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SKYIV STUDIO

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公告

引言

我们知道,自然对数的底 e 定义为以下极限值:

e

这个公式很适合于对幂函数的计算进行一些测试,得到的结果是 e 的近似值,不用担心当 n 很大时计算结果会溢出。

测试程序

下面就是 Tester.cs:

 1 using System;
 2 using System.Numerics;
 3 using System.Diagnostics;
 4 using Skyiv.Extensions;
 5 
 6 namespace Skyiv.Test
 7 {
 8   sealed class Tester
 9   {
10     string Standard(long n)
11     { // n == 10^m
12       if (n > 100000) return "Skip";
13       var s = BigInteger.Pow(n + 1, (int)n).ToString();
14       s = s.Substring(0, Math.Min(31, s.Length));
15       return s[0] + "." + s.Substring(1);
16     }
17     
18     string Direct(long n)
19     {
20       if (n > 1000000000) return "Skip";
21       var y = 1m;
22       for (var x = 1 + 1m / n; n > 0; n--) y *= x;
23       return y.ToString();
24     }
25     
26     string Binary(long n)
27     {
28       var y = 1m;
29       for (var x = 1 + 1m / n; n != 0; x *= x, n >>= 1)
30         if ((n & 1) != 0) y *= x;
31       return y.ToString();
32     }
33     
34     string ExpLog(long n)
35     {
36       return (1 + 1m / n).Pow(n).ToString();
37     }
38     
39     void Out(string name, Func<long, string> func, long n)
40     {
41       var timer = Stopwatch.StartNew();
42       var y = func(n);
43       timer.Stop();
44       Console.WriteLine("{0,-32} {1} {2}", y, timer.Elapsed, name);
45     }
46     
47     void Run(int max)
48     {
49       for (var m = 0; m <= max; m++)
50       {
51         var n = (long)Math.Pow(10, m);
52         Console.WriteLine(string.Format("- {0:D2}:{1:N0} ", m, n).PadRight(58, '-'));
53         Out("Standard", Standard, n);
54         Out("Direct", Direct, n);
55         Out("Binary", Binary, n);
56         Out("ExpLog", ExpLog, n);
57       }
58     }
59   
60     static void Main()
61     {
62       new Tester().Run(18);
63     }
64   }
65 }

这个程序使用四种方法来计算幂函数:

  1. 第 10 至 16 行的 Standard 方法使用 BigInteger.Pow 方法来计算幂函数。这个计算结果(在有效数字范围内)是准确值,作为其他方法的标准。
  2. 第 18 至 24 行的 Direct 方法直接将 x 乘上 n 遍来计算幂函数,是最没技术含量的暴力方法。时间复杂度是 O(N)。
  3. 第 26 至 32 行的 Binary 方法将 n 视为二进制数,根据其为 1 的位来计算幂函数。这是经典的算法,时间复杂度是 O(logN)。FCL 的 BigInteger.Pow 方法也是使用这个算法。
  4. 第 34 至 37 行的 ExpLog 方法使用 decimal 的扩展方法 Pow 来计算幂函数,是通过对数函数和指数函数来计算的:an。理论上说,时间复杂度是 O(1)。

decimal 的扩展方法

下面就是 DecimalExtensions.cs:

 1 using System;
 2 
 3 namespace Skyiv.Extensions
 4 {
 5   static class DecimalExtensions
 6   {
 7     static readonly int[] mask = { 1, 2, 4, 8, 16, 32, 64 };
 8     static readonly decimal ln10 = 2.3025850929940456840179914547m;
 9     static readonly decimal lnr = 0.2002433314278771112016301167m;
10     static readonly decimal expmax = 66.542129333754749704054283659m;
11     static readonly decimal[] exps =
12     {
13       2.71828182845904523536028747135m, // exp(1)
14       7.38905609893065022723042746058m, // exp(2)
15       54.5981500331442390781102612029m, // exp(4)
16       2980.95798704172827474359209945m, // exp(8)
17       8886110.52050787263676302374078m, // exp(16)
18       78962960182680.6951609780226351m, // exp(32)
19       6235149080811616882909238708.93m  // exp(64)
20     };
21 
22     public static decimal Log10(this decimal x)
23     {
24       return Log(x) / ln10;
25     }
26 
27     public static decimal Log(this decimal x)
28     {
29       if (x <= 0) throw new ArgumentException("Must be positive");
30       int k = 0, l = 0;
31       for (; x >= 1.10527199m; k++) x /= 10;
32       for (; x <= 0.1m; k--) x *= 10;        // ( 0.1000, 1.10527199 )
33       for (; x < 0.9047m; l--) x *= 1.2217m; // [ 0.9047, 1.10527199 )
34       return k * ln10 + l * lnr + Logarithm((x - 1) / (x + 1));
35     }
36     
37     static decimal Logarithm(decimal y)
38     { // y in ( -0.05-, 0.05+ ), return ln((1+y)/(1-y))
39       decimal v = 1, y2 = y * y, t = y2, z = t / 3;
40       for (var i = 3; z != 0; z = (t *= y2) / (i += 2)) v += z;
41       return v * y * 2;
42     }
43     
44     public static decimal Exp(this decimal x)
45     {
46       if (x > expmax) throw new OverflowException("overflow");
47       if (x < -66) return 0;
48       var n = (int)decimal.Round(x);
49       if (n > 66) n--;
50       decimal z = 1, y = Exponential(x - n);
51       for (int m = (n < 0) ? -n : n, i = 0; i < mask.Length; i++)
52         if ((m & mask[i]) != 0) z *= exps[i];
53       return (n < 0) ? (y / z) : (y * z);
54     }
55     
56     static decimal Exponential(decimal q)
57     { // q (almost) in [ -0.5, 0.5 ]
58       decimal y = 1, t = q;
59       for (var i = 1; t != 0; t *= q / ++i) y += t;
60       return y;
61     }
62     
63     public static decimal Pow(this decimal x, decimal y)
64     {
65       if (x == 0 && y > 0) return 0;
66       if (y == 0 && x != 0) return 1;
67       return Exp(y * Log(x));
68     }
69   }
70 }

