Interpolation with spline functions

If we want to interpolate a function f on $\sigma_{n}^{}$ by a spline function s of degree l, then s must verify s(x_k) = y_k = f (x_k) for all 0 $\geq$ k $\geq$ n. Hence there are n + 1 conditions, and l - 1 degrees of liberty. We can therefore add l - 1 conditions, these conditions are on the derivatives of s at a and b.

Hermite interpolation, natural interpolation and periodic interpolation are three kinds of interpolation obtained by specifying three kinds of constraints. The unicity of the solution of the interpolation problem can be proved for each kind of constraints.

If l is odd (l = 2m - 1), there are 2m - 2 degrees of liberty. The constraints are defined by :

• Hermite interpolation
  $$\forall$$1 $$\leq$$ j $$\leq$$ m - 1,    s^(j)(a) = f^(j)(a), s^(j)(b) = f^(j)(b)

• Natural interpolation
  $$\forall$$m $$\leq$$ j $$\leq$$ 2m - 2,    s^(j)(a) = s^(j)(b) = 0

• periodic interpolation
  $$\forall$$1 $$\leq$$ j $$\leq$$ 2m - 2,    s^(j)(a) = s^(j)(b)

If l is even (l = 2m), there are 2m - 1 degrees of liberty. The constraints are defined by :

• Hermite interpolation
  $$\forall$$1 $$\leq$$ j $$\leq$$ m - 1,    s^(j)(a) = f^(j)(a), s^(j)(b) = f^(j)(b)
  and
  s^(m)(a) = f^(m)(a)

• Natural interpolation
  $$\forall$$m $$\leq$$ j $$\leq$$ 2m - 2,    s^(j)(a) = s^(j)(b) = 0
  and
  s^(2m-1)(a) = 0

• Periodic interpolation
  $$\forall$$1 $$\leq$$ j $$\leq$$ 2m - 1,    s^(j)(a) = s^(j)(b)

A natural spline is a spline function which verifies the natural interpolation constraints.

spline takes as arguments a list of abscissa (by increasing order), a list of ordinates, a variable name, and a degree.
spline returns the natural spline function (with the specified degree and crossing points) as a list of polynomials, each polynomial being valid on an interval.

Examples:

1. a natural spline of degree 3, crossing through the points x₀ = 0, y₀ = 1, x₁ = 1, y₁ = 3 and x₂ = 2, y₂ = 0, input :
   spline([0,1,2],[1,3,0],x,3)
   Output is a list of two polynomial expressions of x :
   [- 5*x3/4 + 13*x/4 + 1,    5*(x - 1)3/4 - 15*(x - 1)2/4 + (x - 1)/ - 2 + 3]
   defined respectivly on the intervals [0, 1] and [1, 2].
2. a 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, input :
   spline([0,1,2,3],[1,3,0,-1],x,4)
   Output is a list of three polynomial functions of x :
   [(- 62*x4 + 304*x)/121 + 1,
   (201*(x - 1)4 - 248*(x - 1)3 - 372*(x - 1)2 + 56*(x - 1))/121 + 3,
   (- 139*(x - 2)4 + 556*(x - 2)3 + 90*(x - 2)2 + - 628*(x - 2))/121]
   defined respectivly on the intervals [0, 1], [1, 2] and [2, 3].
3. The natural spline interpolation of cos on [0,$\pi$/2, 3$\pi$/2], input :
   spline([0,pi/2,3*pi/2],cos([0,pi/2,3*pi/2]),x,3)
   Output :
   [((3*$$\pi$$3 + (- 7*$$\pi$$2)*x + 4*x3)*1/3)/($$\pi$$3),
   ((15*$$\pi$$3 + (- 46*$$\pi$$2)*x + 36*$$\pi$$*x2 - 8*x3)*1/12)/($$\pi$$3)]