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