• 検索結果がありません。

A chaotic orbit of the logistic map calculated in multiple precision and the orbit that can be

N/A
N/A
Protected

Academic year: 2021

シェア "A chaotic orbit of the logistic map calculated in multiple precision and the orbit that can be"

Copied!
8
0
0

読み込み中.... (全文を見る)

全文

(1)

7

˴ʿˀˍʪˊʶࠃਉʂਪఝૠظޔɿʜʟ ՕஞɼऐɼʓɾɯʟՕஞ

A chaotic orbit of the logistic map calculated in multiple precision and the orbit that can be

considered true

งलஞ໫ƄƄƄƄƄƄƄୟл ࢣƄ

YABUKI, Michiro ƄƄ TSUCHIYA, Takashi

෗ॠ੒ӁࣣൎӁೱ ෗ॠ੒Ӂ෕ีָ࠯

์޶

2छʂਪఝૠ˰ʫ˝˰˱ɻɖʟMPFRʦๆɘƅऐʂՕஞɼʓɾɭɧɼʂɻɡ ʟՕஞʦิɜʟˀˍˊ˞ॅɼƅщॅೱʂ˚ˊˏॅɼʂԟؙʦ෗ʝɟɿɭʟƆɱ ʂٌїʦʖɼɿƅщॅೱʂ˚ˊˏॅɠࢢɾɣƅ੒ɡɾڬܴʦߖɷɳՕஞɻɖɷɺ ʖƅ˴ʿˀˍʪˊʶࠃਉʂഇԻयɿɞɥʟ˗˺ːມηƅɭɾʣɵƅʲʱˀଯʩˏ

˰ʶˆʂມηɿɞɘɺʃƅ಩ɮऐʂՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠ ɿʜʞƅOteoɼRosɠߟɫɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ

݂੒ખɼɾʟ๲ศʦ෗ʝɟɿɭʟƆ

1

ʃɬʕɿ

ʲʱˀ๲໱ɿɞɘɺ݂ʖ࣏ɫɣ٥֋ɩɘʠɺɘʟذɼٿɜʄƅȘߘٴಒভآܴഇ൘ ଟߧʂયɻʖ݂ʖ੺ࡸɻɖʞƅșߘʂಒভآ۽ɫɟߖɳɾɘ˴ʿˀˍʪˊʶࠃਉ

x

n+1

= ax

n

(1 − x

n

) (1)

ɻɖʡɚƆ

x

nʃذʂࣨੁബॅɻɖʞƅ੺Λ׫ԝ௅ʂખʦࠟʟƆ

a

ʃҙೱɟʝक़ڥɻɡ ʟ˘˰˨Ɵˆɻƅ૳࣢

0 < a ≤ 4

ʂ౮ΞɠิɜʝʠʟƆ

n

ʃ،ʞറɫʂѾॅʦॅɜʟ य़ॅബॅɻɖʟƆɧʂذʃ

a

ʂബшɿЫɬɺƅɱʂʓɟɥʂ੺ࡸɩɟʝʃ২ਉɻɡɾ ɘ೾ށɾइ೮ʦ٪ɯʟƆ

ʲʱˀʂ஦૎ʃࡾՃِࣥʂಘࢡɾβɘɿ਻ɭʟЁ೏ɾΜਣड़ɻɖʟƆߧ

(1)

ʃƅʺ

˺˛ˬƟˆɻΆɚʂʖԑ੺ɻɖʟɳʕƅΜਣड़ʦթ໱ɭʟݘɿಒ࣢ɿʜɣޡʣʠʟƆ ɫɟɫƅʺ˺˛ˬƟˆɻظޔɩʠʟ˴ʿˀˍʪˊʶࠃਉʂʲʱˀՕஞʃƅ਻Ыɭʟࡾ

Ճِࣥɿ਻ɭʟॅӁଯɾऐʂՕஞɼʃβʣɪʟʦணɾɘƆࡓɷɺƅʺ˺˛ˬƟˆɿʜ ʞޔࡥɩʠɳՕஞʃƅɘɮʠʃऐʂՕஞɟʝ੒ɡɣ๸ʠɺɫʒɚƆ

ඕ ໱ ഓ ɻ ʃ ƅ

2

छ ʂ ਪ ఝ ૠ ظ ޔ ˰ ʫ ˝ ˰ ˱ ɻ ɖ ʟ

MPFR(Multiple-Precision Floating-Point Reliable Library)[3]

