Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 89

we revert to the local Hermite splines described in the previous subsection. If, in
addition, a purely even function is to be represented, the approximating function
takes the form

c 2 z 2 c 4 z 4 c 6 z 6 s ( z ) = c 0 + + + , (1.576)

where the four free constants, c 0 , c 2 , c 4 , c 6 , are used to match the function and
its derivatives at the end points. If a purely odd function is to be represented, the
approximating function takes the form

c 3 z 3 c 5 z 5 c 7 z 7

= + + +

s ( z ) c 1 z , (1.577)

again leaving the four free constants, c 1 , c 3 , c 5 , c 7 , to match the function and its
derivatives at the end points. In either case, the approximating function is written

i ( z ) L

i +

+ f i ψ

i ( z )

+ f i + 1 ψ

L
i +

s ( z ) = f i η + f i + 1 η

1 ( z ) 1 ( z ).

(1.578)

In the even case, the supporting polynomials are sixth degree and defined in
terms of the canonical Hermite cubics by

R R L L

η

i ( z )

= η

0 (ζ i ),

η

i + 1 ( z )

= η

0 (ζ i + 1 ),

(1.579)

1
ζ i ( z i ) ψ

1
ζ i + 1 ( z i + 1 ) ψ

R R L L

ψ

i ( z )

=

0 (ζ i ),ψ i + 1 ( z )

=

0 (ζ i + 1 ),

(1.580)

with

z 2 z i z 2 z i + 1

− −

ζ i ( z ) =

z i + 1 − z i , i + 1 ( z )

=

z i + 1 − z i ,

(1.581)

2 z
z i + 1 −

ζ i ( z ) = ζ i + 1 ( z ) =

z i .

(1.582)

The term ζ i maps the interval ( z i , z i + 1 ) onto the interval (0,1) of the ζ-axis, while
ζ i + 1 maps the interval ( z i , z i + 1 ) onto the interval ( − 1,0) of the ζ-axis. The corres-

ponding derivatives are

η R = ζ i ( z )η 0 (ζ i ), η L = ζ i + 1 ( z )η 0 (ζ i + 1 ), ( z )

i + 1 ( z )

(1.583)

i

and

z z

ψ R

z i ψ 0 (ζ i ), ψ L z i + 1 ψ 0 (ζ i + 1 ).

= =

( z )

1 ( z )

(1.584)

i i +