B. The Akima Interpolation

The Akima interpolation [36] is a continuously differentiable sub-spline interpolation. It is built from piecewise third order polynomials. Only data from the next neighbor points is used to determine the coefficients of the interpolation polynomial. There is no need to solve large equation systems and therefore this interpolation method is computationally very efficient. Because no functional form for the whole curve is assumed and only a small number of points is taken into account this method does not lead to unnatural wiggles in the resulting curve. The monotonicity of the specified data points is not necessarily retained by the resulting interpolation function. By additional constraints on the estimated derivatives a monotonicity preserving interpolation function can be constructed [42].

For a set of data points

$$\leq \leq$$ (B1)

the interpolation function is defined as

$$\leq \leq$$ (B2)

To determine the coefficients a₀, a₁, a₂, and a₃ of the interpolation polynomial for each interval [x_i, x_{i + 1}] the function values s_i and s_{i + 1}, and the first derivatives s_i' and s_{i + 1}' at the end points of the interval are used.

The first derivative s_i' of the interpolation function at x_i is estimated from the data for this point and the next two points on each side of x_i. Using the ratios

$${\frac{s_{j+1} - s_{j}}{x_{j+1} - x_{j}}}$$ (B3)

and the weighting coefficients

w_{i - 1} = | d_{i + 1} - d_i|, (B4)

w_i = | d_{i - 1} - d_{i - 2}|, (B5)

the estimated derivative s_i' is defined as

$${\frac{w_{i - 1}\cdot d_{i - 1} + w_{i}\cdot d_{i}}{w_{i - 1} + w_{i}}}$$ (B6)

Several special cases for s_i' have to be considered.

$$\neq$$ s_i' = d_{i - 1} (B7)

$$\neq$$ s_i' = d_i (B8)

s_i' = d_{i - 1} = d_i         d_{i - 1} = d_i (B9)

$${\frac{d_{i - 1} + d_{i}}{2}} \neq$$ s_i' (B10)

To be able to use (B.6) for calculating the derivatives s₁', s₂', s_{k - 1}', and s_k' additional ratios d_-1, d₀, d_k, and d_{k + 1} have to be estimated.

d_-1 = 2 ^. d₀ - d₁ (B11)

d₀ = 2 ^. d₁ - d₂ (B12)

d_k = 2 ^. d_{k - 1} - d_{k - 2} (B13)

d_{k + 1} = 2 ^. d_k - d_{k - 1} (B14)

The order of the interpolation function reduces to 2 for these intervals.

Similar algorithms can be used for the interpolation of two-dimensional data on rectangular grids [43] and on unstructured grids[44] by bicubic and cubic polynomials, respectively.