7
˴ʿˀˍʪˊʶࠃਉʂਪఝૠظޔɿʜʟ ՕஞɼऐɼʓɾɯʟՕஞ
A chaotic orbit of the logistic map calculated in multiple precision and the orbit that can be
considered true
งलஞƄƄƄƄƄƄƄୟл ࢣƄ
YABUKI, Michiro ƄƄ TSUCHIYA, Takashi
ॠӁࣣൎӁೱ ॠӁีָ
์
2छʂਪఝૠ˰ʫ˝˰˱ɻɖʟMPFRʦๆɘƅऐʂՕஞɼʓɾɭɧɼʂɻɡ ʟՕஞʦิɜʟˀˍˊ˞ॅɼƅщॅೱʂ˚ˊˏॅɼʂԟؙʦʝɟɿɭʟƆɱ ʂٌїʦʖɼɿƅщॅೱʂ˚ˊˏॅɠࢢɾɣƅɡɾڬܴʦߖɷɳՕஞɻɖɷɺ ʖƅ˴ʿˀˍʪˊʶࠃਉʂഇԻयɿɞɥʟ˗˺ːມηƅɭɾʣɵƅʲʱˀଯʩˏ
˰ʶˆʂມηɿɞɘɺʃƅɮऐʂՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠ ɿʜʞƅOteoɼRosɠߟɫɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ
݂ખɼɾʟศʦʝɟɿɭʟƆ
1
ʃɬʕɿʲʱˀɿɞɘɺ݂ʖ࣏ɫɣ٥ɩɘʠɺɘʟذɼٿɜʄƅȘߘٴಒভآܴഇ൘ ଟߧʂયɻʖ݂ʖࡸɻɖʞƅșߘʂಒভآ۽ɫɟߖɳɾɘ˴ʿˀˍʪˊʶࠃਉ
x
n+1= ax
n(1 − x
n) (1)
ɻɖʡɚƆ
x
nʃذʂࣨੁബॅɻɖʞƅΛԝʂખʦࠟʟƆa
ʃҙೱɟʝक़ڥɻɡ ʟ˘˰˨Ɵˆɻƅ0 < a ≤ 4
ʂ౮ΞɠิɜʝʠʟƆn
ʃ،ʞറɫʂѾॅʦॅɜʟ य़ॅബॅɻɖʟƆɧʂذʃa
ʂബшɿЫɬɺƅɱʂʓɟɥʂࡸɩɟʝʃ২ਉɻɡɾ ɘށɾइ೮ʦ٪ɯʟƆʲʱˀʂʃࡾՃِࣥʂಘࢡɾβɘɿɭʟЁɾΜਣड़ɻɖʟƆߧ
(1)
ʃƅʺ˺˛ˬƟˆɻΆɚʂʖԑɻɖʟɳʕƅΜਣड़ʦթɭʟݘɿಒɿʜɣޡʣʠʟƆ ɫɟɫƅʺ˺˛ˬƟˆɻظޔɩʠʟ˴ʿˀˍʪˊʶࠃਉʂʲʱˀՕஞʃƅЫɭʟࡾ
ՃِࣥɿɭʟॅӁଯɾऐʂՕஞɼʃβʣɪʟʦணɾɘƆࡓɷɺƅʺ˺˛ˬƟˆɿʜ ʞޔࡥɩʠɳՕஞʃƅɘɮʠʃऐʂՕஞɟʝɡɣʠɺɫʒɚƆ
ඕ ഓ ɻ ʃ ƅ
2
छ ʂ ਪ ఝ ૠ ظ ޔ ˰ ʫ ˝ ˰ ˱ ɻ ɖ ʟMPFR(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
งलஞƄƄƄƄƄƄƄୟл ࢣƄ
YABUKI, Michiro ƄƄ TSUCHIYA, Takashi
ॠӁࣣൎӁೱ ॠӁีָ
