11.2.3 Natural interpolation

The spline command finds the natural spline.

• spline takes four arguments:
  • L_x, a list of abscissas (in increasing order).
  • L_y, a list of ordinates (the same length as L_x).
  • x, a variable name.
  • l, an integer for the degree.
• spline(L_x,L_y,x,l) returns the natural spline function s of degree l, where s(L_x,j)=L_y,j for j=0,…,length(L_x), as a list of polynomials, each polynomial being valid on the corresponding interval determined by L_x.

Examples

Find the natural spline of degree 3, crossing through the points x₀=0,y₀=1, x₁=1,y₁=3 and x₂=2, y₂=0.

Where the first polynomial, −5/4 x³+13/4 x+1, is defined on the interval [0,1] (the first interval defined by the list [0,1,2]) and the second polynomial 5/4 (x−1)³−15/4 (x−1)²−x−1/2+3 is defined on the interval [1,2], the second interval defined by the list [0,1,2].

Find the natural spline of degree 4, crossing through the points x₀=0,y₀=1, x₁=1,y₁=3, x₂=2, y₂=0 and x₃=3, y₃=−1.

Output is a list of three polynomial functions of x, defined respectively on the intervals [0,1], [1,2] and [2,3].

Find the natural spline interpolation of cos on [0,π/2,3π/2].