Loss of significance

Under extinction (English cancellation ) is understood in the numerical loss of accuracy in the subtraction of nearly equal size floating point numbers.

Examples

Numerical example

We subtract the numbers and from each other and as a result

Come now and already from previous calculations, the low-order positions will be affected by rounding errors. Voices now but the most significant digits of agreement and so the significant digits to delete from, and the difference arises solely from rounding errors.

Suppose in and the first three digits are correct, and all lower order digits corrupted by rounding errors. Rounds we the numbers to their correct digits, the result is

While there is no single correct digit more from the first, supposedly exact calculation.

Suppose in and the first four digits are still correct, it follows

Whereas we have above with an absolute error of and thus is is a relative error of 11.1%.

Example: Algorithm of Archimedes to the circular speed calculation

Proved Archimedes of Syracuse, that behaves the circumference of a circle to its diameter as well as the area of the circle to the square of the radius. He called this (now known as circle number ) ratio is not π, but gave instructions on how one can approach with the help of one and circumscribed polygons, the ratio up to an arbitrarily high accuracy, probably one of the oldest numerical methods of history. And he led the calculation up to 96 -gon with the following result by:

As can be seen from the numerical example, Archimedes had no chance of extinction at all only undertaken in the 96 -gon.

In today's language to start with directly computable side lengths of a unit circle () inscribed polygons, such as the Zweieck, the triangle, the square or hexagon.

Then whose side length is easily derivable with the auxiliary line and two-time application of the theorem of Pythagoras ( and ) for polygons with double number of corners:

With the four basic arithmetic operations and the extraction of roots so you can calculate starting with the Zweieck the side length and perimeter of a polygon inscribed and thus indirectly an approximation for. In practice, however, the result is disappointing. The following table shows, starting with n = 2, the distance between the side edge of the circle to the center S, the side lengths of the circumscribed and inscribed n- sided polygon and the faces and, should the converge towards the unit circle. The calculation was carried out in C with double IEEE 754 and thus about 15 decimal places. The numerical values are also with any calculator, which dominates the square roots, traceable:

2 1,000 E 00 2.00e 00 Inf Inf 0.00000000000000 4 2.929e -01 1.41e 00 2.00e 00 2.00000000000000 4.00000000000000 8 7.612e -02 7.65e -01 8.28e -01 2.82842712474619 3.31370849898476 16 1.921e -02 3.90e -01 3.98e -01 3.06146745892072 3.18259787807453 32 4.815e -03 1.96e -01 1.97e -01 3.12144515225805 3.15172490742926 64 1.205e -03 9.81e -02 9.83e -02 3.13654849054593 3.14411838524589 128 3.012e -04 4.91e -02 4.91e -02 3.14033115695474 3.14222362994244 256 7.530e -05 2.45e -02 2.45e -02 3.14127725093262 3.14175036916881 512 1.882e -05 1.23E -02 1.23E -02 3.14151380114509 3.14163208070397 1024 4.706e -06 6.14e -03 6.14e -03 3.14157294036989 3.14160251025961 2048 1.177e -06 3.07e -03 3.07e -03 3.14158772527060 3.14159511774302 4096 2.941e -07 1.53e -03 1.53e -03 3.14159142155216 3.14159326967027 8192 7.353e -08 7.67e -04 7.67e -04 3.14159234553025 3.14159280755978 1.638e 04 1.838e -08 3.83e -04 3.83e -04 3.14159257570956 3.14159269121694 3.277e 04 4.596e -09 1.92e -04 1.92e -04 3.14159264036917 3.14159266924601 6.554e 04 1.149e -09 9.59e -05 9.59e -05 3.14159264171161 3.14159264893082 1.311e 05 2.872e -10 4.79e -05 4.79e -05 3.14159260647332 3.14159260827812 2.621e 05 7.181e -11 2.40e -05 2.40e -05 3.14159291071407 3.14159291116527 5.243e 05 1.795e -11 1.20e -05 1.20e -05 3.14159169662728 3.14159169674009 1.049e 06 4.488e -12 5.99e -06 5.99e -06 3.14159655369072 3.14159655371892 2.097e 06 1.122e -12 3.00e -06 3.00e -06 3.14159655370129 3.14159655370834 4.194e 06 2.804e -13 1.50e -06 1.50e -06 3.14151884046467 3.14151884046643 8.389e 06 7.017e -14 7.49e -07 7.49e -07 3.14120796828205 3.14120796828249 1.678e 07 1.754e -14 3.75e -07 3.75e -07 3.14245127249408 3.14245127249419 3.355e 07 4.441e -15 1.87e -07 1.87e -07 3.14245127249412 3.14245127249415 6.711e 07 1.110e -15 9.42e -08 9.42e -08 3.16227766016838 3.16227766016838 1.342e 08 2.220e -16 4.71e -08 4.71e -08 3.16227766016838 3.16227766016838 2.684e 08 0.000e 00 2.11e -08 2.11e -08 2.82842712474619 2.82842712474619 5.369e 08 0.000e 00 0.00E 00 0.00E 00 0.00000000000000 0.00000000000000 It is clearly seen at the beginning of the convergence to pi. After reaching about half the number of digits in the 32768 -gon, however, makes the extinction in the subtraction of nearly equal numbers 2 and noticeable. The result is again inaccurate and wrong at the end ( 2-2000 ... 000xxx = 0).

