{"id":1085866,"date":"2025-01-08T13:20:15","date_gmt":"2025-01-08T05:20:15","guid":{"rendered":"https:\/\/docs.pingcode.com\/ask\/ask-ask\/1085866.html"},"modified":"2025-01-08T13:20:18","modified_gmt":"2025-01-08T05:20:18","slug":"%e5%a6%82%e4%bd%95%e7%94%a8python%e7%b2%be%e7%a1%ae%e6%b1%82%e5%9c%86%e5%91%a8%e7%8e%87-2","status":"publish","type":"post","link":"https:\/\/docs.pingcode.com\/ask\/1085866.html","title":{"rendered":"\u5982\u4f55\u7528python\u7cbe\u786e\u6c42\u5706\u5468\u7387"},"content":{"rendered":"

\"\u5982\u4f55\u7528python\u7cbe\u786e\u6c42\u5706\u5468\u7387\"<\/p>\n

\u5982\u4f55\u7528 Python \u7cbe\u786e\u6c42\u5706\u5468\u7387<\/strong><\/p>\n<\/p>\n

\u7528 Python \u7cbe\u786e\u6c42\u5706\u5468\u7387\u53ef\u4ee5\u901a\u8fc7\u591a\u79cd\u65b9\u6cd5\u5b9e\u73b0\uff0c\u5982\u8499\u7279\u5361\u7f57\u65b9\u6cd5\u3001\u83b1\u5e03\u5c3c\u8328\u516c\u5f0f\u3001Chudnovsky \u7b97\u6cd5\u7b49<\/strong>\u3002\u5176\u4e2d\uff0cChudnovsky \u7b97\u6cd5<\/strong>\u56e0\u5176\u9ad8\u6548\u548c\u7cbe\u786e\u5ea6\u9ad8\uff0c\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8e\u8ba1\u7b97\u5706\u5468\u7387\u3002\u672c\u6587\u5c06\u8be6\u7ec6\u4ecb\u7ecd\u5982\u4f55\u4f7f\u7528 Chudnovsky \u7b97\u6cd5\u6765\u8ba1\u7b97\u5706\u5468\u7387\u3002<\/p>\n<\/p>\n

Chudnovsky \u7b97\u6cd5\u662f\u4e00\u79cd\u7528\u4e8e\u5feb\u901f\u8ba1\u7b97\u5706\u5468\u7387\u7684\u7b97\u6cd5\uff0c\u7531 Chudnovsky \u5144\u5f1f\u4e8e 1988 \u5e74\u63d0\u51fa\u3002\u8be5\u7b97\u6cd5\u57fa\u4e8e Ramanujan \u7684 \u03c0 \u7ea7\u6570\u5c55\u5f00\uff0c\u5e76\u7ed3\u5408\u4e86\u9ad8\u6548\u7684\u6570\u503c\u8ba1\u7b97\u65b9\u6cd5\uff0c\u4f7f\u5176\u5728\u8ba1\u7b97\u5706\u5468\u7387\u65f6\u5177\u6709\u6781\u9ad8\u7684\u7cbe\u786e\u5ea6\u548c\u6548\u7387\u3002<\/p>\n<\/p>\n

\u4e00\u3001Chudnovsky \u7b97\u6cd5\u7b80\u4ecb<\/h3>\n<\/p>\n

Chudnovsky \u7b97\u6cd5\u901a\u8fc7\u4ee5\u4e0b\u7ea7\u6570\u6765\u8ba1\u7b97 \u03c0\uff1a<\/p>\n<\/p>\n

[ \\frac{1}{\\pi} = 12 \\sum_{k=0}^{\\infty} \\frac{(-1)^k (6k)! (545140134k + 13591409)}{(3k)! (k!)^3 (640320)^{3k+3\/2}} ]<\/p>\n<\/p>\n

\u8be5\u7ea7\u6570\u6536\u655b\u901f\u5ea6\u975e\u5e38\u5feb\uff0c\u8ba1\u7b97\u51e0\u767e\u4f4d\u751a\u81f3\u51e0\u5343\u4f4d\u7684\u5706\u5468\u7387\u90fd\u975e\u5e38\u9ad8\u6548\u3002<\/p>\n<\/p>\n

