C#，《小白学程序》第二十六课：大数乘法（BigInteger Multiply）的Toom-Cook 3算法及源程序

深度混淆2023-11-28 1:08

凑数的，仅供参考。

1 文本格式

/// <summary>

/// 《小白学程序》第二十六课：大数（BigInteger）的Toom-Cook 3乘法

/// Toom-Cook 3-Way Multiplication

/// </summary>

/// <param name="a"></param>

/// <param name="b"></param>

/// <returns></returns>

public static string toom_cook3_multiply(string a, string b)

{

int n = Math.Max(a.Length, b.Length);

int[] ra = string_to_digitals(a, n);

int[] rb = string_to_digitals(b, n);

toom_cook3_process_00(ra, rb, out int[] rz);

toom_cook3_carry(rz, n * 2);

return digitals_to_string(rz);

}

/// <summary>

/// 短数字的乘法（常规乘法，小学生算法）

/// </summary>

/// <param name="a"></param>

/// <param name="b"></param>

/// <param name="z"></param>

private static void toom_cook3_normal(int[] a, int[] b, ref int[] z)

{

int n = a.Length;

for (int j = 0; j < n; j++)

{

for (int i = 0; i < n; i++)

{

z[j + i] += a[i] * b[j];

}

}

}

/// <summary>

/// 完全按原始C++代码改写；运行成功；

/// </summary>

/// <param name="a"></param>

/// <param name="b"></param>

/// <param name="z"></param>

private static void toom_cook3_process_01(int[] a, int[] b, out int[] z)

{

int n = a.Length;

int n1 = n * 1 / 3;

int n2 = n * 2 / 3;

int n3 = n * 3 / 3;

int n4 = n * 4 / 3;

z = new int[n * 2];

if (n <= 9)

{

toom_cook3_normal(a, b, ref z);

return;

}

// int *a0 = &a[0];

// Multiplicand / right side array pointer

int a0 = 0;

// int *a1 = &a[tLen / 3];

// Multiplicand / central array pointer

int a1 = n1;

// int *a2 = &a[tLen * 2/ 3];

// Multiplicand / left side array pointer

int a2 = n2;// n * 2 / 3;

// int *b0 = &b[0];

// Multiplier / right side array pointer

int b0 = 0;

// int *b1 = &b[tLen / 3];

// Multiplier / central array pointer

int b1 = n1;

// int *b2 = &b[tLen * 2 / 3];

// Multiplier / left side array pointer

int b2 = n2;// n * 2 / 3;

// int *c0 = &z[(tLen / 3) * 0];

int[] c0 = new int[n2];

int[] c1 = new int[n2];

// int *c2 = &z[(tLen / 3) * 2];

int[] c2 = new int[n2];

int[] c3 = new int[n2];

// int *c4 = &z[(tLen / 3) * 4];

int[] c4 = new int[n2];

int[] a_m2 = new int[n1]; // a(-2)

int[] a_m1 = new int[n1]; // a(-1)

int[] a_0 = new int[n1]; // a(0)

int[] a_1 = new int[n1]; // a(1)

int[] a_inf = new int[n1]; // a(inf)

int[] b_m2 = new int[n1]; // b-2)

int[] b_m1 = new int[n1]; // b-1)

int[] b_0 = new int[n1]; // b(0)

int[] b_1 = new int[n1]; // b(1)

int[] b_inf = new int[n1]; // b(inf)

// ==== a(-2) = 4 * a2 - 2 * a1 + a0, b(-2) = 4 * b2 - 2 * b1 + b0

for (int i = 0; i < n1; i++)

{

a_m2[i] = (a[a2 + i] << 2) - (a[a1 + i] << 1) + a[a0 + i];

b_m2[i] = (b[b2 + i] << 2) - (b[b1 + i] << 1) + b[b0 + i];

}

// ==== c(-2) = a(-2) * b(-2)

toom_cook3_process_01(a_m2, b_m2, out int[] c_m2);

// ==== a(-1) = a2 - a1 + a0, b(-1) = b2 - b1 + b0

for (int i = 0; i < n1; i++)

{

a_m1[i] = a[a2 + i] - a[a1 + i] + a[a0 + i];

b_m1[i] = b[b2 + i] - b[b1 + i] + b[b0 + i];

