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

(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෕ีָ࠯

์޶

1

_ʃɬʕɿ

x

_n₊₁

= ax

_n

(1 − x

_n

) (1)

ɻɖʡɚƆ

x

a

0 < a ≤ 4

n

a

(1)

ʃƅʺ

2

छ ʂ ਪ ఝ ૠ ظ ޔ ˰ ʫ ˝ ˰ ˱ ɻ ɖ ʟ

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

˴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෕ีָ࠯

์޶

1

_ʃɬʕɿ

x

_n₊₁

= ax

_n

(1 − x

_n

) (1)

ɻɖʡɚƆ

x

a

0 < a ≤ 4

n

a

(1)

ʃƅʺ

2

छ ʂ ਪ ఝ ૠ ظ ޔ ˰ ʫ ˝ ˰ ˱ ɻ ɖ ʟ

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

˴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෕ีָ࠯

์޶

1

_ʃɬʕɿ

x

_n₊₁

= ax

_n

(1 − x

_n

) (1)

ɻɖʡɚƆ

x

a

0 < a ≤ 4

n

a

(1)

ʃƅʺ

2

छ ʂ ਪ ఝ ૠ ظ ޔ ˰ ʫ ˝ ˰ ˱ ɻ ɖ ʟ

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

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

_max_{ʂ಺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

Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

_n₍_m₎ɼՒࡧɭʟɧɼɿɭʟƆԷࡵ३୞ʦ

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

_{ɼ಺ՒɭʟƆ}

)

2

次に、

Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

_{ɼ಺ՒɭʟƆ}

)

2 Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

ɼ಺ՒɭʟƆ

)

2 Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

ɼ಺ՒɭʟƆ

)

2 Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

_{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

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

ɼ಺ՒɭʟƆ

)

2 Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

_{ɼ಺ՒɭʟƆ}

)

2 Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

_{ɼ಺ՒɭʟƆ}

)

2 Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

_{ɼ಺ՒɭʟƆ}

)

2 Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

_{ɼ಺ՒɭʟƆ}

)

2 Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

_{ɼ಺ՒɭʟƆ}

)

2 Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

_{ɼ಺ՒɭʟƆ}

)

2 Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

_{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

_max_{ʂࡾՃખΜਣड़}

।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

_max_{ʂખʂβɘʃƅช}

1000

ˀˍˊ˞ɿ਻ɫɺ

8

ˀˍˊ˞ଟ୞ʂβɘ ɻɖʟƆ

3.3 n

_max_ʂ

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

_max_ɻɖʟƆय

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

_max_ʃ

a = 4 . 0

ʂߗɿ݂ࢡɼɾʟɧɼʃബʣʝɾɘƆय

1

ʃࡾՃખ

x

₀

= 0 . 1

ɿɸɘ ɺߟɫɳɠƅ

1000

ʂΪɾʟࡾՃખʂՕஞɿɸɘɺʖ

n

_max_ʃ

a = 4 . 0

ʂߗɿ݂ࢡɼɾ ʟɧɼʃബʣʝɾɘƆ

3.4 n

_max_ʂ֛฻ڬܴ

d

_Μਣड़

ɧʠʒɻও਻֛฻ڬܴɼɫɺ

d = 0 . 0001

ʦๆɘɺɡɳƆ୻ীʂɧɼɾɠʝƅɧʂ

ِࣥɿʜʞ

n

_max_{ʃΪɾɷɺɣʟƆ}

Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

_{ɼ಺ՒɭʟƆ}

)

2 Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

_{ɼ಺ՒɭʟƆ}

)

2 Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

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)

r = 5000

ƅࡾՃખ

x

₀

= 0 . 1

ƅও਻ڬܴ

d = 0 . 0001

ƅ

a = 4 . 0

ɼɫƅ

m = 53(

щॅೱ

53bit

ƅɭɾʣɵఝ३୞

)

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

n

3.2 n

(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

1000

ˀˍˊ˞ɿ਻ɫɺ

8 3.3 n

_max_ʂ

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

1 bit

ॅɻƅ࡙߫ʃ

n

_max_ɻɖʟƆय

1

ɿɞɥʟ

a = 3 . 74

೒פɼ

a = 3 . 85

fully developed chaos

ɼٿʣʠʟ

a = 4 . 0

ɿɞɘɺƅ

n

a = 4 . 0

ɧɧɻʃ

m = 1000

ʂࣟ܄ɼ

m = 1500

m

_ɿɞɘɺʖƅ

n

_max_ʃ

a = 4 . 0

1

ʃࡾՃખ

x

₀

= 0 . 1

ɿɸɘ ɺߟɫɳɠƅ

1000

n

_max_ʃ

a = 4 . 0

3.4 n

_max_ʂ֛฻ڬܴ

d

_Μਣड़

d = 0 . 0001

ِࣥɿʜʞ

n

_max_{ʃΪɾɷɺɣʟƆ}

3 Oteo

ɼ

Ros

2 MPFR

MPFUN[1]

ƅ

Mathematica

ʘڄɣʃ

Unix

ʂ

bc

GPL(General Public License)

GMP(GNU Multiple Precision Arithmetic Library) [2]

ƅտʇƅ

GMP

ʦˠƟˀɼɫɳ

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

MPFR

ʃ

GMP

(

)

IEEE754-1985[4]

3 MPFR

bit

m bit

ʦๆɘɳࣟ܄ʂ

n

x

r bit

ɼɫɺƅ

m bit

฻ڬܴ

d

n

3.1 n

(1)

(

∗

_{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)

r = 5000

ƅࡾՃખ

x

₀

= 0 . 1

ƅও਻ڬܴ

d = 0 . 0001

ƅ

a = 4 . 0

ɼɫƅ

m = 53(

щॅೱ

53bit

ƅɭɾʣɵఝ३୞

)

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

n

3.2 n

(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

1000

ˀˍˊ˞ɿ਻ɫɺ

8 3.3 n

_max_ʂ

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

1 bit

ॅɻƅ࡙߫ʃ

n

_max_ɻɖʟƆय

1

ɿɞɥʟ

a = 3 . 74

೒פɼ

a = 3 . 85

fully developed chaos

ɼٿʣʠʟ

a = 4 . 0

ɿɞɘɺƅ

n

a = 4 . 0

ɧɧɻʃ

m = 1000

ʂࣟ܄ɼ

m = 1500

m

_ɿɞɘɺʖƅ

n

_max_ʃ

a = 4 . 0

1

ʃࡾՃખ

x

₀

= 0 . 1

ɿɸɘ ɺߟɫɳɠƅ

1000

n

_max_ʃ

a = 4 . 0

3.4 n

_max_ʂ֛฻ڬܴ

d

_Μਣड़

d = 0 . 0001

ِࣥɿʜʞ

n

_max_{ʃΪɾɷɺɣʟƆ} 事は確かめてある。