First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 3. INTERPOLATION 124
• Clamped boundary: S 0(x ′ 0 )=f ′ (x 0 ) and S n−1(x ′ n )=f ′ (x n )
Each boundary choice gives another two equations, bringing the total number of equations to
4n. There are 4n unknowns as well. Do these systems of equations have a unique solution?
The answer is yes, and a proof can be found in Burden, Faires, Burden [4]. The spline
obtained from the first boundary choice is called a natural spline, and the other one is
called a clamped spline.
Example 64. Find the natural cubic spline that interpolates the data (0, 0), (1, 1), (2, 0).
Solution. We have two cubic polynomials to determine:
S 0 (x) =a 0 + b 0 x + c 0 x 2 + d 0 x 3
S 1 (x) =a 1 + b 1 x + c 1 x 2 + d 1 x 3
The interpolation equations are:
S 0 (0) = 0 ⇒ a 0 =0
S 0 (1) = 1 ⇒ a 0 + b 0 + c 0 + d 0 =1
S 1 (1) = 1 ⇒ a 1 + b 1 + c 1 + d 1 =1
S 1 (2) = 0 ⇒ a 1 +2b 1 +4c 1 +8d 1 =0
We need the derivatives of the polynomials for the other equations:
The smoothness conditions are:
S ′ 0(x) =b 0 +2c 0 x +3d 0 x 2
S ′ 1(x) =b 1 +2c 1 x +3d 1 x 2
S 0 ′′ (x) =2c 0 +6d 0 x
S 1 ′′ (x) =2c 1 +6d 1 x
The natural boundary conditions are:
S ′ 0(1) = S ′ 1(1) ⇒ b 0 +2c 0 +3d 0 = b 1 +2c 1 +3d 1
S ′′
0 (1) = S ′′
1 (1) ⇒ 2c 0 +6d 0 =2c 1 +6d 1
S ′′
0 (0) = 0 ⇒ 2c 0 =0
S ′′
1 (2) = 0 ⇒ 2c 1 +12d 1 =0