Computational Physics

576 Appendix A

Such a table can be used to construct an interpolation polynomial forxt. First we
note that

x± 1 =x 0 ±δx± 1 / 2

x± 2 =x 0 ± 2 δx± 1 / 2 +δ^2 x± 1

x± 3 =x 0 ± 3 δx± 1 / 2 + 3 δ^2 x± 1 ±δ^3 x± 3 / 2 (A.57)

and so on. In these expressions the binomial coefficients are recognised. From these
equations, it can be directly seen that

xt=x 0 +tδx 1 / 2 +

t(t− 1 )

2

δ^2 x 1 +

t(t− 1 )(t− 2 )

3!

δ^3 x 3 / 2 +··· (A.58)

is a polynomial which coincides withxtfort=1, 2, 3. One can also build higher
order polynomials in the same fashion. For negativet-values, this formula reads

xt=x 0 +tδx− 1 / 2 +

t(t+ 1 )

2

δ^2 x− 1 +

t(t+ 1 )(t+ 2 )

3!

δ^3 x− 3 / 2 +··· (A.59)

These formulas are calledNewton interpolation formulas.
We now use this table to integrate differential equations of type(A.55). Therefore
we consider a difference table, not forx, but forx ̇:

− 2 x ̇− 2 δ^2 x ̇− 2

δ ̇x− 3 / 2 δ^3 x ̇− 3 / 2

− 1 x ̇− 1 δ^2 x ̇− 1

δ ̇x− 1 / 2

0 x ̇ 0

1
For simplicity we did not make this table too ‘deep’ (up to third differences).
Suppose the equation has been integrated up tot=1, sox 1 is known, and we would
like to calculatex 2. This is possible by adding an extra row to the lower end of the
table. This can be done because from(A.55), ̇x 1 =f(x 1 ,1)and the table can be
extended as follows:

− 2 x ̇− 2 δ^2 x ̇− 2

δ ̇x− 3 / 2 δ^3 x ̇− 3 / 2

− 1 x ̇− 1 δ^2 x ̇− 1

δ ̇x− 1 / 2 δ^3 x ̇− 1 / 2

0 x ̇ 0 δ^2 x ̇ 0

δ ̇x 1 / 2

1 x ̇ 1