\u4e8c\u3001\u5b9e\u73b0 Chudnovsky \u7b97\u6cd5<\/h3>\n<\/p>\n

1\u3001\u5b89\u88c5\u5fc5\u8981\u7684\u5e93<\/h4>\n<\/p>\n

\u5728\u5f00\u59cb\u5b9e\u73b0\u7b97\u6cd5\u4e4b\u524d\uff0c\u6211\u4eec\u9700\u8981\u5b89\u88c5 mpmath<\/code> \u5e93\uff0c\u8fd9\u4e2a\u5e93\u4e13\u95e8\u7528\u4e8e\u9ad8\u7cbe\u5ea6\u7684\u6d6e\u70b9\u6570\u8ba1\u7b97\u3002<\/p>\n<\/p>\n

pip install mpmath<\/p>\n
<\/code><\/pre>\n<\/p>\n
2\u3001\u5b9e\u73b0\u4ee3\u7801<\/h4>\n<\/p>\n
\u4e0b\u9762\u662f\u4f7f\u7528 mpmath<\/code> \u5e93\u6765\u5b9e\u73b0 Chudnovsky \u7b97\u6cd5\u7684\u4ee3\u7801\uff1a<\/p>\n<\/p>\n
from mpmath import mp<\/p>\n
def chudnovsky_algorithm(precision):<\/p>\n
    # \u8bbe\u7f6e\u8ba1\u7b97\u7cbe\u5ea6<\/p>\n
    mp.dps = precision + 2  # \u591a\u52a02\u4f4d\u4ee5\u63d0\u9ad8\u8ba1\u7b97\u7cbe\u5ea6<\/p>\n
    # \u5e38\u91cf\u5b9a\u4e49<\/p>\n
    C = 426880 * mp.sqrt(10005)<\/p>\n
    def chudnovsky_term(k):<\/p>\n
        numerator = mp.factorial(6 * k) * (545140134 * k + 13591409)<\/p>\n
        denominator = mp.factorial(3 * k) * (mp.factorial(k) <strong> 3) * (-262537412640768000) <\/strong> k<\/p>\n
        return numerator \/ denominator<\/p>\n
    # \u8ba1\u7b97 \u03c0 \u7684\u5012\u6570<\/p>\n
    series_sum = mp.mpf(0)<\/p>\n
    k = 0<\/p>\n
    while True:<\/p>\n
        term = chudnovsky_term(k)<\/p>\n
        series_sum += term<\/p>\n
        if mp.fabs(term) < mp.mpf(10)  (-precision):<\/p>\n
            break<\/p>\n
        k += 1<\/p>\n
    pi = C \/ series_sum<\/p>\n
    return str(pi)[:precision + 2]  # \u5207\u6389\u591a\u4f59\u7684\u4f4d\u6570<\/p>\n
\u8ba1\u7b97\u5706\u5468\u7387\u7684\u524d1000\u4f4d<\/strong><\/h2>\n
precision = 1000<\/p>\n
pi = chudnovsky_algorithm(precision)<\/p>\n
print(pi)<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u4e09\u3001\u4ee3\u7801\u8be6\u7ec6\u89e3\u91ca<\/h3>\n<\/p>\n
1\u3001\u8bbe\u7f6e\u8ba1\u7b97\u7cbe\u5ea6<\/h4>\n<\/p>\n
mp.dps = precision + 2  # \u591a\u52a02\u4f4d\u4ee5\u63d0\u9ad8\u8ba1\u7b97\u7cbe\u5ea6<\/p>\n
<\/code><\/pre>\n<\/p>\n
mp.dps<\/code> \u8bbe\u7f6e\u4e86\u8ba1\u7b97\u7684\u7cbe\u5ea6\uff0c\u8fd9\u91cc\u52a0\u4e86 2 \u4f4d\u4ee5\u786e\u4fdd\u6700\u7ec8\u7ed3\u679c\u7684\u7cbe\u786e\u6027\u3002<\/p>\n<\/p>\n
2\u3001\u5e38\u91cf\u5b9a\u4e49<\/h4>\n<\/p>\n
