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→Why is Octave's floating-point computation wrong?
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John W. Eaton, David Bateman, Søren Hauberg, Rik Wehbring ({{Release Year}}). | John W. Eaton, David Bateman, Søren Hauberg, Rik Wehbring ({{Release Year}}). | ||
GNU Octave version {{Release}} manual: a high-level interactive language for numerical computations. | GNU Octave version {{Release}} manual: a high-level interactive language for numerical computations. | ||
URL https:// | URL https://octave.org/doc/v{{Release}}/ | ||
A [https://en.wikipedia.org/wiki/BibTeX BibTeX] entry for [https://en.wikipedia.org/wiki/LaTeX LaTeX] users is: | A [https://en.wikipedia.org/wiki/BibTeX BibTeX] entry for [https://en.wikipedia.org/wiki/LaTeX LaTeX] users is: | ||
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author = {John W. Eaton and David Bateman and S{\o}ren Hauberg and Rik Wehbring}, | author = {John W. Eaton and David Bateman and S{\o}ren Hauberg and Rik Wehbring}, | ||
year = <span>{</span>{{Release Year}}}, | year = <span>{</span>{{Release Year}}}, | ||
url = {https:// | url = {[https://octave.org/doc/v{{Release}}/ https://octave.org/doc/v{{Release}}/]}, | ||
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==Why is Octave's floating-point computation wrong?== | ==Why is Octave's floating-point computation wrong?== | ||
Floating-point arithmetic is an approximation '''in binary''' to arithmetic on real or complex numbers. Just like you cannot represent 1/3 exactly in decimal arithmetic (0.333333... is only a rough approximation to 1/3), you cannot represent some fractions like <math>1/10</math> exactly in base 2. In binary, the representation to one tenth is <math>0.0\overline{0011}_b</math> where the bar indicates that it repeats infinitely (like how <math>1/6 = 0.1\overline{6}_d</math> in decimal). Because this infinite repetition cannot be represented exactly with a finite number of digits, rounding errors occur for values that appear to be exact in decimal but are in fact approximations in binary, such as for example how 0.3 - 0.2 - 0.1 is not equal to zero. | Floating-point arithmetic is an approximation '''in binary''' to arithmetic on real or complex numbers. Just like you cannot represent 1/3 exactly in decimal arithmetic (0.333333... is only a rough approximation to 1/3 for any finite number of 3s), you cannot represent some fractions like <math>1/10</math> exactly in base 2. In binary, the representation to one tenth is <math>0.0\overline{0011}_b</math> where the bar indicates that it repeats infinitely (like how <math>1/6 = 0.1\overline{6}_d</math> in decimal). Because this infinite repetition cannot be represented exactly with a finite number of digits, rounding errors occur for values that appear to be exact in decimal but are in fact approximations in binary, such as for example how 0.3 - 0.2 - 0.1 is not equal to zero. | ||
In addition, some advanced operations are computed by approximation and there is no guarantee for them to be accurate, see [https://en.wikipedia.org/wiki/Rounding#Table-maker.27s_dilemma Table-maker's dilemma] for further references. Their results are system-dependent. | In addition, some advanced operations are computed by approximation and there is no guarantee for them to be accurate, see [https://en.wikipedia.org/wiki/Rounding#Table-maker.27s_dilemma Table-maker's dilemma] for further references. Their results are system-dependent. | ||