Cubic Interpolation

or

Substituting the expressions for c and d in (2.55), we obtain the spline in the interval [x_i-1` x_i] as

.........................1.4

Note that the spline second derivatives Mi-1, M_i are unknowns and are to be determined. Setting i = i 1 in (1.4),

we get the spline valid in the interval [x_i, x_{i 1}] as

.........................1.5

where h_{i 1} = x_{i 1} – x_i. Differentiating (1.4) and (1.5), we get

.........................1.6

valid in the interval [x_i-1, x_i], and

..................1.7

valid in the interval [x_i, x_{i 1}].

Now, we require that the derivative F′(x) be continuous at x = x_i. Hence, the left hand and right hand derivatives of F′(x) at x = x_i must be equal, that is,

Using (1.6) and (1.7), we obtain