์
2छʂਪఝૠ˰ʫ˝˰˱ɻɖʟMPFRʦๆɘƅऐʂՕஞɼʓɾɭɧɼʂɻɡ ʟՕஞʦิɜʟˀˍˊ˞ॅɼƅщॅೱʂ˚ˊˏॅɼʂԟؙʦʝɟɿɭʟƆɱ ʂٌїʦʖɼɿƅщॅೱʂ˚ˊˏॅɠࢢɾɣƅɡɾڬܴʦߖɷɳՕஞɻɖɷɺ ʖƅ˴ʿˀˍʪˊʶࠃਉʂഇԻयɿɞɥʟ˗˺ːມηƅɭɾʣɵƅʲʱˀଯʩˏ
˰ʶˆʂມηɿɞɘɺʃƅɮऐʂՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠ ɿʜʞƅOteoɼRosɠߟɫɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ
݂ખɼɾʟศʦʝɟɿɭʟƆ
1
ʃɬʕɿʲʱˀɿɞɘɺ݂ʖ࣏ɫɣ٥ɩɘʠɺɘʟذɼٿɜʄƅȘߘٴಒভآܴഇ൘ ଟߧʂયɻʖ݂ʖࡸɻɖʞƅșߘʂಒভآ۽ɫɟߖɳɾɘ˴ʿˀˍʪˊʶࠃਉ
x
n+1= ax
n(1 − x
n) (1)
ɻɖʡɚƆ
x
nʃذʂࣨੁബॅɻɖʞƅΛԝʂખʦࠟʟƆa
ʃҙೱɟʝक़ڥɻɡ ʟ˘˰˨Ɵˆɻƅ0 < a ≤ 4
ʂ౮ΞɠิɜʝʠʟƆn
ʃ،ʞറɫʂѾॅʦॅɜʟ य़ॅബॅɻɖʟƆɧʂذʃa
ʂബшɿЫɬɺƅɱʂʓɟɥʂࡸɩɟʝʃ২ਉɻɡɾ ɘށɾइ೮ʦ٪ɯʟƆʲʱˀʂʃࡾՃِࣥʂಘࢡɾβɘɿɭʟЁɾΜਣड़ɻɖʟƆߧ
(1)
ʃƅʺ˺˛ˬƟˆɻΆɚʂʖԑɻɖʟɳʕƅΜਣड़ʦթɭʟݘɿಒɿʜɣޡʣʠʟƆ ɫɟɫƅʺ˺˛ˬƟˆɻظޔɩʠʟ˴ʿˀˍʪˊʶࠃਉʂʲʱˀՕஞʃƅЫɭʟࡾ
ՃِࣥɿɭʟॅӁଯɾऐʂՕஞɼʃβʣɪʟʦணɾɘƆࡓɷɺƅʺ˺˛ˬƟˆɿʜ ʞޔࡥɩʠɳՕஞʃƅɘɮʠʃऐʂՕஞɟʝɡɣʠɺɫʒɚƆ
ඕ ഓ ɻ ʃ ƅ
2
छ ʂ ਪ ఝ ૠ ظ ޔ ˰ ʫ ˝ ˰ ˱ ɻ ɖ ʟMPFR(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
งलஞƄƄƄƄƄƄƄୟл ࢣƄ
YABUKI, Michiro ƄƄ TSUCHIYA, Takashi
ॠӁࣣൎӁೱ ॠӁีָ
์
2छʂਪఝૠ˰ʫ˝˰˱ɻɖʟMPFRʦๆɘƅऐʂՕஞɼʓɾɭɧɼʂɻɡ ʟՕஞʦิɜʟˀˍˊ˞ॅɼƅщॅೱʂ˚ˊˏॅɼʂԟؙʦʝɟɿɭʟƆɱ ʂٌїʦʖɼɿƅщॅೱʂ˚ˊˏॅɠࢢɾɣƅɡɾڬܴʦߖɷɳՕஞɻɖɷɺ ʖƅ˴ʿˀˍʪˊʶࠃਉʂഇԻयɿɞɥʟ˗˺ːມηƅɭɾʣɵƅʲʱˀଯʩˏ
