欧拉项目第2题Julia与Python实现异常：数值结果差异求助

问题描述 #

为求解欧拉项目第2题（计算小于400万的斐波那契数列偶数项之和），分别编写了Python和Julia实现代码。Python版本输出正确结果60696，但Julia版本因数值溢出产生包含负数的斐波那契数列，最终得到错误结果-4840，需排查问题根源。

Python实现代码 #

import numpy as np

# Function to generate the Fibonacci series
def fibonacci_fun(fib_series_fun, num_terms_fun):

    # Initialize the first value in series
    first_num_fun = 1

    # Append the first value to series
    fib_series_fun.append(first_num_fun)

    # Initialize the value preceeding the first number; technically always 0 but 1 in this problem
    num_bef_start_fun = 1

    # Add the first_num to the num_bef_start value and append it
    next_num_fun = first_num_fun + num_bef_start_fun
    fib_series_fun.append(next_num_fun)

    # While we do not have num_terms number of elements in Fibonacci series, repeat
    while len(fib_series_fun) != num_terms_fun:

        # Add the previous 2 numbers of the series to itself
        new_num_fun = fib_series_fun[len(fib_series_fun)-2] + fib_series_fun[len(fib_series_fun)-1];
        
        fib_series_fun.append(new_num_fun)

    return fib_series_fun

# Main function
def main():

    # Define the number of terms in Fibonacci series
    num_terms = 25

    # Define empty array of Int16 to store Fibonacci sereis values
    fib_series = []

    # Run the fibonacci_fun to get the Fibonacci series
    fib_series = fibonacci_fun(fib_series, num_terms)
    print(fib_series)

    fib_series = np.array(fib_series)

    # Values less than 4 million
    values_less_than = fib_series[fib_series < 4000000]

    # Get only the even values in the Fibonacci series
    even_values_less_than_fib_series = values_less_than[values_less_than % 2 == 0]
    print(even_values_less_than_fib_series)

    sum_of_fib_series = sum(even_values_less_than_fib_series)
    print("The sum of even values in the Fibonacci series is %d.\n" %sum_of_fib_series)

    print("Process finished!")

main()

Python输出结果 #

[1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393]

[    2     8    34   144   610  2584 10946 46368]
The sum of even values in the Fibonacci series is 60696.
Process finished!

Julia实现代码 #

using Printf

# Function to generate the Fibonacci series
function fibonacci_fun(fib_series_fun, num_terms_fun)

    # Initialize the first value in series
    first_num_fun = 1

    # Append the first value to series
    append!(fib_series_fun, first_num_fun)

    # Initialize the value preceeding the first number; technically always 0 but 1 in this problem
    num_bef_start_fun = 1

    # Add the first_num to the num_bef_start value and append it
    next_num_fun = first_num_fun + num_bef_start_fun
    append!(fib_series_fun, next_num_fun)

    # While we do not have num_terms number of elements in Fibonacci series, repeat
    while length(fib_series_fun) != num_terms_fun

        # Add the previous 2 numbers of the series to itself
        new_num_fun = fib_series_fun[length(fib_series_fun)-1] + fib_series_fun[length(fib_series_fun)];
        
        append!(fib_series_fun, new_num_fun)
    end

    return fib_series_fun

end

# Main function
function main()

    # Define the number of terms in Fibonacci series
    num_terms = 25

    # Define empty array of Int16 to store Fibonacci sereis values
    global fib_series = Int16[]

    # Run the fibonacci_fun to get the Fibonacci series
    fib_series = fibonacci_fun(fib_series, num_terms)
    println(fib_series)

    # Values less than 4 million
    values_less_than = fib_series[fib_series .< 4000000]

    # Get only the even values in the Fibonacci series
    even_values_less_than_fib_series = values_less_than[values_less_than .% 2 .== 0]
    println(even_values_less_than_fib_series)

    sum_of_fib_series = sum(even_values_less_than_fib_series)
    @printf("The sum of even values in the Fibonacci series is %d.\n", sum_of_fib_series)

    println("Process finished!")

end

main()

Julia输出结果 #

Int16[1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, -19168, 9489, -9679]

Int16[2, 8, 34, 144, 610, 2584, 10946, -19168]
The sum of even values in the Fibonacci series is -4840.
Process finished!

问题根源分析 #

Julia代码出现错误的核心原因是整数类型溢出：

• 代码中使用Int16类型存储斐波那契数，而Int16的取值范围是*[-32768, 32767]*。当斐波那契数增长到28657后，下一个数46368已经超出了Int16的最大值32767。
• Julia默认对有符号整数溢出采用环绕处理（即模2^{16运算），46368减去2}16（65536）后得到-19168，后续所有计算都基于错误的负数继续，最终导致求和结果错误。
• 而Python中的int类型是任意精度整数，无论数值多大都不会出现溢出，因此能正确计算所有项。

修复方案 #

方案1：更换为更大的整数类型 #

将Julia代码中数组的类型从Int16改为Int64（Julia默认整数类型，取值范围远大于400万），或者BigInt（任意精度整数，适合超大数值）。修改后的代码片段：

# 替换原有的Int16[]定义
global fib_series = Int64[]

方案2：优化生成逻辑（更贴合题目需求） #

题目要求计算小于400万的斐波那契偶数项之和，无需固定生成25项。可以修改循环条件，当生成的数值超过400万时停止，既避免溢出，又减少不必要的计算：

function fibonacci_fun(fib_series_fun)
    a, b = 1, 2
    push!(fib_series_fun, a)
    push!(fib_series_fun, b)
    while true
        next = a + b
        if next >= 4000000
            break
        end
        push!(fib_series_fun, next)
        a, b = b, next
    end
    return fib_series_fun
end

# 主函数中调用方式修改为
fib_series = fibonacci_fun(Int64[])

修复后，Julia代码将输出与Python一致的正确结果60696。

内容的提问来源于stack exchange，提问作者Shashwat Maharjan