}

// ==== c(-1) = a(-1) * b(-1)

toom_cook3_process_01(a_m1, b_m1, out int[] c_m1);

// ==== a(0) = a0, b(0) = b0

for (int i = 0; i < n1; i++)

{

a_0[i] = a[a0 + i];

b_0[i] = b[b0 + i];

}

// ==== c(0) = a(0) * b(0)

toom_cook3_process_01(a_0, b_0, out int[] c_0);

// ==== a(1) = a2 + a1 + a0, b(1) = b2 + b1 + b0

for (int i = 0; i < n1; i++)

{

a_1[i] = a[a2 + i] + a[a1 + i] + a[a0 + i];

b_1[i] = b[b2 + i] + b[b1 + i] + b[b0 + i];

}

// ==== c(1) = a(1) * b(1)

toom_cook3_process_01(a_1, b_1, out int[] c_1);

// ==== a(inf) = a2, b(inf) = b2

for (int i = 0; i < n1; i++)

{

a_inf[i] = a[a2 + i];

b_inf[i] = b[b2 + i];

}

// ==== c(inf) = a(inf) * b(inf)

toom_cook3_process_01(a_inf, b_inf, out int[] c_inf);

// ==== c4 = 6 * c(inf) / 6

for (int i = 0; i < n2; i++)

{

c4[i] = c_inf[i];

}

// ==== c3 = -c(-2) + 3 * c(-1) - 3 * c(0) + c(1) + 12 * c(inf) / 6

for (int i = 0; i < n2; i++)

{

c3[i] = -c_m2[i];

c3[i] += (c_m1[i] << 1) + c_m1[i];

c3[i] -= (c_0[i] << 1) + c_0[i];

c3[i] += c_1[i];

c3[i] += (c_inf[i] << 3) + (c_inf[i] << 2);

c3[i] /= 6;

}

// ==== c2 = 3 * c(-1) - 6 * c(0) + 3 * c(1) - 6 * c(inf) / 6

for (int i = 0; i < n2; i++)

{

c2[i] = (c_m1[i] << 1) + c_m1[i];

c2[i] -= (c_0[i] << 2) + (c_0[i] << 1);

c2[i] += (c_1[i] << 1) + c_1[i];

c2[i] -= (c_inf[i] << 2) + (c_inf[i] << 1);

c2[i] /= 6;

}

// ==== c1 = c(-2) - 6 * c(-1) + 3 * c(0) + 2 * c(1) - 12 * c(inf) / 6

for (int i = 0; i < n2; i++)

{

c1[i] = c_m2[i];

c1[i] -= (c_m1[i] << 2) + (c_m1[i] << 1);

c1[i] += (c_0[i] << 1) + c_0[i];

c1[i] += (c_1[i] << 1);

c1[i] -= (c_inf[i] << 3) + (c_inf[i] << 2);

c1[i] /= 6;

}

// ==== c0 = 6 * c(0) / 6

for (int i = 0; i < n2; i++)

{

c0[i] = c_0[i];

}

// ==== z = c4 * x^4 + c3 * x^3 + c2 * x^2 + c1 * x + c0

for (int i = 0; i < n2; i++)

{

z[i + n4] += c4[i];

z[i + n3] += c3[i];

z[i + n2] += c2[i];

z[i + n1] += c1[i];

z[i] += c0[i];

}

}

/// <summary>

/// 乘积和的进位计算

/// </summary>

/// <param name="a"></param>

/// <param name="n"></param>

/// <exception cref="Exception"></exception>

private static void toom_cook3_carry(int[] a, int n)

{

int cr = 0;

for (int i = 0; i < n; i++)

{

a[i] += cr;

if (a[i] < 0)

{

cr = -(-(a[i] + 1) / 10 + 1);

}

else

{

cr = a[i] / 10;

}

a[i] -= cr * 10;

}

if (cr != 0)

{

// Overflow

throw new Exception("OVERFLOW! cr=" + cr);