ʦๆɘɺƅʺ˺˛ˬƟˆɿʜʞظޔɩʠʟ˴

˴ʿˀˍʪˊʶࠃਉʂਪఝૠظޔɿʜʟ ՕஞɼऐɼʓɾɯʟՕஞ

A chaotic orbit of the logistic map calculated in multiple precision and the orbit that can be

considered true

งलஞ໫ƄƄƄƄƄƄƄୟл ࢣƄ

YABUKI, Michiro ƄƄ TSUCHIYA, Takashi

෗ॠ੒ӁࣣൎӁೱ ෗ॠ੒Ӂ෕ีָ࠯

์޶

2छʂਪఝૠ˰ʫ˝˰˱ɻɖʟMPFRʦๆɘƅऐʂՕஞɼʓɾɭɧɼʂɻɡ ʟՕஞʦิɜʟˀˍˊ˞ॅɼƅщॅೱʂ˚ˊˏॅɼʂԟؙʦ෗ʝɟɿɭʟƆɱ ʂٌїʦʖɼɿƅщॅೱʂ˚ˊˏॅɠࢢɾɣƅ੒ɡɾڬܴʦߖɷɳՕஞɻɖɷɺ ʖƅ˴ʿˀˍʪˊʶࠃਉʂഇԻयɿɞɥʟ˗˺ːມηƅɭɾʣɵƅʲʱˀଯʩˏ

˰ʶˆʂມηɿɞɘɺʃƅ಩ɮऐʂՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠ ɿʜʞƅOteoɼRosɠߟɫɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ

݂੒ખɼɾʟ๲ศʦ෗ʝɟɿɭʟƆ

1

ʃɬʕɿ

ʲʱˀ๲໱ɿɞɘɺ݂ʖ࣏ɫɣ٥֋ɩɘʠɺɘʟذɼٿɜʄƅȘߘٴಒভآܴഇ൘ ଟߧʂયɻʖ݂ʖ੺ࡸɻɖʞƅșߘʂಒভآ۽ɫɟߖɳɾɘ˴ʿˀˍʪˊʶࠃਉ

x

n+1

= ax

n

(1 − x

n

) (1)

ɻɖʡɚƆ

x

nʃذʂࣨੁബॅɻɖʞƅ੺Λ׫ԝ௅ʂખʦࠟʟƆ

a

ʃҙೱɟʝक़ڥɻɡ ʟ˘˰˨Ɵˆɻƅ૳࣢

0 < a ≤ 4

ʂ౮ΞɠิɜʝʠʟƆ

n

ʃ،ʞറɫʂѾॅʦॅɜʟ य़ॅബॅɻɖʟƆɧʂذʃ

a

ʂബшɿЫɬɺƅɱʂʓɟɥʂ੺ࡸɩɟʝʃ২ਉɻɡɾ ɘ೾ށɾइ೮ʦ٪ɯʟƆ

ʲʱˀʂ஦૎ʃࡾՃِࣥʂಘࢡɾβɘɿ਻ɭʟЁ೏ɾΜਣड़ɻɖʟƆߧ

(1)

ʃƅʺ

˺˛ˬƟˆɻΆɚʂʖԑ੺ɻɖʟɳʕƅΜਣड़ʦթ໱ɭʟݘɿಒ࣢ɿʜɣޡʣʠʟƆ ɫɟɫƅʺ˺˛ˬƟˆɻظޔɩʠʟ˴ʿˀˍʪˊʶࠃਉʂʲʱˀՕஞʃƅ਻Ыɭʟࡾ

Ճِࣥɿ਻ɭʟॅӁଯɾऐʂՕஞɼʃβʣɪʟʦணɾɘƆࡓɷɺƅʺ˺˛ˬƟˆɿʜ ʞޔࡥɩʠɳՕஞʃƅɘɮʠʃऐʂՕஞɟʝ੒ɡɣ๸ʠɺɫʒɚƆ

ඕ ໱ ഓ ɻ ʃ ƅ

2

छ ʂ ਪ ఝ ૠ ظ ޔ ˰ ʫ ˝ ˰ ˱ ɻ ɖ ʟ

MPFR(Multiple-Precision Floating-Point Reliable Library)[3]