C = 426880 * mp.sqrt(10005)<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u8be5\u5e38\u91cf\u662f Chudnovsky \u516c\u5f0f\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u90e8\u5206\uff0c\u7528\u4e8e\u8ba1\u7b97 \u03c0 \u7684\u5012\u6570\u3002<\/p>\n<\/p>\n
3\u3001\u8ba1\u7b97 Chudnovsky \u7ea7\u6570\u7684\u6bcf\u4e00\u9879<\/h4>\n<\/p>\n
def chudnovsky_term(k):<\/p>\n
    numerator = mp.factorial(6 * k) * (545140134 * k + 13591409)<\/p>\n
    denominator = mp.factorial(3 * k) * (mp.factorial(k) <strong> 3) * (-262537412640768000) <\/strong> k<\/p>\n
    return numerator \/ denominator<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u8fd9\u4e2a\u51fd\u6570\u8ba1\u7b97\u4e86 Chudnovsky \u7ea7\u6570\u7684\u7b2c k \u9879\u3002\u516c\u5f0f\u4e2d\u7684\u5206\u5b50\u548c\u5206\u6bcd\u5206\u522b\u8ba1\u7b97\u5e76\u8fd4\u56de\u5b83\u4eec\u7684\u5546\u3002<\/p>\n<\/p>\n
4\u3001\u8ba1\u7b97\u7ea7\u6570\u7684\u548c<\/h4>\n<\/p>\n
series_sum = mp.mpf(0)<\/p>\n
k = 0<\/p>\n
while True:<\/p>\n
    term = chudnovsky_term(k)<\/p>\n
    series_sum += term<\/p>\n
    if mp.fabs(term) < mp.mpf(10)  (-precision):<\/p>\n
        break<\/p>\n
    k += 1<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u901a\u8fc7\u8fed\u4ee3\u8ba1\u7b97\u7ea7\u6570\u7684\u6bcf\u4e00\u9879\uff0c\u76f4\u5230\u5f53\u524d\u9879\u5c0f\u4e8e\u6307\u5b9a\u7cbe\u5ea6\u7684\u5012\u6570\u4e3a\u6b62\u3002mp.fabs(term) < mp.mpf(10)  (-precision)<\/code> \u7528\u4e8e\u5224\u65ad\u5f53\u524d\u9879\u662f\u5426\u8db3\u591f\u5c0f\uff0c\u4ee5\u7ec8\u6b62\u8fed\u4ee3\u3002<\/p>\n<\/p>\n
5\u3001\u8ba1\u7b97 \u03c0<\/h4>\n<\/p>\n
pi = C \/ series_sum<\/p>\n
return str(pi)[:precision + 2]  # \u5207\u6389\u591a\u4f59\u7684\u4f4d\u6570<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u6700\u540e\uff0c\u901a\u8fc7\u5e38\u91cf C \u9664\u4ee5\u7ea7\u6570\u548c\u5f97\u5230 \u03c0 \u7684\u8fd1\u4f3c\u503c\uff0c\u5e76\u8fd4\u56de\u7cbe\u786e\u5230\u6307\u5b9a\u4f4d\u6570\u7684\u7ed3\u679c\u3002<\/p>\n<\/p>\n
\u56db\u3001\u5176\u4ed6\u8ba1\u7b97 \u03c0 \u7684\u65b9\u6cd5<\/h3>\n<\/p>\n
\u9664\u4e86 Chudnovsky \u7b97\u6cd5\uff0c\u8fd8\u6709\u5176\u4ed6\u51e0\u79cd\u5e38\u7528\u7684\u65b9\u6cd5\u6765\u8ba1\u7b97\u5706\u5468\u7387\uff1a<\/p>\n<\/p>\n
1\u3001\u83b1\u5e03\u5c3c\u8328\u516c\u5f0f<\/h4>\n<\/p>\n
\u83b1\u5e03\u5c3c\u8328\u516c\u5f0f\u901a\u8fc7\u4ee5\u4e0b\u7ea7\u6570\u8ba1\u7b97 \u03c0\uff1a<\/p>\n<\/p>\n
[ \\pi = 4 \\sum_{k=0}^{\\infty} \\frac{(-1)^k}{2k + 1} ]<\/p>\n<\/p>\n
