II. FINITE DIFFERENCE METHOD 1 Difference formulae

ǫ = 2.22 £ 10 −16 . As h decreases, the global error is reduced till a certain
value of h, then it increases. Such a behaviour is due to the fact that the
values of e 1±h are computed approximately, and for small h cancellations take
place.
h Dce x ¡ e x
10 −1 0.004532
10 −2 0.000453
10 −3 0.0000000453
10 −4 0.00000000453
10 −8 -0.0000000165
10 −9 0.0000000740
10 −10 0.000000224
10¡14 -0.002172
Remark II.1A Let us denote by ˜ f(x§h) the approximate values of f(x§h)
and let us suppose that
j ˜ f(x § h) ¡ f(x § h)j · ǫ
where ǫ only depends on the precision of the used machine. Then, denoting
˜Dc the numerically computed central difference,
j ˜ Dcf(x) ¡ f ′ (x)j · j ˜ Dcf(x) ¡ Dcf(x)j + jDcf(x) ¡ f ′ (x)j · ǫ h2
+
h 6 f (3) (ξ)
The bound describes the real situation, there is a divergent component of
the error as h ! 0. In other words, for decreasing h rounding errors can
prevail on truncation error, with unreliable results. The situation is described
in Figure II.1. An optimal choice of h corresponds to the minimum of the
function ǫ
h
+ h2
6 max jf (3) j.
2 Finite difference method
Does science make the Creator superfluous? No. The sentence ”God created the
world” is not an outmoded scientific statement. It is a theological statement that is
concerned with the relation of the world to God. Being created is a lasting quality
in things and a fundamental truth about them. (YouCat 41)
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