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Lab demo about numerical result

Example 1

radiusH 251012 -^* m * :

radius Hydrogen Atom

ExactMeter 1 :

exact from hypothtical caliper

MeasuredMeter 1015 -^± ≡

11015 -^+ 1.0000000000000015 ≡

11015 -^- 0.999999999999999

a sheet of hydrogen atoms is 12500 thicker than the min/max error deviation

251012 -^* (21015 -^(* / 12500

Example 2

C(x) = Cπ segment coefficient

x C 1 π / 12 x * - acos 12 x * - (4 x * 2 x * (2^- sqrt * - (* :

Area π D 2^* 4 / x C + :

Diameter is known ± 10e-15 x is measured ± 10e-15 π is true to 11 millions decimals. Accuracy of area of the circular segment: what will it be ? Answer: 5 true decimals ... C(x)is ± 0.00001

For practical use, if an integral that belongs to Engineering project, is explicit: it is infinetely accurate. Otherwise , its accuracy is unknown. It depends upon the numerical integrator, a safe assumption is 5 decimals. The converse is vice versa: the Ridder derivative is assumed 5 decimals.

Example 3

Accuracy of

5.6 Zeta 1.02338 ≡

5610 / Zeta 1.023375479227029991089041788103 ≡

Accuracy of

[....]

... Smath

1.023375479227029991089041788103 1.02337547922703

Example 4

1 x / x 12 int 0.693147180579475

1 x / x 12 int Adaptive ≡ Exact to 15 D ≡

2 ln 0.693147180559945

RelativeError 12 ln 1 x / x 12 int / - : 2.817615 1011 -^*

Example 5

The diameter of a turbine shaft is 1 meter, what is the circular area ?

circular π 1 (2^* 4 / : 0.785398163397448

The diameter of a turbine shaft is 1.0 meter, what is the circular area ? It does not have any measured diameter between shoulder and thumb. If you argue it is truly 1.0 meter then make true for computation 1.0 = 1 1.0 is unspecified

circular π 1.0 (2^* 4 / : Error ≡ 1.0 is unspecified ≡

Example 6

How accurate exp(1) ?

1 exp 2.71828182845905

How accurate exp(1.0) ?

1.0 exp unknown ≡

2 sqrt (2 ln^1.27153712971414

How accurate ?

0.123456789 0.123456789^0.772398384

You can't have more decimals than those specified

How accurate ?

2 sqrt (0.123456789^1.043715416

How accurate ?

3 sqrt (0.123^1.07

3 sqrt (1231000 /^

How accurate ?

3 sqrt (1231000 /^eval 1.0698994325152