ʦๆɘɺƅʺ˺˛ˬƟˆɿʜʞظޔɩʠʟ˴

˴ʿˀˍʪˊʶࠃਉʂਪఝૠظޔɿʜʟ ՕஞɼऐɼʓɾɯʟՕஞ

A chaotic orbit of the logistic map calculated in multiple precision and the orbit that can be

considered true

งलஞ໫ƄƄƄƄƄƄƄୟл ࢣƄ

YABUKI, Michiro ƄƄ TSUCHIYA, Takashi

෗ॠ੒ӁࣣൎӁೱ ෗ॠ੒Ӂ෕ีָ࠯

์޶

2छʂਪఝૠ˰ʫ˝˰˱ɻɖʟMPFRʦๆɘƅऐʂՕஞɼʓɾɭɧɼʂɻɡ ʟՕஞʦิɜʟˀˍˊ˞ॅɼƅщॅೱʂ˚ˊˏॅɼʂԟؙʦ෗ʝɟɿɭʟƆɱ ʂٌїʦʖɼɿƅщॅೱʂ˚ˊˏॅɠࢢɾɣƅ੒ɡɾڬܴʦߖɷɳՕஞɻɖɷɺ ʖƅ˴ʿˀˍʪˊʶࠃਉʂഇԻयɿɞɥʟ˗˺ːມηƅɭɾʣɵƅʲʱˀଯʩˏ

˰ʶˆʂມηɿɞɘɺʃƅ಩ɮऐʂՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠ ɿʜʞƅOteoɼRosɠߟɫɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ

݂੒ખɼɾʟ๲ศʦ෗ʝɟɿɭʟƆ

1

ʃɬʕɿ

ʲʱˀ๲໱ɿɞɘɺ݂ʖ࣏ɫɣ٥֋ɩɘʠɺɘʟذɼٿɜʄƅȘߘٴಒভآܴഇ൘ ଟߧʂયɻʖ݂ʖ੺ࡸɻɖʞƅșߘʂಒভآ۽ɫɟߖɳɾɘ˴ʿˀˍʪˊʶࠃਉ

x

n+1

= ax

n

(1 − x

n

) (1)

ɻɖʡɚƆ

x

nʃذʂࣨੁബॅɻɖʞƅ੺Λ׫ԝ௅ʂખʦࠟʟƆ

a

ʃҙೱɟʝक़ڥɻɡ ʟ˘˰˨Ɵˆɻƅ૳࣢

0 < a ≤ 4

ʂ౮ΞɠิɜʝʠʟƆ

n

ʃ،ʞറɫʂѾॅʦॅɜʟ य़ॅബॅɻɖʟƆɧʂذʃ

a

ʂബшɿЫɬɺƅɱʂʓɟɥʂ੺ࡸɩɟʝʃ২ਉɻɡɾ ɘ೾ށɾइ೮ʦ٪ɯʟƆ

ʲʱˀʂ஦૎ʃࡾՃِࣥʂಘࢡɾβɘɿ਻ɭʟЁ೏ɾΜਣड़ɻɖʟƆߧ

(1)

ʃƅʺ

˺˛ˬƟˆɻΆɚʂʖԑ੺ɻɖʟɳʕƅΜਣड़ʦթ໱ɭʟݘɿಒ࣢ɿʜɣޡʣʠʟƆ ɫɟɫƅʺ˺˛ˬƟˆɻظޔɩʠʟ˴ʿˀˍʪˊʶࠃਉʂʲʱˀՕஞʃƅ਻Ыɭʟࡾ

Ճِࣥɿ਻ɭʟॅӁଯɾऐʂՕஞɼʃβʣɪʟʦணɾɘƆࡓɷɺƅʺ˺˛ˬƟˆɿʜ ʞޔࡥɩʠɳՕஞʃƅɘɮʠʃऐʂՕஞɟʝ੒ɡɣ๸ʠɺɫʒɚƆ

ඕ ໱ ഓ ɻ ʃ ƅ

2

छ ʂ ਪ ఝ ૠ ظ ޔ ˰ ʫ ˝ ˰ ˱ ɻ ɖ ʟ

MPFR(Multiple-Precision Floating-Point Reliable Library)[3]

