1} - x