这个程序的详细说明请见参考资料[5]和[6]。

编译和运行

在 Arch Linux 操作系统的 Mono 环境下编译和运行:

work$ dmcs -r:System.Numerics.dll Tester.cs DecimalExtensions.cs
work$ mono Tester.exe
- 00:1 ---------------------------------------------------
2.                               00:00:00.0085818 Standard
2                                00:00:00.0033230 Direct
2                                00:00:00.0002739 Binary
2.0000000000000000000000000005   00:00:00.0049157 ExpLog
- 01:10 --------------------------------------------------
2.5937424601                     00:00:00.0015421 Standard
2.5937424601000000000000000000   00:00:00.0000146 Direct
2.5937424601000000000000000000   00:00:00.0000092 Binary
2.5937424600999999999999999977   00:00:00.0000488 ExpLog
- 02:100 -------------------------------------------------
2.704813829421526093267194710807 00:00:00.0006872 Standard
2.7048138294215260932671947112   00:00:00.0000735 Direct
2.7048138294215260932671947103   00:00:00.0000234 Binary
2.7048138294215260932671947257   00:00:00.0000330 ExpLog
- 03:1,000 -----------------------------------------------
2.716923932235892457383088121947 00:00:00.0277308 Standard
2.7169239322358924573830881229   00:00:00.0007167 Direct
2.7169239322358924573830881218   00:00:00.0000159 Binary
2.7169239322358924573830883380   00:00:00.0000310 ExpLog
- 04:10,000 ----------------------------------------------
2.718145926825224864037664674913 00:00:03.3247007 Standard
2.7181459268252248640376646760   00:00:00.0068304 Direct
2.7181459268252248640376646665   00:00:00.0000191 Binary
2.7181459268252248640376679109   00:00:00.0000276 ExpLog
- 05:100,000 ---------------------------------------------
2.718268237174489668035064824426 00:07:56.2341075 Standard
2.7182682371744896680350648397   00:00:00.0686007 Direct
2.7182682371744896680350643783   00:00:00.0000222 Binary
2.7182682371744896680350286262   00:00:00.0000255 ExpLog
- 06:1,000,000 -------------------------------------------
Skip                             00:00:00.0000008 Standard
2.7182804693193768838197997202   00:00:00.6837104 Direct
2.7182804693193768838198166432   00:00:00.0000241 Binary
2.7182804693193768838199803836   00:00:00.0000213 ExpLog
- 07:10,000,000 ------------------------------------------
Skip                             00:00:00.0000009 Standard
2.7182816925449662711985502083   00:00:06.8334721 Direct
2.7182816925449662711985623547   00:00:00.0000289 Binary
2.7182816925449662712010419841   00:00:00.0000221 ExpLog
- 08:100,000,000 -----------------------------------------
Skip                             00:00:00.0000009 Standard
2.7182818148676362176529774118   00:01:08.3492423 Direct
2.7182818148676362176523859621   00:00:00.0000409 Binary
2.7182818148676362176710998015   00:00:00.0000230 ExpLog
- 09:1,000,000,000 ---------------------------------------
Skip                             00:00:00.0000007 Standard
2.7182818270999043223766453801   00:11:23.4187574 Direct
2.7182818270999043223770801045   00:00:00.0000442 Binary
2.7182818270999043220142064477   00:00:00.0000215 ExpLog
- 10:10,000,000,000 --------------------------------------
Skip                             00:00:00.0000007 Standard
Skip                             00:00:00.0000008 Direct
2.7182818283231311436196542093   00:00:00.0000349 Binary
2.7182818283231311439407330619   00:00:00.0000172 ExpLog
- 11:100,000,000,000 -------------------------------------
Skip                             00:00:00.0000008 Standard
Skip                             00:00:00.0000010 Direct
2.7182818284454538261539965115   00:00:00.0000398 Binary
2.7182818284454538262180262237   00:00:00.0000176 ExpLog
- 12:1,000,000,000,000 -----------------------------------
Skip                             00:00:00.0000010 Standard
Skip                             00:00:00.0000007 Direct
2.7182818284576860942863185484   00:00:00.0000403 Binary
2.7182818284576860944460582886   00:00:00.0000174 ExpLog
- 13:10,000,000,000,000 ----------------------------------
Skip                             00:00:00.0000009 Standard
Skip                             00:00:00.0000007 Direct
2.7182818284589093212295138270   00:00:00.0000436 Binary
2.7182818284589093212688645227   00:00:00.0000176 ExpLog
- 14:100,000,000,000,000 ---------------------------------
Skip                             00:00:00.0000009 Standard
Skip                             00:00:00.0000009 Direct
2.7182818284590316438350187680   00:00:00.0000480 Binary
2.7182818284590452353602874714   00:00:00.0000112 ExpLog
- 15:1,000,000,000,000,000 -------------------------------
Skip                             00:00:00.0000009 Standard
Skip                             00:00:00.0000009 Direct
2.7182818284590431765145511000   00:00:00.0000522 Binary
2.7182818284590452353602874714   00:00:00.0000114 ExpLog
- 16:10,000,000,000,000,000 ------------------------------
Skip                             00:00:00.0000009 Standard
Skip                             00:00:00.0000006 Direct
2.7182818284590335325626228124   00:00:00.0000547 Binary
2.7182818284590452353602874714   00:00:00.0000109 ExpLog
- 17:100,000,000,000,000,000 -----------------------------
Skip                             00:00:00.0000010 Standard
Skip                             00:00:00.0000006 Direct
2.7182818284590296936415060358   00:00:00.0000567 Binary
2.7182818284590452353602874714   00:00:00.0000108 ExpLog
- 18:1,000,000,000,000,000,000 ---------------------------
Skip                             00:00:00.0000010 Standard
Skip                             00:00:00.0000006 Direct
2.7182818284590434884535909399   00:00:00.0000615 Binary
2.7182818284590452353602874714   00:00:00.0000108 ExpLog
work$ echo 'scale=30;e(1)' | bc -lq
2.718281828459045235360287471352