ʦๆɘɺƅʺ˺˛ˬƟˆɿʜʞظޔɩʠʟ˴

˴ʿˀˍʪˊʶࠃਉʂਪఝૠظޔɿʜʟ ՕஞɼऐɼʓɾɯʟՕஞ

A chaotic orbit of the logistic map calculated in multiple precision and the orbit that can be

considered true

งलஞ໫ƄƄƄƄƄƄƄୟл ࢣƄ

YABUKI, Michiro ƄƄ TSUCHIYA, Takashi

෗ॠ੒ӁࣣൎӁೱ ෗ॠ੒Ӂ෕ีָ࠯

์޶

2छʂਪఝૠ˰ʫ˝˰˱ɻɖʟMPFRʦๆɘƅऐʂՕஞɼʓɾɭɧɼʂɻɡ ʟՕஞʦิɜʟˀˍˊ˞ॅɼƅщॅೱʂ˚ˊˏॅɼʂԟؙʦ෗ʝɟɿɭʟƆɱ ʂٌїʦʖɼɿƅщॅೱʂ˚ˊˏॅɠࢢɾɣƅ੒ɡɾڬܴʦߖɷɳՕஞɻɖɷɺ ʖƅ˴ʿˀˍʪˊʶࠃਉʂഇԻयɿɞɥʟ˗˺ːມηƅɭɾʣɵƅʲʱˀଯʩˏ

˰ʶˆʂມηɿɞɘɺʃƅ಩ɮऐʂՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠ ɿʜʞƅOteoɼRosɠߟɫɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ

݂੒ખɼɾʟ๲ศʦ෗ʝɟɿɭʟƆ

1

ʃɬʕɿ

ʲʱˀ๲໱ɿɞɘɺ݂ʖ࣏ɫɣ٥֋ɩɘʠɺɘʟذɼٿɜʄƅȘߘٴಒভآܴഇ൘ ଟߧʂયɻʖ݂ʖ੺ࡸɻɖʞƅșߘʂಒভآ۽ɫɟߖɳɾɘ˴ʿˀˍʪˊʶࠃਉ

x

n+1

= ax

n

(1 − x

n

) (1)

ɻɖʡɚƆ

x

nʃذʂࣨੁബॅɻɖʞƅ੺Λ׫ԝ௅ʂખʦࠟʟƆ

a

ʃҙೱɟʝक़ڥɻɡ ʟ˘˰˨Ɵˆɻƅ૳࣢

0 < a ≤ 4

ʂ౮ΞɠิɜʝʠʟƆ

n

ʃ،ʞറɫʂѾॅʦॅɜʟ य़ॅബॅɻɖʟƆɧʂذʃ

a

ʂബшɿЫɬɺƅɱʂʓɟɥʂ੺ࡸɩɟʝʃ২ਉɻɡɾ ɘ೾ށɾइ೮ʦ٪ɯʟƆ

ʲʱˀʂ஦૎ʃࡾՃِࣥʂಘࢡɾβɘɿ਻ɭʟЁ೏ɾΜਣड़ɻɖʟƆߧ

(1)

ʃƅʺ

˺˛ˬƟˆɻΆɚʂʖԑ੺ɻɖʟɳʕƅΜਣड़ʦթ໱ɭʟݘɿಒ࣢ɿʜɣޡʣʠʟƆ ɫɟɫƅʺ˺˛ˬƟˆɻظޔɩʠʟ˴ʿˀˍʪˊʶࠃਉʂʲʱˀՕஞʃƅ਻Ыɭʟࡾ

Ճِࣥɿ਻ɭʟॅӁଯɾऐʂՕஞɼʃβʣɪʟʦணɾɘƆࡓɷɺƅʺ˺˛ˬƟˆɿʜ ʞޔࡥɩʠɳՕஞʃƅɘɮʠʃऐʂՕஞɟʝ੒ɡɣ๸ʠɺɫʒɚƆ

ඕ ໱ ഓ ɻ ʃ ƅ

2

छ ʂ ਪ ఝ ૠ ظ ޔ ˰ ʫ ˝ ˰ ˱ ɻ ɖ ʟ

MPFR(Multiple-Precision Floating-Point Reliable Library)[3]

ʦๆɘɺƅʺ˺˛ˬƟˆɿʜʞظޔɩʠʟ˴

1

(2)

8

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

高い

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

次に、

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ

ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ

ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

x

n+1

= ( a ∗ x

n

) ∗ (1 . 0 − x

n

) (L1) x

n+1

= a ∗ ( x

n

∗ (1 . 0 − x

n

)) (L2) x

n+1

= ( a − a ∗ x

n

) ∗ x

n

(L3) x

n+1

= a ∗ ( x

n

− x

n

∗ x

n

) (L4) x

n+1

= a ∗ x

n

− ( a ∗ x

n

) ∗ x

n

(L5) x

n+1

= a ∗ x

n

− a ∗ ( x

n

∗ x

n

) (L6)

