240
21 Piecewise Interpolation Polynomials
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
-0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
-0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
-1
-0.5
0
0.5
1
Fig. 21.2 The lowest four Lagrange interpolation polynomials. The nodes are equidistant in [0, 1].
Thus for two nodes, (0, +1), the interpolation polynomials are linear; for three nodes, (0, 0.5, +1)
quadratic and so on. On top from left to right: 1st and 2nd order and bottom from left to right
3rd and 4th order of the Lagrange interpolation polynomial. Obviously each of the interpolation
polynomials fulfills L n,k ( ˜
x i ) = δ k,i
function and in addition its first derivative exactly at the nodal points. Therefore we
have to take into account the first derivative via
˜
x|ψ =
α
Φ α ( ˜
x) ˜
x α |ψ + ¯
Φ α ( ˜
x)
d
d ˜
x
˜
x|ψ ˜
x= ˜
x α
(21.5)
=
α
Φ α ( ˜
x)ψ α + ¯
Φ α ( ˜
x) ¯
ψ α
,
and for the derivative of the wave function
d
d ˜
x ˜
x|ψ
d
d ˜
x
˜
x|ψ =
α
Φ
α ( ˜
x)ψ α + ¯
Φ
α ( ˜
x) ¯
ψ α
,
which leads to
Φ α ( ˜
x β ) = δ αβ ⇒ for n nodes n equations
(21.6a)
d
d ˜
x
¯
Φ α ( ˜
x β ) = δ αβ ⇒ for n nodes n equations
(21.6b)
d
d ˜
x
Φ α ( ˜
x β ) = 0
and
¯
Φ α ( ˜
x β ) = 0.
(21.6c)