}

}

2 代码格式

cs 复制代码 
/// <summary>
/// 《小白学程序》第二十六课：大数（BigInteger）的Toom-Cook 3乘法
/// Toom-Cook 3-Way Multiplication
/// </summary>
/// <param name="a"></param>
/// <param name="b"></param>
/// <returns></returns>
public static string toom_cook3_multiply(string a, string b)
{
    int n = Math.Max(a.Length, b.Length);
    int[] ra = string_to_digitals(a, n);
    int[] rb = string_to_digitals(b, n);
    toom_cook3_process_00(ra, rb, out int[] rz);
    toom_cook3_carry(rz, n * 2);
    return digitals_to_string(rz);
}

/// <summary>
/// 短数字的乘法（常规乘法，小学生算法）
/// </summary>
/// <param name="a"></param>
/// <param name="b"></param>
/// <param name="z"></param>
private static void toom_cook3_normal(int[] a, int[] b, ref int[] z)
{
    int n = a.Length;
    for (int j = 0; j < n; j++)
    {
        for (int i = 0; i < n; i++)
        {
            z[j + i] += a[i] * b[j];
        }
    }
}

/// <summary>
/// 完全按原始C++代码改写；运行成功；
/// </summary>
/// <param name="a"></param>
/// <param name="b"></param>
/// <param name="z"></param>
private static void toom_cook3_process_01(int[] a, int[] b, out int[] z)
{
    int n = a.Length;
    int n1 = n * 1 / 3;
    int n2 = n * 2 / 3;
    int n3 = n * 3 / 3;
    int n4 = n * 4 / 3;

    z = new int[n * 2];
    if (n <= 9)
    {
        toom_cook3_normal(a, b, ref z);
        return;
    }

    // int *a0 = &a[0];
    // Multiplicand / right side array pointer
    int a0 = 0;
    // int *a1 = &a[tLen / 3];
    // Multiplicand / central array pointer
    int a1 = n1;
    // int *a2 = &a[tLen * 2/ 3];
    // Multiplicand / left side array pointer
    int a2 = n2;// n * 2 / 3;

    // int *b0 = &b[0];
    // Multiplier / right side array pointer
    int b0 = 0;
    // int *b1 = &b[tLen / 3];
    // Multiplier / central array pointer
    int b1 = n1;
    // int *b2 = &b[tLen * 2 / 3];
    // Multiplier / left side array pointer
    int b2 = n2;// n * 2 / 3;

    // int *c0 = &z[(tLen / 3) * 0];
    int[] c0 = new int[n2];
    int[] c1 = new int[n2];
    // int *c2 = &z[(tLen / 3) * 2];
    int[] c2 = new int[n2];
    int[] c3 = new int[n2];
    // int *c4 = &z[(tLen / 3) * 4];
    int[] c4 = new int[n2];

    int[] a_m2 = new int[n1];  // a(-2)
    int[] a_m1 = new int[n1];  // a(-1)
    int[] a_0 = new int[n1];   // a(0)
    int[] a_1 = new int[n1];   // a(1)
    int[] a_inf = new int[n1]; // a(inf)
    int[] b_m2 = new int[n1];  // b-2)
    int[] b_m1 = new int[n1];  // b-1)
    int[] b_0 = new int[n1];   // b(0)
    int[] b_1 = new int[n1];   // b(1)
    int[] b_inf = new int[n1]; // b(inf)

    // ==== a(-2) = 4 * a2 - 2 * a1 + a0, b(-2) = 4 * b2 - 2 * b1 + b0
    for (int i = 0; i < n1; i++)
    {
        a_m2[i] = (a[a2 + i] << 2) - (a[a1 + i] << 1) + a[a0 + i];
        b_m2[i] = (b[b2 + i] << 2) - (b[b1 + i] << 1) + b[b0 + i];
    }
    // ==== c(-2) = a(-2) * b(-2)
    toom_cook3_process_01(a_m2, b_m2, out int[] c_m2);

    // ==== a(-1) = a2 - a1 + a0, b(-1) = b2 - b1 + b0
    for (int i = 0; i < n1; i++)
    {
        a_m1[i] = a[a2 + i] - a[a1 + i] + a[a0 + i];
        b_m1[i] = b[b2 + i] - b[b1 + i] + b[b0 + i];
    }
    // ==== c(-1) = a(-1) * b(-1)
    toom_cook3_process_01(a_m1, b_m1, out int[] c_m1);