˰ʶˆʂມηɿɞɘɺʃƅɮऐʂՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠ ɿʜʞƅOteoɼRosɠߟɫɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ
݂ખɼɾʟศʦʝɟɿɭʟƆ
1
ʃɬʕɿʲʱˀɿɞɘɺ݂ʖ࣏ɫɣ٥ɩɘʠɺɘʟذɼٿɜʄƅȘߘٴಒভآܴഇ൘ ଟߧʂયɻʖ݂ʖࡸɻɖʞƅșߘʂಒভآ۽ɫɟߖɳɾɘ˴ʿˀˍʪˊʶࠃਉ
x
n+1= ax
n(1 − x
n) (1)
ɻɖʡɚƆ
x
nʃذʂࣨੁബॅɻɖʞƅΛԝʂખʦࠟʟƆa
ʃҙೱɟʝक़ڥɻɡ ʟ˘˰˨Ɵˆɻƅ0 < a ≤ 4
ʂ౮ΞɠิɜʝʠʟƆn
ʃ،ʞറɫʂѾॅʦॅɜʟ य़ॅബॅɻɖʟƆɧʂذʃa
ʂബшɿЫɬɺƅɱʂʓɟɥʂࡸɩɟʝʃ২ਉɻɡɾ ɘށɾइ೮ʦ٪ɯʟƆʲʱˀʂʃࡾՃِࣥʂಘࢡɾβɘɿɭʟЁɾΜਣड़ɻɖʟƆߧ
(1)
ʃƅʺ˺˛ˬƟˆɻΆɚʂʖԑɻɖʟɳʕƅΜਣड़ʦթɭʟݘɿಒɿʜɣޡʣʠʟƆ ɫɟɫƅʺ˺˛ˬƟˆɻظޔɩʠʟ˴ʿˀˍʪˊʶࠃਉʂʲʱˀՕஞʃƅЫɭʟࡾ
ՃِࣥɿɭʟॅӁଯɾऐʂՕஞɼʃβʣɪʟʦணɾɘƆࡓɷɺƅʺ˺˛ˬƟˆɿʜ ʞޔࡥɩʠɳՕஞʃƅɘɮʠʃऐʂՕஞɟʝɡɣʠɺɫʒɚƆ
ඕ ഓ ɻ ʃ ƅ
2
छ ʂ ਪ ఝ ૠ ظ ޔ ˰ ʫ ˝ ˰ ˱ ɻ ɖ ʟMPFR(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
งलஞƄƄƄƄƄƄƄୟл ࢣƄ
YABUKI, Michiro ƄƄ TSUCHIYA, Takashi
ॠӁࣣൎӁೱ ॠӁีָ
์
2छʂਪఝૠ˰ʫ˝˰˱ɻɖʟMPFRʦๆɘƅऐʂՕஞɼʓɾɭɧɼʂɻɡ ʟՕஞʦิɜʟˀˍˊ˞ॅɼƅщॅೱʂ˚ˊˏॅɼʂԟؙʦʝɟɿɭʟƆɱ ʂٌїʦʖɼɿƅщॅೱʂ˚ˊˏॅɠࢢɾɣƅɡɾڬܴʦߖɷɳՕஞɻɖɷɺ ʖƅ˴ʿˀˍʪˊʶࠃਉʂഇԻयɿɞɥʟ˗˺ːມηƅɭɾʣɵƅʲʱˀଯʩˏ
˰ʶˆʂມηɿɞɘɺʃƅɮऐʂՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠ ɿʜʞƅOteoɼRosɠߟɫɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ
݂ખɼɾʟศʦʝɟɿɭʟƆ
1
ʃɬʕɿʲʱˀɿɞɘɺ݂ʖ࣏ɫɣ٥ɩɘʠɺɘʟذɼٿɜʄƅȘߘٴಒভآܴഇ൘ ଟߧʂયɻʖ݂ʖࡸɻɖʞƅșߘʂಒভآ۽ɫɟߖɳɾɘ˴ʿˀˍʪˊʶࠃਉ
x
n+1= ax
n(1 − x
n) (1)
ɻɖʡɚƆ
