Do you know cmath module?

Pycon mini Shizuoka 2020 (online)
https://github.com/HayaoSuzuki/PyCon-mini-Shizuoka-2020

Transcript

1. ܅͸ cmath Λ஌͍ͬͯΔ͔
   Hayao Suzuki
   PyCon mini Shizuoka 2020
   February 29, 2020
   

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2. Contents
   1 ࣗݾ঺հ
   2 cmath ͱ͸Կऀ͔
   3 ෳૉ਺ͱ͸Կ͔
   4 ෳૉ਺ͷۃ࠲ඪදه
   5 ෳૉࢦ਺വ਺
   6 ཭ࢄ Fourier ม׵
   7 1 ͷ n ৐ࠜ
   8 ·ͱΊ
   Hayao (Shizuoka 2020) All about cmath module February 29, 2020 2 / 33
   

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3. ࣗݾ঺հ
   ͓લ୭Α
   ໊લ Hayao Suzukiʢླ໦ɹॣʣ
   Twitter @CardinalXaro
   ϒϩά https://xaro.hatenablog.jp/
   ઐ໳ ਺ֶ (૊߹ͤ࿦ɾάϥϑཧ࿦)
   ֶҐ म࢜ʢ޻ֶʣ
   ɺిؾ௨৴େֶ
   ࢓ࣄ גࣜձࣾΞΠϦοδ
   › εϚʔτϑΥϯΞϓϦͷόοΫΤϯυαʔόʔͷ։ൃ
   Hayao (Shizuoka 2020) All about cmath module February 29, 2020 3 / 33
   

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4. ࣗݾ঺հ
   ٕज़ॻͷࠪಡ
   › ʰEffective Pythonʱ
   ʢΦϥΠϦʔδϟύϯʣ
   › ʰΤϨΨϯτͳ SciPyʱ
   ʢΦϥΠϦʔδϟύϯʣ
   › ʰσʔλαΠΤϯεઃܭϚχϡΞϧʱ
   ʢΦϥΠϦʔδϟύϯʣͳͲ
   › https://xaro.hatenablog.jp/ ʹҰཡ͋Γ·͢ɻ
   ͍ΖΜͳൃද
   › ʮSymPy ʹΑΔ਺ࣜॲཧʯ
   ʢPyCon JP 2018ʣ
   › ʮPython Ͱָ͠Ήॳ౳੔਺࿦ʯ
   ʢPyCon mini Hiroshima 2019ʣ
   ͳͲ
   › https://xaro.hatenablog.jp/ ʹҰཡ͋Γ·͢ɻ
   Hayao (Shizuoka 2020) All about cmath module February 29, 2020 4 / 33
   

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5. ࠓ೔ͷൃද
   όοςϦʔಉࠝ఩ֶʢPEP 206 ΑΓʣ
   Python σΟετϦϏϡʔγϣϯࣗ਎͕ɺผ్μ΢ϯϩʔυ͢Δ͜ͱͳ͘
   ͙͢ʹར༻Ͱ͖Δ๛෋Ͱ൚༻ੑͷߴ͍ඪ४ϥΠϒϥϦΛ࣋ͭ͜ͱɻ
   Python νϡʔτϦΞϧͰ঺հ͞Ε͍ͯΔྫ
   › xmlrpc.client XML-RPC ΫϥΠΞϯτ
   › xmlrpc.server XML-RPC αʔόʔ
   › email ిࢠϝʔϧͱ MIME ॲཧͷͨΊͷύοέʔδ
   › json JSON Τϯίʔμ͓Αͼσίʔμ
   › sqlite3 SQLite σʔλϕʔεʹର͢Δ DB-API 2.0 Πϯλ
   ϑΣʔε
   Hayao (Shizuoka 2020) All about cmath module February 29, 2020 5 / 33
   

