Python中向量矩阵乘积的时间复杂度有什么问题？

我有以下代码对随机向量（dimension

const

）和随机矩阵（dimension

const

x

n

）进行采样，所有条目都有

const/n

位

为了避免舍入错误，

n

始终是

const

的除数，因此，

const/n

始终是整数

因此，函数

mult_vec_mat（u，A）

总是执行

const*const*n

常量/n位整数的乘积。假设Python需要时间

b*log（b）

乘以两个

b

-位整数，该函数的时间复杂度应该是

const*const*n*log（const/n）*const/n=const³*（log（const）-log（n））

因此，当

n

增大时，它应该减小

即使我们假设Python使用

b*b

乘以

b

-位整数，复杂性仍然是

n

中的一个递减函数

但是，从下面的输出可以看出，

mult_vec_mat（u，A）

的运行时间减少，直到

n

的某个值，然后开始增加

你对此有什么解释吗？为什么

mult\u vec\u mat（u，A）

的运行时间不符合我估计的时间复杂度

代码 输出

from random import randint
from time import time

def random_vec(n, bound=2):
    return [randint(0, bound) for _ in xrange(n)]

def random_matrix(nrows, ncols, bound=2):
    A = [0 for _ in xrange(nrows)]
    for i in xrange(nrows):
        A[i] = random_vec(ncols, bound)
    return A

def mult_vec_mat(u, A):
    nrows = len(A)
    ncols = len(A[0])
#    v = [0 for _ in xrange(ncols)]
    for j in xrange(ncols):
        tmp = 0
        for i in xrange(nrows):
            tmp += u[i] * A[i][j]
#        v[j] = tmp
#    return v

def benchmark_prod():

    NUMBER_TESTS = 3
    const = 5040

    for n in [2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72]:
        bitlen = const / n # always an integer

        avg_time = 0
        for k in xrange(NUMBER_TESTS):
            c = random_vec(const, 2**bitlen)
            C = random_matrix(const, n, 2**bitlen)

            start = time()
            mult_vec_mat(c, C)
            end = time()
            avg_time += (end - start)

        print "n =", n
        print "bitlen =", bitlen
        print "n * bitlen =", n*bitlen
        print "const =", const
        print "const * n * bitlen =", const * n * bitlen
        print "         time: ", (avg_time / NUMBER_TESTS), "seconds"
        print "..................................................................."


benchmark_prod()

n = 2
bitlen = 2520
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0886503060659 seconds
...................................................................
n = 3
bitlen = 1680
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0701853434245 seconds
...................................................................
n = 4
bitlen = 1260
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0559728940328 seconds
...................................................................
n = 5
bitlen = 1008
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0478646755219 seconds
...................................................................
n = 6
bitlen = 840
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0430323282878 seconds
...................................................................
n = 7
bitlen = 720
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.041002591451 seconds
...................................................................
n = 8
bitlen = 630
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0406827411652 seconds
...................................................................
n = 9
bitlen = 560
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.039892355601 seconds
...................................................................
n = 10
bitlen = 504
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0358901023865 seconds
...................................................................
n = 12
bitlen = 420
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0406420230865 seconds
...................................................................
n = 14
bitlen = 360
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0487660566966 seconds
...................................................................
n = 15
bitlen = 336
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0454970200857 seconds
...................................................................
n = 16
bitlen = 315
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0551973978678 seconds
...................................................................
n = 18
bitlen = 280
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0513706207275 seconds
...................................................................
n = 20
bitlen = 252
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0515103340149 seconds
...................................................................
n = 21
bitlen = 240
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0477599302928 seconds
...................................................................
n = 24
bitlen = 210
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0475799242655 seconds
...................................................................
n = 28
bitlen = 180
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0549654165904 seconds
...................................................................
n = 30
bitlen = 168
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0527282555898 seconds
...................................................................
n = 35
bitlen = 144
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0609803994497 seconds
...................................................................
n = 36
bitlen = 140
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0620452562968 seconds
...................................................................
n = 40
bitlen = 126
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0735117594401 seconds
...................................................................
n = 42
bitlen = 120
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.078006029129 seconds
...................................................................
n = 45
bitlen = 112
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0749526818593 seconds
...................................................................
n = 48
bitlen = 105
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0841933091482 seconds
...................................................................
n = 56
bitlen = 90
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0847909450531 seconds
...................................................................
n = 60
bitlen = 84
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.0929036140442 seconds
...................................................................
n = 63
bitlen = 80
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.111557722092 seconds
...................................................................
n = 70
bitlen = 72
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.109657287598 seconds
...................................................................
n = 72
bitlen = 70
n * bitlen = 5040
const = 5040
const * n * bitlen = 25401600
         time:  0.118753115336 seconds
...................................................................