The Equation for a Cubic Spline

Figure 7.7: Cubic spline.

• We will create a succession of cubic splines over successive intervals of the data (See Fig. 7.7).
• Each spline must join with its neighbouring cubic polynomials at the knots where they join with the same slope and curvature.
• We write the equation for a cubic polynomial, $g_i(x)$, in the $i^{th}$ interval, between points $(x_i,y_i), (x_{i+1}, y_{i+1})$ (solid line).
• It has this equation:
  $$g_i(x)=a_i(x - x_i)^3 + b_i(x - x_i)^2 + c_i(x - x_i) + d_i$$

• The dashed curves are other cubic spline polynomials.

• Thus, the cubic spline function is of the form
  $$g(x)=g_i(x)~on ~the ~interval [x_i, x_{i+1}],~for~ i = 0,1,\ldots, n-1$$

• and meets these conditions:

  $$g_i(x_i)=y_i,~ i=0,1,\ldots, n - 1~and ~ g_{n-1}(x_n)=y_n$$ (7.1)

  $$g_i(x_{i+1})=g_{i+1}(x_{i+1}),~i=0,1,\ldots,n - 2$$ (7.2)

  $$g_i^{'}(x_{i+1})=g_{i+1}^{'}(x_{i+1}),~i=0,1,\ldots,n - 2$$ (7.3)

  $$g_i^{''}(x_{i+1})=g_{i+1}^{''}(x_{i+1}),~i=0,1,\ldots,n - 2$$ (7.4)

• Equations say that the cubic spline fits to each of the points Eq. 7.1, is continuous Eq. 7.2, and is continuous in slope and curvature Eq. 7.3 and Eq. 7.4, throughout the region spanned by the points.