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6. ࠓ೔ͷൃද
   cmath ͱ͸Կऀ͔
   1 C ݴޠͰ࣮૷͞Εͨߴ଎ͳ math ϥΠϒϥϦ
   2 Ωϡ΢Ϧ (Cucumber) ͷը૾ࣝผͷͨΊͷ਺ֶϥΠϒϥϦ
   3 ෳૉ਺ (Complex Number) ͷܭࢉϥΠϒϥϦ
   Hayao (Shizuoka 2020) All about cmath module February 29, 2020 6 / 33
   

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7. ࠓ೔ͷൃද
   cmath Ϟδϡʔϧ
   › ෳૉ਺ͷͨΊͷ਺ֶؔ਺
   › 9V ి஑΍ΒχΧυి஑ͷΑ͏ͳଘࡏʹࠓɺεϙοτΛ౰ͯΔɻ
   ࠓճ࢖͏΋ͷ
   › Python 3.7.xʢPython 3.8.x Ͱ΋ָ͠Ί·͢ʂʣ
   › MatplotlibʢάϥϑඳըϥΠϒϥϦʣ
   Hayao (Shizuoka 2020) All about cmath module February 29, 2020 7 / 33
   

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8. ࠓ೔ͷൃද
   ܅͸ cmath Λ஌͍ͬͯΔ͔
   › cmath ͱ͸Կऀ͔
   › ෳૉ਺ͱ͸Կ͔
   › ෳૉ਺ͷۃ࠲ඪදه
   › ෳૉࢦ਺വ਺
   › ཭ࢄ Fourier ม׵
   › 1 ͷ n ৐ࠜ
   › ·ͱΊ
   ࢿྉ͸ઃܭਤڞ༗αΠτʹ͋Δʂ
   ࢿྉ͸͢΂ͯ
   https://github.com/HayaoSuzuki/PyCon-mini-Shizuoka-2020/ ʹ
   ͋Γ·͢ɻ
   Hayao (Shizuoka 2020) All about cmath module February 29, 2020 8 / 33
   

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9. ෳૉ਺ͱ͸
   ෳૉ਺ɺ஌ͬͯ·͔͢ʁ
   Hayao (Shizuoka 2020) All about cmath module February 29, 2020 9 / 33
   

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10. ෳૉ਺ͷఆٛ
    ఆٛ (ෳૉ਺)
    i2 = `1 Ͱ͋ΔΑ͏ͳجఈ 1; i Λ࣮࣋ͭ਺ମ R ্ͷ 2 ࣍ݩϕΫτϧۭ
    ؒͷݩΛෳૉ਺ͱݺͿɻ·ͨɺi Λڏ਺୯ҐͱݺͿɻ
    Python Ͱෳૉ਺Λఆٛ͢Δ
    >>> 3 + 5j # Python Ͱ͸ڏ਺୯ҐΛ j ·ͨ͸ J ͱ͢Δ
    (3+5j)
    >>> 1J**2 # ڏ਺୯Ґͷࣗ৐͸-1 ͱͳΔɻ
    (-1+0j)
    >>> 4 + 5j == (5j + 4) # ࣮෦ͱڏ෦͕ͦΕͧΕ౳͍͠
    True
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 10 / 33
    

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11. ෳૉ਺ͱମ
    ମ (Field) == ࢛ଇԋࢉ͕Ͱ͖Δू߹
    ෳૉ਺͸ෳૉ਺ମ C Λͳ͢ɻ
    Python ʹ͓͚Δෳૉ਺ͷ࢛ଇԋࢉ
    >>> 8 - 5j + -5 + 1j # Ճ๏
    (3-4j)
    >>> (1 + 2j) * (1 - 2J) # ৐๏
    (5+0j)
    >>> (97 + 0j) / (4 + 9j) # আ๏
    (3.9999999999999996-9j)
    97 ͸ૉ਺Ͱ͋Δ͕ɺ97 = (4 + 9i)(4 ` 9i) ͱͳΔɻ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 11 / 33
    

