算法导论（第三版）Exercises4.2(第四章二节）

4.2-1（计算结果）

18  14

62  66

4.2-2（Strassen算法计算矩阵乘法）

void multiplyMatrix(int a[], int b[], int n, int result[])
{
    int i, j, dim=n/2;
    int n1=(n/2) * (n/2);
    int a11[n1], a12[n1], a21[n1], a22[n1];
    int b11[n1], b12[n1], b21[n1], b22[n1];

    int s1[n1], s2[n1], s3[n1], s4[n1], s5[n1];
    int s6[n1], s7[n1], s8[n1], s9[n1], s10[n1];

    int p1[n1], p2[n1], p3[n1], p4[n1];
    int p5[n1], p6[n1], p7[n1];

    int c11[n1], c12[n1], c21[n1], c22[n1];

    if(n == 1)
    {
        result[0] = a[0] * b[0];
        return;
    }
    if(n%2 != 0)
    {
        printf("wrong array!
");
        return;
    }

    divideMatrix(a, n, a11, a12, a21, a22);
    divideMatrix(b, n, b11, b12, b21, b22);
    subtractMatrix(b12, b22, dim, s1);
    addMatrix(a11, a12, dim, s2);
    addMatrix(a21, a22, dim, s3);
    subtractMatrix(b21, b11, dim, s4);
    addMatrix(a11, a22, dim, s5);
    addMatrix(b11, b22, dim, s6);
    subtractMatrix(a12, a22, dim, s7);
    addMatrix(b21, b22, dim, s8);
    subtractMatrix(a11, a21, dim, s9);
    addMatrix(b11, b12, dim, s10);

    multiplyMatrix(a11, s1, dim, p1);
    multiplyMatrix(s2, b22, dim, p2);
    multiplyMatrix(s3, b11, dim, p3);
    multiplyMatrix(a22, s4, dim, p4);
    multiplyMatrix(s5, s6, dim, p5);
    multiplyMatrix(s7, s8, dim, p6);
    multiplyMatrix(s9, s10, dim, p7);

    addMatrix(p5, p4, dim, c11);
    subtractMatrix(c11, p2, dim, c11);
    addMatrix(c11, p6, dim, c11);
    addMatrix(p1, p2, dim, c12);
    addMatrix(p3, p4, dim, c21);
    addMatrix(p5, p1, dim, c22);
    subtractMatrix(c22, p3, dim, c22);
    subtractMatrix(c22, p7, dim, c22);

    combineMatrix(c11, c12, c21, c22, dim, result);
}

void addMatrix(int a[], int b[], int n, int result[])
{
    int i;
    for(i=0; i<n*n; i++) result[i] = a[i] + b[i];
}

void subtractMatrix(int a[], int b[], int n, int result[])
{
    int i;
    for(i=0; i<n*n; i++) result[i] = a[i] - b[i];
}

void divideMatrix(int a[], int n, int a11[], int a12[], int a21[], int a22[])
{
    int i, mid, j, k, h;
    mid = n / 2;
    for(i=0; i<mid; i++)
    {
        for(j=0; j<mid; j++)
        {
            h = i * mid + j;
            k = i * n + j;
            a11[h] = a[k];
            a12[h] = a[k+mid];
            a21[h] = a[k+n*n/2];
            a22[h] = a[k+n*n/2+mid];
        }
    }
}

void combineMatrix(int a11[], int a12[], int a21[], int a22[], int n, int result[])
{
    int i, j, h, k, dim, mid;
    mid = n;
    dim = mid * 2;
    for(i=0; i<mid; i++)
    {
        for(j=0; j<mid; j++)
        {
            h = i * mid + j;
            k = i * dim + j;
            result[k] = a11[h];
            result[k+mid] = a12[h];
            result[k+n*n*2] = a21[h];
            result[k+n*n*2+mid] = a22[h];
        }
    }
}

4.2-3

当矩阵的n不是2的指数时，添加0补足，即可按上述算法运行

4.2-4

21

4.2-5

72x72的那个最佳，比Strassen算法好

4.2-6

当knXn矩阵时，按4.2-3的方法处理，同样nXkn也一样，可以看出nXkn计算时间短

4.2-7(用数组存储复数）

void multiplyComplex(double c1[], double c2[], double result[])
{
    double p1, p2, p3;
    p1 = c1[0] + c1[1];
    p2 = c2[0] + c2[1];
    p1 = p1 * p2;
    p2 = c1[0] * c2[0];
    p3 = c1[1] * c2[1];
    result[0] = p2 - p3;
    result[1] = p1 - p2 - p3;
}