ȝɸʂߧʂβɘʦ૛ʍʟɳʕɿƅԷࡵ३୞

r = 5000

ƅࡾՃખ

x

0

= 0 . 1

ƅও਻ڬܴ

d = 0 . 0001

ƅ

a = 4 . 0

ɼɫƅ

m = 53(

щॅೱ

53bit

ƅɭɾʣɵఝ३୞

)

ɼɭʟɼƅɭʍ ɺʂ಺ߧɻ

n

max

= 37

ɠணʝʠɳƆʒɳƅ

m = 1000

ɼɭʟɼƅ಺ߧ

(L1)(L2)(L3)

ɻ

n

max

= 985

ƅ಺ߧ

(L4)(L5)(L6)

ɻ

n

max

= 988

ƅ

m = 2000

ɼɭʟɼƅ಺ߧ

(L1)(L2)(L3)

ɻ

n

max

= 1984

ƅ಺ߧ

(L4)(L5)(L6)

ɻ

n

max

= 1993

ɼɾɷɳƆʜɷ ɺƅɧɧɻʃ

n

maxʂખʂ಺ߧɿʜʟβɘɿɸɘɺʃթ໱ɫɾɘƆ

3.2 n

maxʂࡾՃખΜਣड़

।ҷɩʂࡾՃખΜਣड़ʦ٪ʟɳʕɿƅ಺ߧ

(L1)

ƅ

r = 5000

ƅ

m = 1000

ƅ

d = 0 . 0001

ƅ

a = 4 . 0

ɼɫƅࡾՃખ

0 . 033 ≤ x

0

≤ 0 . 990

ƅ܌ʓ

0 . 033

ɼɭʟɼƅ

983 ≤ n

max

≤ 991

ɠணʝʠɳƆ

n

maxʂખʂβɘʃƅช

1000

ˀˍˊ˞ɿ਻ɫɺ

8

ˀˍˊ˞ଟ୞ʂβɘ ɻɖʟƆ

3.3 n

maxʂ

a

Μਣड़

಺ߧ

(L1)

ƅ

r = 10000

ƅ

m = 1000

ɞʜʇ

m = 1500

ƅ

d = 0 . 0001

ƅ

x

0

= 0 . 1

ƅɼɫƅ

3 . 60 ≤ a ≤ 4 . 00

ɼɫɳࣟ܄ʂٌїʦय

1

ɿߟɭƆЮ߫ʃщॅೱʂ

bit

ॅɻƅ࡙߫ʃ

n

maxɻɖʟƆय

1

ɿɞɥʟ

a = 3 . 74

೒פɼ

a = 3 . 85

೒פɻʂ˛Ɵʶʃ৸ʂೱഇɿ ɖɳʞƅ࠵ՃՕஞɿɞɵɘʟɳʕખɠ।ҷɻɖʞਞɥɺɘʟƆʒɳƅ

fully developed chaos

ɼٿʣʠʟ

a = 4 . 0

ɿɞɘɺƅ

n

maxʃ݂ࢡɼɾɷɺɘʟƆɭɾʣɵ

a = 4 . 0

ɿ ɞɘɺ݂ʖ৮ɣऐʂՕஞɼʂܴɠټʠʟɧɼɿɾʟƆ

ɧɧɻʃ

m = 1000

ʂࣟ܄ɼ

m = 1500

ʂࣟ܄ʦߟɫɳɠƅ਩ʂ

m

ɿɞɘɺʖƅ

n

maxʃ

a = 4 . 0

ʂߗɿ݂ࢡɼɾʟɧɼʃബʣʝɾɘƆय

1

ʃࡾՃખ

x

0

= 0 . 1

ɿɸɘ ɺߟɫɳɠƅ

1000

ʂΪɾʟࡾՃખʂՕஞɿɸɘɺʖ

n

maxʃ

a = 4 . 0

ʂߗɿ݂ࢡɼɾ ʟɧɼʃബʣʝɾɘƆ

3.4 n

maxʂ֛฻ڬܴ

d

Μਣड़

ɧʠʒɻও਻֛฻ڬܴɼɫɺ

d = 0 . 0001

ʦๆɘɺɡɳƆ୻ীʂɧɼɾɠʝƅɧʂ

ِࣥɿʜʞ

n

maxʃΪɾɷɺɣʟƆ

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ

ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

x

n+1

= ( a ∗ x

n

) ∗ (1 . 0 − x

n

) (L1) x

n+1

= a ∗ ( x

n

∗ (1 . 0 − x

n

)) (L2) x

n+1

= ( a − a ∗ x

n

) ∗ x

n

(L3) x

n+1

= a ∗ ( x

n

− x

n

∗ x

n

) (L4) x

n+1

= a ∗ x

n

− ( a ∗ x

n

) ∗ x

n

(L5) x

n+1

= a ∗ x

n

− a ∗ ( x

n

∗ x

n

) (L6)