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12. ෳૉ਺ͱॱং
    ෳૉ਺ମ͸ॱংମͰ͸ͳ͍
    ࣮਺ͷΑ͏ͳશॱংؔ܎ΛఆٛͰ͖ͳ͍ʂ
    Python ΋ෳૉ਺ମ͸ॱংମͰ͸ͳ͍͜ͱΛ஌͍ͬͯΔ
    >>> -100 - 100j < 65536 + 256j # ӈล͕େ͖ͦ͏ʹࢥ͑Δ͕...
    Traceback (most recent call last):
    File "", line 1, in
    TypeError: '<' not supported between
    instances of 'complex' and 'complex'
    ॱংମʹ͓͚Δฏํݩ͸ඇෛͰ͋Δɻ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 12 / 33
    

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13. ෳૉ਺ͷڞ໾
    ෳૉ਺ͷڞ໾
    ෳૉ਺ z = x + iy ʹରͯ͠ —
    z = x ` iy Λ z ͷڞ໾ͱݺͿɻ
    Python ʹ͓͚Δෳૉ਺ͷڞ໾
    >>> z = 5 - 3j
    >>> z.conjugate() # complex ܕͷϝιουͱͯ͠
    (5+3j)
    >>> z * (z.conjugate() / abs(z)**2) # ٯݩΛߏ੒͢Δ
    (1+5.551115123125783e-17j)
    ڞ໾ͷઆ໌͸Ͳ͜ʹ͋Δʁ
    ૊ΈࠐΈܕ΍ cmath Ͱ͸ͳ͘ numbers ϞδϡʔϧͰઆ໌͞Ε͍ͯΔɻ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 13 / 33
    

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14. ෳૉ਺ͷۃ࠲ඪදه
    ෳૉ਺ฏ໘
    ෳૉ਺ z = x + iy Λ 2 ࣍ݩ࣮਺ฏ໘ R2 ্ͷ఺ (x; y) ͱΈͳ͢͜ͱ
    ͕Ͱ͖Δɻ͜ΕΛෳૉ਺ฏ໘ͱ͍͏ɻ
    ෳૉ਺ͷۃ࠲ඪܗࣜ
    ෳૉ਺ฏ໘্ͷ఺ z = x + iy(x; y 2 R) Λ࣮෦ x ͱڏ෦ y ͷ૊
    (x; y) Ͱ͸ͳ͘ݪ఺͔Βͷڑ཭ r ͱภ֯ „ ͷ૊ (r; „) Ͱ΋ఆٛͰ͖Δɻ
    ͜ΕΛෳૉ਺ͷۃ࠲ඪܗࣜͱ͍͏ɻ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 14 / 33
    

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15. ෳૉ਺ͷۃ࠲ඪදه
    ඦฉ͸ҰݟʹવΓ
    Figure: ෳૉ਺ฏ໘ʢWikipedia ͔ΒҾ༻ʣ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 15 / 33
    

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16. Python ʹ͓͚Δෳૉ਺ͷۃ࠲ඪදه
    Python ʹ͓͚Δෳૉ਺ͷ࢛ଇԋࢉ
    >>> import cmath # ਅଧొ৔
    >>> z = 1 + 2j # ௚ަ࠲ඪ͔Βۃ࠲ඪʹม׵͢Δ
    >>> r, phi = cmath.polar(z)
    >>> r, phi # r = abs(z), phi = cmath.phase(z)
    (2.23606797749979, 1.1071487177940904)
    >>> w = cmath.rect(r, phi) # ۃ࠲ඪ͔Β௚ަ࠲ඪʹม׵͢Δ
    >>> w
    (1.0000000000000002+2j)
    >>> cmath.isclose(z, w) # == Ͱ͸ͳ͘ isclose Λ࢖͏
    True
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 16 / 33
    