x
nʃذʂࣨੁബॅɻɖʞƅΛԝʂખʦࠟʟƆa
ʃҙೱɟʝक़ڥɻɡ ʟ˘˰˨Ɵˆɻƅ0 < a ≤ 4
ʂ౮ΞɠิɜʝʠʟƆn
ʃ،ʞറɫʂѾॅʦॅɜʟ य़ॅബॅɻɖʟƆɧʂذʃa
ʂബшɿЫɬɺƅɱʂʓɟɥʂࡸɩɟʝʃ২ਉɻɡɾ ɘށɾइ೮ʦ٪ɯʟƆʲʱˀʂʃࡾՃِࣥʂಘࢡɾβɘɿɭʟЁɾΜਣड़ɻɖʟƆߧ
(1)
ʃƅʺ˺˛ˬƟˆɻΆɚʂʖԑɻɖʟɳʕƅΜਣड़ʦթɭʟݘɿಒɿʜɣޡʣʠʟƆ ɫɟɫƅʺ˺˛ˬƟˆɻظޔɩʠʟ˴ʿˀˍʪˊʶࠃਉʂʲʱˀՕஞʃƅЫɭʟࡾ
ՃِࣥɿɭʟॅӁଯɾऐʂՕஞɼʃβʣɪʟʦணɾɘƆࡓɷɺƅʺ˺˛ˬƟˆɿʜ ʞޔࡥɩʠɳՕஞʃƅɘɮʠʃऐʂՕஞɟʝɡɣʠɺɫʒɚƆ
ඕ ഓ ɻ ʃ ƅ
2
छ ʂ ਪ ఝ ૠ ظ ޔ ˰ ʫ ˝ ˰ ˱ ɻ ɖ ʟMPFR(Multiple-Precision Floating-Point Reliable Library)[3]
ʦๆɘɺƅʺ˺˛ˬƟˆɿʜʞظޔɩʠʟ˴1
8
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
高いʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
次に、
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
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)
ȝɸʂߧʂβɘʦʍʟɳʕɿƅԷࡵ३
r = 5000
ƅࡾՃખx
0= 0 . 1
ƅওڬܴd = 0 . 0001
ƅa = 4 . 0
ɼɫƅm = 53(
щॅೱ53bit
ƅɭɾʣɵఝ३)
ɼɭʟɼƅɭʍ ɺʂߧɻn
max= 37
ɠணʝʠɳƆʒɳƅm = 1000
ɼɭʟɼƅߧ(L1)(L2)(L3)
ɻn
max= 985
ƅߧ(L4)(L5)(L6)
ɻn
max= 988
ƅm = 2000
ɼɭʟɼƅߧ(L1)(L2)(L3)
ɻn
max= 1984
ƅߧ(L4)(L5)(L6)
ɻn
max= 1993
ɼɾɷɳƆʜɷ ɺƅɧɧɻʃn
maxʂખʂߧɿʜʟβɘɿɸɘɺʃթɫɾɘƆ3.2 n
maxʂࡾՃખΜਣड़।ҷɩʂࡾՃખΜਣड़ʦ٪ʟɳʕɿƅߧ
(L1)
ƅr = 5000
ƅm = 1000
ƅd = 0 . 0001
ƅa = 4 . 0
ɼɫƅࡾՃખ0 . 033 ≤ x
0≤ 0 . 990
ƅ܌ʓ0 . 033