ȝɸʂߧʂβɘʦ૛ʍʟɳʕɿƅԷࡵ३୞

r = 5000

ƅࡾՃખ

x

0

= 0 . 1

ƅও਻ڬܴ

d = 0 . 0001

ƅ

a = 4 . 0

ɼɫƅ

m = 53(

щॅೱ

53bit

ƅɭɾʣɵఝ३୞

)

ɼɭʟɼƅɭʍ ɺʂ಺ߧɻ

n

max

= 37

ɠணʝʠɳƆʒɳƅ

m = 1000

ɼɭʟɼƅ಺ߧ

(L1)(L2)(L3)

ɻ

n

max

= 985

ƅ಺ߧ

(L4)(L5)(L6)

ɻ

n

max

= 988

ƅ

m = 2000

ɼɭʟɼƅ಺ߧ

(L1)(L2)(L3)

ɻ

n

max

= 1984

ƅ಺ߧ

(L4)(L5)(L6)

ɻ

n

max

= 1993

ɼɾɷɳƆʜɷ ɺƅɧɧɻʃ

n

maxʂખʂ಺ߧɿʜʟβɘɿɸɘɺʃթ໱ɫɾɘƆ

3.2 n

maxʂࡾՃખΜਣड़

।ҷɩʂࡾՃખΜਣड़ʦ٪ʟɳʕɿƅ಺ߧ

(L1)

ƅ

r = 5000

ƅ

m = 1000

ƅ

d = 0 . 0001

ƅ

a = 4 . 0

ɼɫƅࡾՃખ

0 . 033 ≤ x

0

≤ 0 . 990

ƅ܌ʓ

0 . 033

ɼɭʟɼƅ

983 ≤ n

max

≤ 991

ɠணʝʠɳƆ

n

maxʂખʂβɘʃƅช

1000

ˀˍˊ˞ɿ਻ɫɺ

8

ˀˍˊ˞ଟ୞ʂβɘ ɻɖʟƆ

3.3 n

maxʂ

a

Μਣड़

಺ߧ

(L1)

ƅ

r = 10000

ƅ

m = 1000

ɞʜʇ

m = 1500

ƅ

d = 0 . 0001

ƅ

x

0

= 0 . 1

ƅɼɫƅ

3 . 60 ≤ a ≤ 4 . 00

ɼɫɳࣟ܄ʂٌїʦय

1

ɿߟɭƆЮ߫ʃщॅೱʂ

bit

ॅɻƅ࡙߫ʃ

n

maxɻɖʟƆय

1

ɿɞɥʟ

a = 3 . 74

೒פɼ

a = 3 . 85

೒פɻʂ˛Ɵʶʃ৸ʂೱഇɿ ɖɳʞƅ࠵ՃՕஞɿɞɵɘʟɳʕખɠ।ҷɻɖʞਞɥɺɘʟƆʒɳƅ

fully developed chaos

ɼٿʣʠʟ

a = 4 . 0

ɿɞɘɺƅ

n

maxʃ݂ࢡɼɾɷɺɘʟƆɭɾʣɵ

a = 4 . 0

ɿ ɞɘɺ݂ʖ৮ɣऐʂՕஞɼʂܴɠټʠʟɧɼɿɾʟƆ

ɧɧɻʃ

m = 1000

ʂࣟ܄ɼ

m = 1500

ʂࣟ܄ʦߟɫɳɠƅ਩ʂ

m

ɿɞɘɺʖƅ

n

maxʃ

a = 4 . 0

ʂߗɿ݂ࢡɼɾʟɧɼʃബʣʝɾɘƆय

1

ʃࡾՃખ

x

0

= 0 . 1

ɿɸɘ ɺߟɫɳɠƅ

1000

ʂΪɾʟࡾՃખʂՕஞɿɸɘɺʖ

n

maxʃ

a = 4 . 0

ʂߗɿ݂ࢡɼɾ ʟɧɼʃബʣʝɾɘƆ

3.4 n

maxʂ֛฻ڬܴ

d

Μਣड़

ɧʠʒɻও਻֛฻ڬܴɼɫɺ

d = 0 . 0001

ʦๆɘɺɡɳƆ୻ীʂɧɼɾɠʝƅɧʂ

ِࣥɿʜʞ

n

maxʃΪɾɷɺɣʟƆ

3

ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३୞ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ