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17. ࢦ਺വ਺
    ࢦ਺വ਺ cmath.exp(x)ʢެࣜυΩϡϝϯτΑΓʣ
    e Λࣗવର਺ͷఈͱͯ͠ɺe ͷ x ৐Λฦ͠·͢ɻ
    ࣗવର਺ͷఈͷෳૉ਺৐ͬͯԿʁʁʁ
    › ࣗવ਺৐ ! Θ͔Δ
    › ੔਺৐ ! Θ͔Δ
    › ༗ཧ਺৐ ! ·ͩΘ͔Δ
    › ࣮਺৐ ! ·ͩ͜ΕͳΒ...
    › ෳૉ਺৐ ! ͜Ε΋͏Θ͔ΜͶ͐ͳ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 17 / 33
    

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18. ࢦ਺വ਺Λ८Δ๯ݥ
    ۃݶͰఆٛ͢Ε͹͍͍Μͩʂ
    ez := lim
    n!1
    1 +
    z
    n
    !n
    :
    ແݶڃ਺Ͱఆٛ͢Ε͹͍͍Μͩʂ
    ez :=
    1
    X
    n=0
    zn
    n!
    :
    ࣮വ਺Ͱఆٛ͢Ε͹͍͍Μͩʂ
    ez := ex(cos y + i sin y)
    where z = x + iy.
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 18 / 33
    

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19. ࢦ਺വ਺Λ८Δ๯ݥ
    ۃݶ஋ʹΑΔఆٛ
    def exp_by_limit(z: complex, n: int = 10) -> complex:
    N = 10 ** n
    return pow(1 + z / N, N)
    ແݶڃ਺ʹΑΔఆٛ
    def exp_by_series(z: complex, n: int = 30) -> complex:
    return sum(pow(z, i) / factorial(i) for i in range(n))
    ࣮വ਺ʹΑΔఆٛ
    def exp_by_real_func(z: complex) -> complex:
    x, y, i = z.real, z.imag, (0 + 1j)
    return math.exp(x) * (math.cos(y) + math.sin(y) * i)
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 19 / 33
    

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20. ݟͤͯ΋Β͓͏͔ɺcmath.exp ͷҖྗͱ΍ΒΛ
    ࣗ࡞ͷؔ਺Ͱ Euler ͷ౳ࣜ eiı = `1 Λܭࢉͯ͠ΈΔ
    >>> z = cmath.pi * 1j
    >>> exp_by_limit(z)
    (-1.0004936019770099+1.0340526558763341e-07j)
    >>> exp_by_series(z)
    (-1.0000000000000002+3.461777852236587e-16j)
    >>> exp_by_real_func(z)
    (-1+1.2246467991473532e-16j)
    cmath.exp Ͱ Euler ͷ౳ࣜ eiı = `1 Λܭࢉͯ͠ΈΔ
    >>> cmath.exp(z)
    (-1+1.2246467991473532e-16j)
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 20 / 33
    

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21. ཭ࢄ Fourier ม׵
    ཭ࢄ Fourier ม׵
    ෳૉവ਺ f(x) ͷ཭ࢄ Fourier ม׵ F (t) ͸
    F (t) =
    N`1
    X
    n=0
    f(n)e`i 2ınt
    N
    Ͱ༩͑ΒΕΔɻ
    ཭ࢄ Fourier ม׵ͷԠ༻෼໺͸ 2 ͭ
    › ৴߸ղੳ
    › σʔλѹॖ
    › ͦͷଞॾʑ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 21 / 33
    

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22. ཭ࢄ Fourier ม׵
    ྫɿ৴߸ղੳ
    Figure: ಾͷ৴߸
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 22 / 33
    

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23. ཭ࢄ Fourier ม׵
    ྫɿ৴߸ͷαϯϓϦϯά
    Figure: ಾͷ৴߸ΛαϯϓϦϯά͢Δ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 23 / 33
    