\u8fd9\u4e2a\u516c\u5f0f\u867d\u7136\u7b80\u5355\uff0c\u4f46\u6536\u655b\u901f\u5ea6\u8f83\u6162\uff0c\u9700\u8981\u8ba1\u7b97\u5927\u91cf\u9879\u624d\u80fd\u5f97\u5230\u9ad8\u7cbe\u5ea6\u7684\u7ed3\u679c\u3002<\/p>\n<\/p>\n
def leibniz_formula(precision):<\/p>\n
    pi = 0<\/p>\n
    for k in range(precision):<\/p>\n
        pi += ((-1)k) \/ (2*k + 1)<\/p>\n
    return 4 * pi<\/p>\n
precision = 1000000<\/p>\n
pi = leibniz_formula(precision)<\/p>\n
print(f"Leibniz Formula: {pi}")<\/p>\n
<\/code><\/pre>\n<\/p>\n
2\u3001\u8499\u7279\u5361\u7f57\u65b9\u6cd5<\/h4>\n<\/p>\n
\u8499\u7279\u5361\u7f57\u65b9\u6cd5\u662f\u4e00\u79cd\u57fa\u4e8e\u6982\u7387\u7684\u8ba1\u7b97\u65b9\u6cd5\uff0c\u901a\u8fc7\u968f\u673a\u751f\u6210\u70b9\u6765\u4f30\u8ba1 \u03c0 \u7684\u503c\u3002<\/p>\n<\/p>\n
import random<\/p>\n
def monte_carlo_pi(precision):<\/p>\n
    inside_circle = 0<\/p>\n
    total_points = precision<\/p>\n
    for _ in range(total_points):<\/p>\n
        x = random.uniform(0, 1)<\/p>\n
        y = random.uniform(0, 1)<\/p>\n
        if x<strong>2 + y<\/strong>2 <= 1:<\/p>\n
            inside_circle += 1<\/p>\n
    pi = (inside_circle \/ total_points) * 4<\/p>\n
    return pi<\/p>\n
precision = 1000000<\/p>\n
pi = monte_carlo_pi(precision)<\/p>\n
print(f"Monte Carlo Method: {pi}")<\/p>\n
<\/code><\/pre>\n<\/p>\n
3\u3001\u5c3c\u5c14\u68ee\u516c\u5f0f<\/h4>\n<\/p>\n
\u5c3c\u5c14\u68ee\u516c\u5f0f\u662f\u53e6\u4e00\u79cd\u7ea7\u6570\u5c55\u5f00\u7684\u516c\u5f0f\uff0c\u7528\u4e8e\u8ba1\u7b97 \u03c0\uff1a<\/p>\n<\/p>\n
[ \\pi = 3 + \\sum_{k=1}^{\\infty} \\frac{(-1)^{k-1}}{16^k} \\left( \\frac{4}{8k+1} – \\frac{2}{8k+4} – \\frac{1}{8k+5} – \\frac{1}{8k+6} \\right) ]<\/p>\n<\/p>\n
def nilakantha_formula(precision):<\/p>\n
    pi = 3<\/p>\n
    for k in range(1, precision):<\/p>\n
        pi += ((-1)<strong>(k-1)) \/ (16<\/strong>k) * (4\/(8*k+1) - 2\/(8*k+4) - 1\/(8*k+5) - 1\/(8*k+6))<\/p>\n
    return pi<\/p>\n
precision = 1000<\/p>\n
pi = nilakantha_formula(precision)<\/p>\n
print(f"Nilakantha Formula: {pi}")<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u4e94\u3001\u603b\u7ed3<\/h3>\n<\/p>\n