ɼɭʟɼƅ983 ≤ n
max≤ 991
ɠணʝʠɳƆn
maxʂખʂβɘʃƅช1000
ˀˍˊ˞ɿɫɺ8
ˀˍˊ˞ଟʂβɘ ɻɖʟƆ3.3 n
maxʂa
Μਣड़ߧ
(L1)
ƅr = 10000
ƅm = 1000
ɞʜʇm = 1500
ƅd = 0 . 0001
ƅx
0= 0 . 1
ƅɼɫƅ3 . 60 ≤ a ≤ 4 . 00
ɼɫɳࣟ܄ʂٌїʦय1
ɿߟɭƆЮ߫ʃщॅೱʂbit
ॅɻƅ࡙߫ʃn
maxɻɖʟƆय1
ɿɞɥʟa = 3 . 74
פɼa = 3 . 85
פɻʂ˛Ɵʶʃ৸ʂೱഇɿ ɖɳʞƅ࠵ՃՕஞɿɞɵɘʟɳʕખɠ।ҷɻɖʞਞɥɺɘʟƆʒɳƅfully developed chaos
ɼٿʣʠʟa = 4 . 0
ɿɞɘɺƅn
maxʃ݂ࢡɼɾɷɺɘʟƆɭɾʣɵa = 4 . 0
ɿ ɞɘɺ݂ʖ৮ɣऐʂՕஞɼʂܴɠټʠʟɧɼɿɾʟƆɧɧɻʃ
m = 1000
ʂࣟ܄ɼm = 1500
ʂࣟ܄ʦߟɫɳɠƅʂm
ɿɞɘɺʖƅn
maxʃa = 4 . 0
ʂߗɿ݂ࢡɼɾʟɧɼʃബʣʝɾɘƆय1
ʃࡾՃખx
0= 0 . 1
ɿɸɘ ɺߟɫɳɠƅ1000
ʂΪɾʟࡾՃખʂՕஞɿɸɘɺʖn
maxʃa = 4 . 0
ʂߗɿ݂ࢡɼɾ ʟɧɼʃബʣʝɾɘƆ3.4 n
maxʂ֛ڬܴd
Μਣड़ɧʠʒɻও֛ڬܴɼɫɺ
d = 0 . 0001
ʦๆɘɺɡɳƆীʂɧɼɾɠʝƅɧʂِࣥɿʜʞ
n
maxʃΪɾɷɺɣʟƆʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜ ʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
2
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
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)
ȝɸʂߧʂβɘʦʍʟɳʕɿƅԷࡵ३
r = 5000
ƅࡾՃખx
0= 0 . 1
ƅওڬܴd = 0 . 0001
ƅa = 4 . 0
ɼɫƅm = 53(
щॅೱ53bit
ƅɭɾʣɵఝ३)
ɼɭʟɼƅɭʍ ɺʂߧɻn
max= 37
ɠணʝʠɳƆʒɳƅm = 1000
ɼɭʟɼƅߧ(L1)(L2)(L3)
ɻn
max= 985
ƅߧ(L4)(L5)(L6)
ɻn
max= 988
ƅm = 2000
ɼɭʟɼƅߧ(L1)(L2)(L3)
ɻn
max= 1984
