Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 87

1.0 0.15

0.10

0.8

0.05

0.6

0

0.4

-0.05

0.2

-0.10

-0.15

0 -1

0
ξ

1 -1 0 1

ξ

Figure 1.6 The canonical cubics.

1.6.2 Local Hermite splines
In contrast to natural splines , for which the approximating cubic depends on all
of the function values, ( f 1 ,..., f N ), at the N nodes, ( x 1 ,..., x N ), along the x -axis,
local or Hermite spline interpolation depends only on the local values of the func-
tion and its derivatives ( f 1 ,..., f N ). On the interval ( x i , x i + 1 ), the function is rep-
resented by the approximating cubic, s ( x ), with

i ( x ) i

f i ψ

i ( x )

f i + 1 ψ

i

s ( x ) = f i η + f i + 1 η

1 ( x )

+ +

1 ( x ).

(1.569)

+ +

The representation depends only on the four quantities f i , f i + 1 , f i , f i + 1 , the function

R

values and derivatives at the end points. The cubic η

i ( x ) has vanishing function

value and derivative at x = x i + 1 , but vanishing derivative and unit function value at

L

x = x i , while the cubicη

i + 1 ( x ) has vanishing function value and derivative at x

= x i ,

R

but vanishing derivative and unit function value at x = x i + 1 . The cubic ψ

i ( x )has

vanishing function value and derivative at x = x i + 1 , but vanishing function value

L

and unit derivative at x = x i , while the cubic ψ

i + 1 ( x ) has vanishing function value

and derivative at x = x i , but vanishing function value and unit derivative at x = x i + 1 .

R
i

L R

i

L
i + 1

Thus η and η

i + 1 support the function at the ends of the interval, while ψ

and ψ

support the derivatives at the ends of the interval.
We may define canonical cubics on the interval ( 1,1) of the ξ-axis with van-

ishing function values and derivatives at the end points. Support for the function is
provided by the canonical cubic η 0 (ξ), which has unit function value and vanishing
derivative at the midpoint, ξ =

−

0, and is shown on the left of Figure 1.6. Support
for the derivative is provided by the canonical cubic ψ 0 (ξ), which has vanishing
function value and unit derivative at the midpoint, ξ =

0, and is shown on the right

of Figure 1.6.