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24. ཭ࢄ Fourier ม׵
    ཭ࢄ Fourier ม׵ʢߴ଎ Fourier ม׵Ͱ͸ͳ͍ʂʣ
    Fs = [
    sum(
    sampling_fs[n]
    * cmath.exp(-2j * cmath.pi * k * n / sampling_freq)
    for n in range(sampling_freq)
    )
    for k in range(sampling_freq)
    ]
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 24 / 33
    

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25. ཭ࢄ Fourier ม׵
    ৴߸ͷप೾਺εϖΫτϧ
    plt.stem(
    range(-sampling_freq // 2, sampling_freq // 2),
    list(map(abs, Fs)),
    use_line_collection=True,
    )
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 25 / 33
    

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26. ཭ࢄ Fourier ม׵
    ྫɿ৴߸ͷप೾਺εϖΫτϧ
    Figure: ಾͷ৴߸ͷप೾਺εϖΫτϧ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 26 / 33
    

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27. ཭ࢄ Fourier ม׵
    ཭ࢄ Fourier ม׵
    ٯ཭ࢄ Fourier ม׵͸
    f(n) =
    1
    N
    N`1
    X
    n=0
    F (t)ei 2ınt
    N
    Ͱ༩͑ΒΕΔɻ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 27 / 33
    

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28. ཭ࢄ Fourier ม׵
    ٯ཭ࢄ Fourier ม׵
    # ٯ཭ࢄ Fourier ม׵
    inverse_Fs = [
    sum(
    (1 / sampling_freq)
    * Fs[n]
    * cmath.exp(2j * cmath.pi * k * n / sampling_freq)
    for n in range(sampling_freq)
    )
    for k in range(sampling_freq)
    ]
    # ࣮෦͚ͩऔΓग़͢
    real_attr = operator.attrgetter("real")
    inverse_Fs_real = list(map(real_attr, inverse_Fs))
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 28 / 33
    

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29. ཭ࢄ Fourier ม׵
    ྫɿ৴߸Λ෮ݩ͢Δ
    Figure: ٯ཭ࢄ Fourier ม׵Ͱݩͷ৴߸Λ෮ݩ͢Δ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 29 / 33
    

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30. 1 ͷ n ৐ࠜ
    1 ͷ n ৐ࠜ
    1 ͷ n ৐ࠜ͸
    ei 2ık
    n (k = 0; : : : ; n ` 1)
    Ͱ༩͑ΒΕΔɻ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 30 / 33
    

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31. 1 ͷ n ৐ࠜ
    1 ͷ n ৐ࠜͰਖ਼ n ֯ܗΛ࡞ਤ͢Δ
    N = 7
    roots_of_one = [
    cmath.exp(((2 * cmath.pi * k) / N) * 1j)
    for k in range(N + 1)
    ]
    angles = list(map(cmath.phase, roots_of_one))
    length = list(map(abs, roots_of_one))
    plt.polar(angles, length)
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 31 / 33
    

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32. 1 ͷ n ৐ࠜ
    1 ͷ 7 ৐ࠜʹΑΔਖ਼ࣣ֯ܗͷ࡞ਤ
    Figure: ਖ਼ࣣ֯ܗ
    ਖ਼ࣣ֯ܗ͸ఆنͱίϯύεʹΑͬͯ࡞ਤ͕Ͱ͖ͳ͍ɻ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 32 / 33
    

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33. ·ͱΊ
    ·ͱΊ
    › cmath ͸ෳૉ਺ͷͨΊͷ਺ֶϥΠϒϥϦͰ͋Δɻ
    › ඪ४ϥΠϒϥϦͷൣғͰ΋཭ࢄ Fourier ม׵͸Ͱ͖Δɻ
    › ܅͚ͩͷ࠷ڧͷ cmath ΞϓϦέʔγϣϯΛ࣮૷͠Α͏ʂ
    Hayao (Shizuoka 2020) All about cmath module February 29, 2020 33 / 33
    

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