ƅߧ(L4)(L5)(L6)
ɻn
max= 1993
ɼɾɷɳƆʜɷ ɺƅɧɧɻʃn
maxʂખʂߧɿʜʟβɘɿɸɘɺʃթɫɾɘƆ3.2 n
maxʂࡾՃખΜਣड़।ҷɩʂࡾՃખΜਣड़ʦ٪ʟɳʕɿƅߧ
(L1)
ƅr = 5000
ƅm = 1000
ƅd = 0 . 0001
ƅa = 4 . 0
ɼɫƅࡾՃખ0 . 033 ≤ x
0≤ 0 . 990
ƅ܌ʓ0 . 033
ɼɭʟɼƅ983 ≤ n
max≤ 991
ɠணʝʠɳƆn
maxʂખʂβɘʃƅช1000
ˀˍˊ˞ɿɫɺ8
ˀˍˊ˞ଟʂβɘ ɻɖʟƆ3.3 n
maxʂa
Μਣड़ߧ
(L1)
ƅr = 10000
ƅm = 1000
ɞʜʇm = 1500
ƅd = 0 . 0001
ƅx
0= 0 . 1
ƅɼɫƅ3 . 60 ≤ a ≤ 4 . 00
ɼɫɳࣟ܄ʂٌїʦय1
ɿߟɭƆЮ߫ʃщॅೱʂbit
ॅɻƅ࡙߫ʃn
maxɻɖʟƆय1
ɿɞɥʟa = 3 . 74
פɼa = 3 . 85
פɻʂ˛Ɵʶʃ৸ʂೱഇɿ ɖɳʞƅ࠵ՃՕஞɿɞɵɘʟɳʕખɠ।ҷɻɖʞਞɥɺɘʟƆʒɳƅfully developed chaos
ɼٿʣʠʟa = 4 . 0
ɿɞɘɺƅn
maxʃ݂ࢡɼɾɷɺɘʟƆɭɾʣɵa = 4 . 0
ɿ ɞɘɺ݂ʖ৮ɣऐʂՕஞɼʂܴɠټʠʟɧɼɿɾʟƆɧɧɻʃ
m = 1000
ʂࣟ܄ɼm = 1500
ʂࣟ܄ʦߟɫɳɠƅʂm
ɿɞɘɺʖƅn
maxʃa = 4 . 0
ʂߗɿ݂ࢡɼɾʟɧɼʃബʣʝɾɘƆय1
ʃࡾՃખx
0= 0 . 1
ɿɸɘ ɺߟɫɳɠƅ1000
ʂΪɾʟࡾՃખʂՕஞɿɸɘɺʖn
maxʃa = 4 . 0
ʂߗɿ݂ࢡɼɾ ʟɧɼʃബʣʝɾɘƆ3.4 n
maxʂ֛ڬܴd
Μਣड़ɧʠʒɻও֛ڬܴɼɫɺ
d = 0 . 0001
ʦๆɘɺɡɳƆীʂɧɼɾɠʝƅɧʂِࣥɿʜʞ
n
maxʃΪɾɷɺɣʟƆ3
ʿˀˍʪˊʶࠃਉʂՕஞƅɭɾʣɵɖʟยځ३ɿʜʟظޔٌїʂՕஞʂ।ҷɩʦʜʞҏ३ɿʜʟՕஞɟʝʠʟʒɻʂˀˍˊ˞ॅɻɭƆɧʂ൘൚ɿʜʞƅ์ɾ ˀˍˊ˞ॅʒɻऐɼʓɾɯʟՕஞʦಾɣщॅೱ˚ˊˏॅʦணʟƆऐɼʓɾɯʟՕஞɟ ʝʠɳՕஞɻɖɷɺʖƅ˴ʿˀˍʪˊʶࠃਉʂ˗˺ːʲʱˀມηɿɞɘɺʃƅɮ ऐɼʓɾɯʟՕஞɼஔɬ˗˺ːɿຄʒʟߏʦߟɭƆɧʠɿʜʞ
Oteo
ɼRos
ɠߟɫ ɳƅఝ३ɿʜʟظޔٌїʂڬܴʃɮ˗˺ːʂ݂ખɼɾʟศɠʝɟɼɾʟƆ2
ਪఝૠظޔˋƟ˲ɼɫɺʂMPFR
