The Equation for a Cubic Spline

Figure 4: Cubic spline.

• We will create a succession of cubic splines over successive intervals of the data (See Fig. 4).
• 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 $(x_i,y_i), (x_{i+1}, y_{i+1})$.
• It looks like the solid curve shown here.
• The dashed curves are other cubic spline polynomials. It has this equation:
  
  $$g_i(x)=a_i(x - x_i)^3 + b_i(x - x_i)^2 + c_i(x - x_i) + d_i$$
  
  

• Thus, the cubic spline function we want 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:
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    $$g_i(x_i)=y_i,~ i=0,1,\ldots, n - 1~and ~ g_{n-1}(x_n)=y_n$$ (1)

    
    

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

    
    

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    $$g_i^{'}(x_{i+1})=g_{i+1}^{'}(x_{i+1}),~i=0,1,\ldots,n - 2$$ (3)

    
    

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    $$g_i^{''}(x_{i+1})=g_{i+1}^{''}(x_{i+1}),~i=0,1,\ldots,n - 2$$ (4)

    
    

• Equations say that the cubic spline fits to each of the points Eq. 1, is continuous Eq. 2, and is continuous in slope and curvature Eq. 3 and Eq. 4, throughout the region spanned by the points.