In many cases, as in this case, one can avoid the extinction, simply by avoiding the affected subtractions. Here succeed with a deformation of the formula in an equivalent form, without subtraction using

With

It follows that:

Of course it is a happy coincidence that in the counter subtraction " lifting off ". Now the bill is as desired:

2.000e 00 1,000 E 00 2.00e 00 Inf Inf 0.00000000000000 4.000e 00 2.929e -01 1.41e 00 2.00e 00 2.00000000000000 4.00000000000000 8.000e 00 7.612e -02 7.65e -01 8.28e -01 2.82842712474619 3.31370849898476 1.600e 01 1.921e -02 3.90e -01 3.98e -01 3.06146745892072 3.18259787807453 3.200e 01 4.815e -03 1.96e -01 1.97e -01 3.12144515225805 3.15172490742926 6.400e 01 1.205e -03 9.81e -02 9.83e -02 3.13654849054594 3.14411838524590 1.280e 02 3.012e -04 4.91e -02 4.91e -02 3.14033115695475 3.14222362994246 2.560e 02 7.530e -05 2.45e -02 2.45e -02 3.14127725093277 3.14175036916897 5.120e 02 1.882e -05 1.23E -02 1.23E -02 3.14151380114430 3.14163208070318 1.024e 03 4.706e -06 6.14e -03 6.14e -03 3.14157294036709 3.14160251025681 2.048e 03 1.177e -06 3.07e -03 3.07e -03 3.14158772527716 3.14159511774959 4.096e 03 2.941e -07 1.53e -03 1.53e -03 3.14159142151120 3.14159326962931 8.192e 03 7.353e -08 7.67e -04 7.67e -04 3.14159234557012 3.14159280759964 1.638e 04 1.838e -08 3.83e -04 3.83e -04 3.14159257658487 3.14159269209225 3.277e 04 4.596e -09 1.92e -04 1.92e -04 3.14159263433856 3.14159266321541 6.554e 04 1.149e -09 9.59e -05 9.59e -05 3.14159264877699 3.14159265599620 1.311e 05 2.872e -10 4.79e -05 4.79e -05 3.14159265238659 3.14159265419140 2.621e 05 7.181e -11 2.40e -05 2.40e -05 3.14159265328899 3.14159265374019 5.243e 05 1.795e -11 1.20e -05 1.20e -05 3.14159265351459 3.14159265362739 1.049e 06 4.488e -12 5.99e -06 5.99e -06 3.14159265357099 3.14159265359919 2.097e 06 1.122e -12 3.00e -06 3.00e -06 3.14159265358509 3.14159265359214 4.194e 06 2.804e -13 1.50e -06 1.50e -06 3.14159265358862 3.14159265359038 8.389e 06 7.017e -14 7.49e -07 7.49e -07 3.14159265358950 3.14159265358994 1.678e 07 1.754e -14 3.75e -07 3.75e -07 3.14159265358972 3.14159265358983 3.355e 07 4.441e -15 1.87e -07 1.87e -07 3.14159265358978 3.14159265358980 6.711e 07 1.110e -15 9.36e -08 9.36e -08 3.14159265358979 3.14159265358980 1.342e 08 2.220e -16 4.68e -08 4.68e -08 3.14159265358979 3.14159265358979 2.684e 08 0.000e 00 2.34e -08 2.34e -08 3.14159265358979 3.14159265358979 Even with the 268435456 -Eck to reach the full accuracy of almost 16 decimal places. The abort signal, the 0 in the second column.

Rule of thumb

Subtracting two - digit, almost equal numbers, the match in the first places, so lost as a result of the actually possible locations. So there are only non-zero points. The information that the first digits have been canceled to zero, will be lost. The accuracy of the result is reduced by these bodies.

The numbers in the last digits differ by only rounding errors, the result has no significance. It should not be included as such in further calculations.