在上述结果中:

  1. 最后一行是 e 的近似值,是使用 Linux 操作系统的高精度计算器 bc 计算的,请见参考资料[4]。
  2. 使用 BigInteger.Pow 计算出来的是准确值。在 05:100,000 这一组中,计算结果达 500,001 个十进制数字。当 n 达到 106 以后,由于计算量太大,已经无法在合理的时间内计算准确值了。
  3. 使用 Direct 计算最慢(除 Standard 外,因为计算量不同)。当 n 达到 1010 以后,由于费时太多,已经不使用 Direct 方法计算了。
  4. 使用 Binary 计算的速度非常快,其精度和 Direct 差不多。这两者答案不同说明 decimal 的乘法不满足结合律。
  5. 使用 ExpLog 计算的速度理论上是最快的,实际的速度和 ExpLog 差不多,因为 n 还不够大。其精度在 n 不是很大时稍差。
  6. 当 n 达到 1014 以后,ExpLog 计算出来的值在 29 个有效数字范围内已经等于 e 值,不再变化了。
  7. 当 n 达到 1014 以后,由于舍入误差的累计,Binary 计算出来的值大约只有 14 个有效数字是可信的,再增大 n 值也不能更逼近 e 值了。也就是说,在逼近 e 值的意义上说,计算结果在有效数字范围内不再变化了。
  8. 要计算的幂函数是增函数,请注意观察上述运行结果是如何体现这一点的。

参数资料

  1. Wikipedia: e (mathematical constant)
  2. MSDN: BigInteger.Pow 方法 (System.Numerics)
  3. MSDN: Math.Pow 方法 (System)
  4. Linux man pages: bc - An arbitrary precision calculator language
  5. 博客园: 计算自然对数的算法
  6. 博客园: 计算指数函数的算法
posted on 2013-03-09 21:19 银河 阅读(...) 评论(...) 编辑 收藏