ʞ਩ҏ३୞ɿʜʟՕஞɟʝ๸ʠʟʒɻʂˀˍˊ˞ॅɻ಺ɭƆɧʂ൘൚ɿʜʞƅ಩์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝ๸ʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅ಩ɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ

Oteo

ɼ

Ros

ɠߟɫ ɳƅఝ३୞ɿʜʟظޔٌїʂڬܴʃ಩ɮ˗˺ː೺ʂ݂੒ખɼɾʟ๲ศɠ෗ʝɟɼɾʟƆ

2

ਪఝૠظޔˋƟ˲ɼɫɺʂ

MPFR

˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ಩์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚ

GPL(General Public License)

ɻజ೘ɩʠɺɘʟƅ

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

MPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]

ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱ଑ɭʟɧɼɿʜʞظޔ३୞ʦޱ଑ɭʟƆ

MPFR

ʃ

GMP

ʦұૌɫɳ˰ʫ˝˰˱ɻƅ

GMP

ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ

(

ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ

)

ƅ३୞ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿ঎଑ɻɡʟɧɼƅ

IEEE754-1985[4]

ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙ୞ʂڽࣘʦयɷɺɘʟɧɼƅ୾ʂβɘɠɖʟƆ

3

ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞ

ਪఝૠظޔɿʜʟظޔʂٌїʦթ໱ɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿ਻ɫɺɽʂଟ୞ʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ಩์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ಩์ɠɖʟƆɧɧɻʃ

MPFR

ʦๆɘɺƅ੒ɡɾщॅೱʂ

bit

ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔ

ٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ

m bit

ʦๆɘɳࣟ܄ʂ

n

ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦ

x

n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

ɿʜʟٌїɠও਻֛

฻ڬܴ

d

Ιࣘ๸ʠɳߗʂˀˍˊ˞ॅ

n

maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿ

n

maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ

3.1 n

maxʂ಺ߧΜਣड़

˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾ಺ټʃߧ

(1)

ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌ

їɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪຋ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟ಺ߧɼɫɺ಺ټя௾ɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ

(

ɧɧɻʃ˞˴ʷ˰˧ʂ಺ټɿ܄ʣɯɺƅঀʂЖޔޫʦ

ɼ಺ՒɭʟƆ

)

2

(3)

x

n+1

= ( a ∗ x

n

) ∗ (1 . 0 − x

n

) (L1) x

n+1

= a ∗ ( x

n

∗ (1 . 0 − x

n

)) (L2) x

n+1

= ( a − a ∗ x

n

) ∗ x

n

(L3) x

n+1

= a ∗ ( x

n

− x

n

∗ x

n

) (L4) x

n+1

= a ∗ x

n

− ( a ∗ x

n

) ∗ x

n

(L5) x

n+1

= a ∗ x

n

− a ∗ ( x

n

∗ x

n

) (L6)

ȝɸʂߧʂβɘʦ૛ʍʟɳʕɿƅԷࡵ३୞

r = 5000

ƅࡾՃખ

x

0

= 0 . 1

ƅও਻ڬܴ

d = 0 . 0001

ƅ

a = 4 . 0

ɼɫƅ

m = 53(

щॅೱ

53bit

ƅɭɾʣɵఝ३୞

)

ɼɭʟɼƅɭʍ ɺʂ಺ߧɻ

n

max

= 37

ɠணʝʠɳƆʒɳƅ

m = 1000

ɼɭʟɼƅ಺ߧ

(L1)(L2)(L3)

ɻ

n

max

= 985

ƅ಺ߧ

(L4)(L5)(L6)

ɻ

n

max

= 988

ƅ

m = 2000

ɼɭʟɼƅ಺ߧ

(L1)(L2)(L3)

ɻ

n

max

= 1984

ƅ಺ߧ

(L4)(L5)(L6)