    // ==== a(0) = a0, b(0) = b0
    for (int i = 0; i < n1; i++)
    {
        a_0[i] = a[a0 + i];
        b_0[i] = b[b0 + i];
    }
    // ==== c(0) = a(0) * b(0)
    toom_cook3_process_01(a_0, b_0, out int[] c_0);

    // ==== a(1) = a2 + a1 + a0, b(1) = b2 + b1 + b0
    for (int i = 0; i < n1; i++)
    {
        a_1[i] = a[a2 + i] + a[a1 + i] + a[a0 + i];
        b_1[i] = b[b2 + i] + b[b1 + i] + b[b0 + i];
    }
    // ==== c(1) = a(1) * b(1)
    toom_cook3_process_01(a_1, b_1, out int[] c_1);

    // ==== a(inf) = a2, b(inf) = b2
    for (int i = 0; i < n1; i++)
    {
        a_inf[i] = a[a2 + i];
        b_inf[i] = b[b2 + i];
    }

    // ==== c(inf) = a(inf) * b(inf)
    toom_cook3_process_01(a_inf, b_inf, out int[] c_inf);

    // ==== c4 = 6 * c(inf) / 6
    for (int i = 0; i < n2; i++)
    {
        c4[i] = c_inf[i];
    }
    // ==== c3 = -c(-2) + 3 * c(-1) - 3 * c(0) + c(1) + 12 * c(inf) / 6
    for (int i = 0; i < n2; i++)
    {
        c3[i] = -c_m2[i];
        c3[i] += (c_m1[i] << 1) + c_m1[i];
        c3[i] -= (c_0[i] << 1) + c_0[i];
        c3[i] += c_1[i];
        c3[i] += (c_inf[i] << 3) + (c_inf[i] << 2);
        c3[i] /= 6;
    }
    // ==== c2 = 3 * c(-1) - 6 * c(0) + 3 * c(1) - 6 * c(inf) / 6
    for (int i = 0; i < n2; i++)
    {
        c2[i] = (c_m1[i] << 1) + c_m1[i];
        c2[i] -= (c_0[i] << 2) + (c_0[i] << 1);
        c2[i] += (c_1[i] << 1) + c_1[i];
        c2[i] -= (c_inf[i] << 2) + (c_inf[i] << 1);
        c2[i] /= 6;
    }
    // ==== c1 = c(-2) - 6 * c(-1) + 3 * c(0) + 2 * c(1) - 12 * c(inf) / 6
    for (int i = 0; i < n2; i++)
    {
        c1[i] = c_m2[i];
        c1[i] -= (c_m1[i] << 2) + (c_m1[i] << 1);
        c1[i] += (c_0[i] << 1) + c_0[i];
        c1[i] += (c_1[i] << 1);
        c1[i] -= (c_inf[i] << 3) + (c_inf[i] << 2);
        c1[i] /= 6;
    }
    // ==== c0 = 6 * c(0) / 6
    for (int i = 0; i < n2; i++)
    {
        c0[i] = c_0[i];
    }
    // ==== z = c4 * x^4 + c3 * x^3 + c2 * x^2 + c1 * x + c0
    for (int i = 0; i < n2; i++)
    {
        z[i + n4] += c4[i];
        z[i + n3] += c3[i];
        z[i + n2] += c2[i];
        z[i + n1] += c1[i];
        z[i] += c0[i];
    }
}


    /// <summary>
    /// 乘积和的进位计算
    /// </summary>
    /// <param name="a"></param>
    /// <param name="n"></param>
    /// <exception cref="Exception"></exception>
    private static void toom_cook3_carry(int[] a, int n)
    {
        int cr = 0;
        for (int i = 0; i < n; i++)
        {
            a[i] += cr;
            if (a[i] < 0)
            {
                cr = -(-(a[i] + 1) / 10 + 1);
            }
            else
            {
                cr = a[i] / 10;
            }
            a[i] -= cr * 10;
        }
        if (cr != 0)
        {
            // Overflow
            throw new Exception("OVERFLOW! cr=" + cr);
        }
    }
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