˴ʿˀˍʪˊʶࠃਉʂऐɿפɘՕஞʦֆʕʟɿʃƅਪఝૠظޔʦ۱ɚ์ɠɖʟƆ ਪఝૠظޔʦ۱ɚʖʂɼɫɺʃƅ
MPFUN[1]
ƅMathematica
ʘڄɣʃUnix
ʂbc
ɾɽɠɖʟƆפ௰ɻʃƅਪఝૠظޔ˰ʫ˝˰˱ɼɫɺƅټݚGPL(General Public License)
ɻజɩʠɺɘʟƅGMP(GNU Multiple Precision Arithmetic Library) [2]
ƅտʇƅGMP
ʦˠƟˀɼɫɳMPFR(Multiple Precision Binary Floating-Point Reliable Library)[3]
ɠɖʟƆɧʠʝșछʂਪఝૠظޔ˰ʫ˝˰˱ʃƅщॅೱʂ˚ˊ ˏॅʦޱɭʟɧɼɿʜʞظޔ३ʦޱɭʟƆMPFR
ʃGMP
ʦұૌɫɳ˰ʫ˝˰˱ɻƅGMP
ɼʃƅˤƟˆ˚˱ˍʪʦڽࣘɩ ɯɳɧɼ(
ٌїɠɱʂ˥ʾ˺ʂ˶ƟːʼʫˁɿΜਣɫɾɘ)
ƅ३ʦബॅɨɼɿ।ҷɿ ɟɸߢศɿɻɡʟɧɼƅIEEE754-1985[4]
ɿࡵ֗ɫɳțɸʂԤʕ˩Ɵːʦߖɸɧ ɼƅظޔਙʂڽࣘʦयɷɺɘʟɧɼƅʂβɘɠɖʟƆ3
ऐɼʓɾɯʟ˴ʿˀˍʪˊʶࠃਉʂՕஞਪఝૠظޔɿʜʟظޔʂٌїʦթɭʟɿʃƅɱʂٌїɠ˴ʿˀˍʪˊʶࠃਉʂऐ ʂՕஞʂખɿɫɺɽʂଟʂ।ҷɩʦߖɸʂɟʂޱವʦߖɸ์ƅɭɾʣɵƅظ ޔɠɘɟɾʟˀˍˊ˞ॅʒɻ।ɫɘՕஞʦߟɭʂɟาʕગʟ์ɠɖʟƆɧɧɻʃ
MPFR
ʦๆɘɺƅɡɾщॅೱʂbit
ૠʦߖɸബॅɿʜʟ˴ʿˀˍʪˊʶࠃਉʂظޔٌїʦԷࡵɼɫɺƅʜʞࢡɩɾщॅೱʦߖɸബॅɿʜʟظޔٌїɼಅҼɭʟƆ ɧɧɻщॅೱɿ
m bit
ʦๆɘɳࣟ܄ʂn
ˀˍˊ˞ʂ˴ʿˀˍʪˊʶࠃਉʂٌїʦx
n(m)ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३ʦr bit
ɼɫɺƅm bit
ɿʜʟٌїɠও֛ڬܴ
d
Ιࣘʠɳߗʂˀˍˊ˞ॅn
maxʦ।ҷɩʂޱವɼɭʟɧɼɿɭʟƆ Ιчɿn
maxɠүِࣥɿʜʞɽʜʂʜɚɿബшɭʟɟʦߟɭƆ3.1 n
maxʂߧΜਣड़˴ʿˀˍʪˊʶࠃਉʂॅӁଯɾټʃߧ
(1)
ɻɖʟƆλ౫ɿظޔՆʜʟظޔɻʃƅ ș۽ʂЖޔʂٌїʃɱʂࡼࢍɿϳׄɩʠɾɘɠƅȚ۽ΙࣘʂЖޔɻʃࡼࢍɿʜʞٌїɠΪɾʟƆɧʂɳʕƅɭʍɺʂظޔࡼࢍʦ۪ɭʟɼƅ˞˴ʷ˰˧ࣘɻʂ˴ʿˀ ˍʪˊʶࠃਉʃƅߘʂȝɸΪɾʟߧɼɫɺټяɼɾʞƅɱʠɲʠظޔٌїɠΪ ɾɷɺɣʟƆ
(
ɧɧɻʃ˞˴ʷ˰˧ʂټɿ܄ʣɯɺƅঀʂЖޔޫʦ∗
ɼՒɭʟƆ)