\u672c\u6587\u8be6\u7ec6\u4ecb\u7ecd\u4e86\u5982\u4f55\u4f7f\u7528 Python \u5b9e\u73b0 Chudnovsky \u7b97\u6cd5\u6765\u7cbe\u786e\u6c42\u5706\u5468\u7387\uff0c\u5e76\u5bf9\u4ee3\u7801\u8fdb\u884c\u4e86\u8be6\u7ec6\u7684\u89e3\u91ca\u3002\u9664\u4e86 Chudnovsky \u7b97\u6cd5\uff0c\u672c\u6587\u8fd8\u4ecb\u7ecd\u4e86\u83b1\u5e03\u5c3c\u8328\u516c\u5f0f\u3001\u8499\u7279\u5361\u7f57\u65b9\u6cd5\u548c\u5c3c\u5c14\u68ee\u516c\u5f0f\u7b49\u5176\u4ed6\u51e0\u79cd\u5e38\u7528\u7684\u65b9\u6cd5\u3002\u901a\u8fc7\u8fd9\u4e9b\u65b9\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u4e0d\u540c\u7684\u573a\u666f\u4e0b\u9009\u62e9\u5408\u9002\u7684\u7b97\u6cd5\u6765\u8ba1\u7b97\u5706\u5468\u7387\u3002<\/p>\n<\/p>\n
\u76f8\u5173\u95ee\u7b54FAQs\uff1a<\/strong><\/h2>\n
 \u5982\u4f55\u4f7f\u7528Python\u8ba1\u7b97\u5706\u5468\u7387\u7684\u4e0d\u540c\u65b9\u6cd5\uff1f<\/strong>
\u5728Python\u4e2d\uff0c\u6709\u591a\u79cd\u65b9\u6cd5\u53ef\u4ee5\u8ba1\u7b97\u5706\u5468\u7387\uff0c\u5305\u62ec\u4f7f\u7528\u6570\u5b66\u5e93\u3001\u6570\u503c\u8ba1\u7b97\u548c\u7b97\u6cd5\u5b9e\u73b0\u3002\u5e38\u7528\u7684\u65b9\u6cd5\u5305\u62ec\u5229\u7528\u83b1\u5e03\u5c3c\u8328\u516c\u5f0f\u3001\u8499\u7279\u5361\u6d1b\u65b9\u6cd5\u4ee5\u53ca\u4f7f\u7528NumPy\u5e93\u4e2d\u7684\u5185\u7f6e\u51fd\u6570\u3002\u83b1\u5e03\u5c3c\u8328\u516c\u5f0f\u901a\u8fc7\u65e0\u7a77\u7ea7\u6570\u903c\u8fd1\u03c0\uff0c\u800c\u8499\u7279\u5361\u6d1b\u65b9\u6cd5\u5219\u901a\u8fc7\u968f\u673a\u53d6\u70b9\u6765\u4f30\u7b97<\/a>\u5706\u7684\u9762\u79ef\uff0c\u4ece\u800c\u63a8\u5bfc\u51fa\u03c0\u7684\u503c\u3002NumPy\u5e93\u63d0\u4f9b\u7684\u5185\u7f6e\u03c0\u5e38\u6570\u4e5f\u662f\u4e00\u79cd\u7b80\u5355\u76f4\u63a5\u7684\u65b9\u5f0f\u3002<\/p>\n
\u5728Python\u4e2d\u5982\u4f55\u63d0\u9ad8\u5706\u5468\u7387\u8ba1\u7b97\u7684\u7cbe\u786e\u5ea6\uff1f<\/strong>
\u8981\u63d0\u9ad8\u5706\u5468\u7387\u7684\u8ba1\u7b97\u7cbe\u5ea6\uff0c\u53ef\u4ee5\u4f7f\u7528\u5927\u6570\u5e93\uff08\u5982decimal<\/code>\u6a21\u5757\uff09\u6765\u5904\u7406\u9ad8\u7cbe\u5ea6\u7684\u6d6e\u70b9\u6570\u8ba1\u7b97\u3002\u6b64\u5916\uff0c\u9009\u62e9\u66f4\u590d\u6742\u7684\u7b97\u6cd5\uff0c\u4f8b\u5982\u9ad8\u65af-\u52d2\u8ba9\u5fb7\u7b97\u6cd5\u6216\u5e03\u4f26\u7279-\u9ea6\u514b\u7c73\u4f26\u7b97\u6cd5\uff0c\u8fd9\u4e9b\u7b97\u6cd5\u53ef\u4ee5\u5728\u8f83\u5c11\u7684\u8fed\u4ee3\u6b21\u6570\u5185\u63d0\u4f9b\u66f4\u9ad8\u7684\u7cbe\u786e\u5ea6\u3002\u901a\u8fc7\u8bbe\u7f6e\u8f83\u9ad8\u7684\u7cbe\u5ea6\u53c2\u6570\uff0c\u53ef\u4ee5\u786e\u4fdd\u8ba1\u7b97\u7ed3\u679c\u66f4\u4e3a\u51c6\u786e\u3002<\/p>\n
\u4f7f\u7528Python\u8ba1\u7b97\u5706\u5468\u7387\u65f6\u53ef\u80fd\u4f1a\u9047\u5230\u54ea\u4e9b\u95ee\u9898\uff1f<\/strong>
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