SPLINES 139
y 2 =pn–1(x 2 ) = α 0 + α 1 (x 2 – x 0 ) + α 2 (x 2 – x 0 ) (x 2 – x 1 )
⇒ α 2
20
21
21
10
10
=^1
(– )
- xx –
yy
xx
yy
xx
⎛
⎝
⎞
⎠
...
yn–1=α 0 + α 1 (xn–1– x 0 ) +... + αn–1(xn–1– x 0 )(xn–1– x 1 )... (xn–1– xn–2) (5.14)
The scheme works using forward substitutions with an advantage that if a new data point (xn,yn) is
introduced, only one unknown αn needs to be determined without altering the previously calculated
coefficients. Note that α 0 depends only on y 0 ,α 1 depends on y 0 and y 1 ,α 2 depends on y 0 ,y 1 and y 2 ,
and so on. This dependence is usually expressed as
αi = y[x 0 ,x 1 ,... , xi] (5.15)
with α 0 = y[x 0 ] = y 0
α 1 = y[x 0 ,x 1 ] =
yy
xx
yx y x
xx
10
10
10
10
=
[] – [ ]
α 2 012
20
21
21
10
10
12 01
20
= [ , , ] =^1
(– )
=
[ , ] – [ , ]
- yx x x
xx
yy
xx
yy
xx
yx x y x x
xx
⎛
⎝
⎞
⎠
Thus, by inspection
αrrr r
r
yx x x x
yx x x y x x x
xx
= [ , , ,... , ] =
[ , ,... , ] – [ , ,... , ]
01 2 –
12 01 –1
0
It is possible to construct similar entities from any consecutive set of data points. Thus, in general
yx x x x
yx x x y x x x
s ss r xx
ss rssr
rs
[ , , ,... , ] =
[ , ,... , ] – [ , ,... , ]
+1 + 2 –
+1 + 2 +1 –1
(5.16)
The expressions y[xs,xs+1,xs+2,... , xr] are known as divided differences and can be computed in the
tabular form (Table 5.1).
Table 5.1 Computation of divided differences
x values y values 1st differences 2nd differences 3rd differences
x 0 y[x 0 ]
y[x 0 ,x 1 ]
x 1 y[x 1 ] y[x 0 ,x 1 , x 2 ]
y[x 1 , x 2 ] y[x 0 ,x 1 ,x 2 , x 3 ]
x 2 y[x 2 ] y[x 1 , x 2 , x 3 ]
y[x 2 ,x 3 ]
x 3 y[x 3 ]
Figure 5.7 gives the geometric interpretation of the divided differences. For the curve that passes
through the specified points (xi,yi),i = 0,... , n– 1, the zeroth divided difference y[xs] = ys represents