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)
ȝɸʂߧʂβɘʦʍʟɳʕɿƅԷࡵ३
r = 5000
ƅࡾՃખx
0= 0 . 1
ƅওڬܴd = 0 . 0001
ƅa = 4 . 0
ɼɫƅm = 53(
щॅೱ53bit
ƅɭɾʣɵఝ३)
ɼɭʟɼƅɭʍ ɺʂߧɻn
max= 37
ɠணʝʠɳƆʒɳƅm = 1000
ɼɭʟɼƅߧ(L1)(L2)(L3)
ɻn
max= 985
ƅߧ(L4)(L5)(L6)
ɻn
max= 988
ƅm = 2000
ɼɭʟɼƅߧ(L1)(L2)(L3)
ɻn
max= 1984
ƅߧ(L4)(L5)(L6)
ɻn
max= 1993
ɼɾɷɳƆʜɷ ɺƅɧɧɻʃn
maxʂખʂߧɿʜʟβɘɿɸɘɺʃթɫɾɘƆ3.2 n
maxʂࡾՃખΜਣड़।ҷɩʂࡾՃખΜਣड़ʦ٪ʟɳʕɿƅߧ
(L1)
ƅr = 5000
ƅm = 1000
ƅd = 0 . 0001
ƅa = 4 . 0
ɼɫƅࡾՃખ0 . 033 ≤ x
0≤ 0 . 990
ƅ܌ʓ0 . 033
ɼɭʟɼƅ983 ≤ n
max≤ 991
ɠணʝʠɳƆn
maxʂખʂβɘʃƅช1000
ˀˍˊ˞ɿɫɺ8
ˀˍˊ˞ଟʂβɘ ɻɖʟƆ3.3 n
maxʂa
Μਣड़ߧ
(L1)
ƅr = 10000
ƅm = 1000
ɞʜʇm = 1500
ƅd = 0 . 0001
ƅx
0= 0 . 1
ƅɼɫƅ3 . 60 ≤ a ≤ 4 . 00
ɼɫɳࣟ܄ʂٌїʦय1
ɿߟɭƆЮ߫ʃщॅೱʂbit
ॅɻƅ࡙߫ʃn
maxɻɖʟƆय1
ɿɞɥʟa = 3 . 74
פɼa = 3 . 85
פɻʂ˛Ɵʶʃ৸ʂೱഇɿ ɖɳʞƅ࠵ՃՕஞɿɞɵɘʟɳʕખɠ।ҷɻɖʞਞɥɺɘʟƆʒɳƅfully developed chaos
ɼٿʣʠʟa = 4 . 0
ɿɞɘɺƅn
maxʃ݂ࢡɼɾɷɺɘʟƆɭɾʣɵa = 4 . 0
ɿ ɞɘɺ݂ʖ৮ɣऐʂՕஞɼʂܴɠټʠʟɧɼɿɾʟƆɧɧɻʃ
m = 1000
ʂࣟ܄ɼm = 1500
ʂࣟ܄ʦߟɫɳɠƅʂm
ɿɞɘɺʖƅn
maxʃa = 4 . 0
ʂߗɿ݂ࢡɼɾʟɧɼʃബʣʝɾɘƆय1
ʃࡾՃખx
0= 0 . 1
ɿɸɘ ɺߟɫɳɠƅ1000
ʂΪɾʟࡾՃખʂՕஞɿɸɘɺʖn
maxʃa = 4 . 0
ʂߗɿ݂ࢡɼɾ ʟɧɼʃബʣʝɾɘƆ3.4 n
maxʂ֛ڬܴd
Μਣड़ɧʠʒɻও֛ڬܴɼɫɺ
d = 0 . 0001
ʦๆɘɺɡɳƆীʂɧɼɾɠʝƅɧʂِࣥɿʜʞ