ɻ

n

max

= 1993

ɼɾɷɳƆʜɷ ɺƅɧɧɻʃ

n

maxʂખʂ಺ߧɿʜʟβɘɿɸɘɺʃթ໱ɫɾɘƆ

3.2 n

maxʂࡾՃખΜਣड़

।ҷɩʂࡾՃખΜਣड़ʦ٪ʟɳʕɿƅ಺ߧ

(L1)

ƅ

r = 5000

ƅ

m = 1000

ƅ

d = 0 . 0001

ƅ

a = 4 . 0

ɼɫƅࡾՃખ

0 . 033 ≤ x

0

≤ 0 . 990

ƅ܌ʓ

0 . 033

ɼɭʟɼƅ

983 ≤ n

max

≤ 991

ɠணʝʠɳƆ

n

maxʂખʂβɘʃƅช

1000

ˀˍˊ˞ɿ਻ɫɺ

8

ˀˍˊ˞ଟ୞ʂβɘ ɻɖʟƆ

3.3 n

maxʂ

a

Μਣड़

಺ߧ

(L1)

ƅ

r = 10000

ƅ

m = 1000

ɞʜʇ

m = 1500

ƅ

d = 0 . 0001

ƅ

x

0

= 0 . 1

ƅɼɫƅ

3 . 60 ≤ a ≤ 4 . 00

ɼɫɳࣟ܄ʂٌїʦय

1

ɿߟɭƆЮ߫ʃщॅೱʂ

bit

ॅɻƅ࡙߫ʃ

n

maxɻɖʟƆय

1

ɿɞɥʟ

a = 3 . 74

೒פɼ

a = 3 . 85

೒פɻʂ˛Ɵʶʃ৸ʂೱഇɿ ɖɳʞƅ࠵ՃՕஞɿɞɵɘʟɳʕખɠ।ҷɻɖʞਞɥɺɘʟƆʒɳƅ

fully developed chaos

ɼٿʣʠʟ

a = 4 . 0

ɿɞɘɺƅ

n

maxʃ݂ࢡɼɾɷɺɘʟƆɭɾʣɵ

a = 4 . 0

ɿ ɞɘɺ݂ʖ৮ɣऐʂՕஞɼʂܴɠټʠʟɧɼɿɾʟƆ

ɧɧɻʃ

m = 1000

ʂࣟ܄ɼ

m = 1500

ʂࣟ܄ʦߟɫɳɠƅ਩ʂ

m

ɿɞɘɺʖƅ

n

maxʃ

a = 4 . 0

ʂߗɿ݂ࢡɼɾʟɧɼʃബʣʝɾɘƆय

1

ʃࡾՃખ

x

0

= 0 . 1

ɿɸɘ ɺߟɫɳɠƅ

1000

ʂΪɾʟࡾՃખʂՕஞɿɸɘɺʖ

n

maxʃ

a = 4 . 0

ʂߗɿ݂ࢡɼɾ ʟɧɼʃബʣʝɾɘƆ

3.4 n

maxʂ֛฻ڬܴ

d

Μਣड़

ɧʠʒɻও਻֛฻ڬܴɼɫɺ

d = 0 . 0001

ʦๆɘɺɡɳƆ୻ীʂɧɼɾɠʝƅɧʂ

ِࣥɿʜʞ

n

maxʃΪɾɷɺɣʟƆ 事は確かめてある。

参照

関連したドキュメント

The first variation of action formula shows that these copies fit together smoothly with the original to form a single orbit, periodic in the reduced configuration space (i.e.

Let T be an additive category and F : T → T an automorphism (a stan- dard construction allows one to replace a category with autoequivalence by a category with automorphism)..

In fact, previous to this, discrete series representations for homogeneous spaces of reductive type have been studied only in the cases of group manifolds, reductive symmetric

Reynolds, “Sharp conditions for boundedness in linear discrete Volterra equations,” Journal of Difference Equations and Applications, vol.. Kolmanovskii, “Asymptotic properties of

A wave bifurcation is a supercritical Hopf bifurcation from a stable steady constant solution to a stable periodic and nonconstant solution.. The bifurcating solution in the case

In this work we give definitions of the notions of superior limit and inferior limit of a real distribution of n variables at a point of its domain and study some properties of

Then it follows immediately from a suitable version of “Hensel’s Lemma” [cf., e.g., the argument of [4], Lemma 2.1] that S may be obtained, as the notation suggests, as the m A

We study infinite words coding an orbit under an exchange of three intervals which have full complexity C (n) = 2n + 1 for all n ∈ N (non-degenerate 3iet words). In terms of