Let us put a cubic spline through the points (0 ; 0) , (1 ; 1 ) (2 ; 0 : 5) and (3 ; 0) .
1.5
1
0.5
±0.5
0.5
1 1.5
2 2.5
3 3.5
±0.5
We shall assume that the spline meets the x ¡ axis at 45 ± will be de…ned by three curves.
at both ends The spline f
1
( x ) = a
1 x 3 f
2
( x ) = a
2 x 3 f
3
( x ) = a
3 x 3
+ b
1 x 2
+ b
2 x 2
+ b
3 x 2
+ c
1 x + d
1
; 0 < x < 1 ;
+ c
2 x + d
2
; 1 < x < 2 ;
+ c
3 x + d
3
; 2 < x < 3 :
At the point (0 ; 0) we have these constraints: f
1
(0) = 0 ; f
1
0 (0) = 1 ;
At (1 ; 1) we get f
1
(1) = 1 ; f
2
(1) = 1 ; f
1
0
00 f
1
(1) = f
2
0
00
(1) ;
(1) = f
2
(1) :

Similarly at (2 ; : 5) and (3 ; 0) we get: f
2
(2) = 0 : 5 ; f
3
(2) = 0 : 5 ; f
2
0
00 f
2
(2) = f
3
0
00
(2) ;
(2) = f
3
(2) ; and f
3
(3) = 0 ; f
3
0 (3) = ¡ 1 ;
This yields the following 12 equations in 12 unknowns.
0 ¢ a
1
+ 0 ¢ b
1
+ 0 ¢ c
1
+ d
1
= 0
3 ¢ 0 ¢ a
1
+ 2 ¢ 0 ¢ b
1
+ c
1
= 1
1 ¢ a
1
+ 1 ¢ b
1
+ 1 ¢ c
1
+ d
1
= 1
1 ¢ a
2
+ 1 ¢ b
2
+ 1 ¢ c
2
+ d
2
= 1
3 ¢ 1 ¢ a
1
+ 2 ¢ 1 ¢ b
1
+ c
1
= 3 ¢ 1 ¢ a
2
+ 2 ¢ 1 ¢ b
2
+ c
2
6 ¢ 1 ¢ a
1
+ 2 ¢ 1 ¢ b
1
= 6 ¢ 1 ¢ a
2
+ 2 ¢ 1 ¢ b
2
8 ¢ a
1
+ 4 ¢ b
1
+ 2 ¢ c
1
+ d
1
= : 5
8 ¢ a
2
+ 4 ¢ b
2
+ 2 ¢ c
2
+ d
2
= : 5
3 ¢ 4 ¢ a
2
+ 2 ¢ 3 ¢ b
2
+ c
2
= 3 ¢ 4 ¢ a
3
+ 2 ¢ 2 ¢ b
3
+ c
3
27 ¢ a
3
+ 9 ¢ b
3
+ 3 ¢ c
3
+ d
3
= 0
3 ¢ 9 ¢ a
3
+ 2 ¢ 3 ¢ b
3
+ c
3
= ¡ 1
2

These equations have the form below where all the unmarked entries are zero.
2 3 2 3 2 3
6
4
6
6
6
6
6
6
6
6
6
6
6
6
6
6
0 0 0 1
0 0 1 0
1 1 1 1
1 1 1 1
3 2 1 0 ¡ 3 ¡ 2 ¡ 1 0
6 2 0 0 ¡ 6 ¡ 2 0 0
8 4 2 1
8 4 2 1
12 4 1 0 ¡ 12 ¡ 4 ¡ 1 0
12 2 0 0 ¡ 12 ¡ 2 0 0
27 9 3 1
27 6 1 0
7
5
7
7
7
7
7
7
7
7
7
7
7
7
7
7
6
4
6
6
6
6
6
6
6
6
6
6
6
6
6
6
7
5
7
7
7
7
7
7
7
7
7
7
7
7
7
7
=
6
4
6
6
6
6
6
6
6
6
6
6
6
6
6
6 b
3 c
3 d
3 c
2 d
2 a
3 d
1 a
2 b
2 a
1 b
1 c
1
7
5
7
7
7
7
7
7
7
7
7
7
7
7
7
7
0
0
¡ 1
: 5
: 5
0
1
0
0
0
1
1
Rearranging the equations might make a better structure for the matrix. For example another arrangement is:
2 3 2 3 2 3
6
4
6
6
6
6
6
6
6
6
6
6
6
6
6
6
0 0 0 1
0 0 1 0
1 1 1 1
1 1 1 1
8 4 2 1
8 4 2 1
27 9 3 1
27 6 1 0
3 2 1 0 ¡ 3 ¡ 2 ¡ 1 0
12 4 1 0 ¡ 12 ¡ 4 ¡ 1 0
6 2 0 0 ¡ 6 ¡ 2 0 0
12 2 0 0 ¡ 12 ¡ 2 0 0
7
5
7
7
7
7
7
7
7
7
7
7
7
7
7
7
6
4
6
6
6
6
6
6
6
6
6
6
6
6
6
6
7
5
7
7
7
7
7
7
7
7
7
7
7
7
7
7
=
6
4
6
6
6
6
6
6
6
6
6
6
6
6
6
6 a
1 b
1 c
1 d
1 a
2 b
2 c
2 d
2 a
3 b
3 c
3 d
3
7
5
7
7
7
7
7
7
7
7
7
7
7
7
7
7
1
: 5
: 5
0
¡ 1
0
0
1
1
0
0
0
The solution is :
3

2 d
2 a
3 b
3 c
3 d
3 a
1 b
1 c
1 d
1 a
2 b
2 c
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
=
6
6
6
6
6
6
6
6
6
6
6
6
6
6
2
¡ :: 73333
: 73333
1
0
: 70000
¡ 3 : 5666
5 : 3000
¡ 1 : 43333
¡ : 56667
4 : 03333
¡ 9 : 90000
8 : 70000
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
The graph of the spline is:
1.5
1
0.5
±0.5
±0.5
0.5
1 1.5
2 2.5
3 3.5
4