Question

使用以下代码，我使用cramer规则计算逆矩阵4x4，但是如何为NxN矩阵扩展此代码？

void PIII_Inverse_4x4(float* src) {
    __m128 minor0,minor1,minor2,minor3;
    __m128 row0,row1,row2,row3;
    __m128 det,tmp1;

    tmp1= _mm_loadh_pi(_mm_loadl_pi(tmp1, (__m64*)(src)), (__m64*)(src+ 4));
    row1= _mm_loadh_pi(_mm_loadl_pi(row1, (__m64*)(src+8)), (__m64*)(src+12));
    row0= _mm_shuffle_ps(tmp1, row1, 0x88);
    row1= _mm_shuffle_ps(row1, tmp1, 0xDD);
    tmp1= _mm_loadh_pi(_mm_loadl_pi(tmp1, (__m64*)(src+ 2)), (__m64*)(src+ 6));
    row3= _mm_loadh_pi(_mm_loadl_pi(row3, (__m64*)(src+10)), (__m64*)(src+14));
    row2= _mm_shuffle_ps(tmp1, row3, 0x88);
    row3= _mm_shuffle_ps(row3, tmp1, 0xDD);

    tmp1= _mm_mul_ps(row2, row3);
    tmp1= _mm_shuffle_ps(tmp1, tmp1, 0xB1);
    minor0= _mm_mul_ps(row1, tmp1);
    minor1= _mm_mul_ps(row0, tmp1);
    tmp1= _mm_shuffle_ps(tmp1, tmp1, 0x4E);
    minor0= _mm_sub_ps(_mm_mul_ps(row1, tmp1), minor0);
    minor1= _mm_sub_ps(_mm_mul_ps(row0, tmp1), minor1);
    minor1= _mm_shuffle_ps(minor1, minor1, 0x4E);

    tmp1= _mm_mul_ps(row1, row2);
    tmp1= _mm_shuffle_ps(tmp1, tmp1, 0xB1);
    minor0= _mm_add_ps(_mm_mul_ps(row3, tmp1), minor0);
    minor3= _mm_mul_ps(row0, tmp1);
    tmp1= _mm_shuffle_ps(tmp1, tmp1, 0x4E);
    minor0= _mm_sub_ps(minor0, _mm_mul_ps(row3, tmp1));
    minor3= _mm_sub_ps(_mm_mul_ps(row0, tmp1), minor3);
    minor3= _mm_shuffle_ps(minor3, minor3, 0x4E);

    tmp1= _mm_mul_ps(_mm_shuffle_ps(row1, row1, 0x4E), row3);
    tmp1= _mm_shuffle_ps(tmp1, tmp1, 0xB1);
    row2= _mm_shuffle_ps(row2, row2, 0x4E);
    minor0= _mm_add_ps(_mm_mul_ps(row2, tmp1), minor0);
    minor2= _mm_mul_ps(row0, tmp1);
    tmp1= _mm_shuffle_ps(tmp1, tmp1, 0x4E);
    minor0= _mm_sub_ps(minor0, _mm_mul_ps(row2, tmp1));
    minor2= _mm_sub_ps(_mm_mul_ps(row0, tmp1), minor2);
    minor2= _mm_shuffle_ps(minor2, minor2, 0x4E);

    tmp1= _mm_mul_ps(row0, row1);
    tmp1= _mm_shuffle_ps(tmp1, tmp1, 0xB1);

    minor2= _mm_add_ps(_mm_mul_ps(row3, tmp1), minor2);
    minor3= _mm_sub_ps(_mm_mul_ps(row2, tmp1), minor3);
    tmp1= _mm_shuffle_ps(tmp1, tmp1, 0x4E);
    minor2= _mm_sub_ps(_mm_mul_ps(row3, tmp1), minor2);
    minor3= _mm_sub_ps(minor3, _mm_mul_ps(row2, tmp1));

    tmp1= _mm_mul_ps(row0, row3);
    tmp1= _mm_shuffle_ps(tmp1, tmp1, 0xB1);
    minor1= _mm_sub_ps(minor1, _mm_mul_ps(row2, tmp1));
    minor2= _mm_add_ps(_mm_mul_ps(row1, tmp1), minor2);
    tmp1= _mm_shuffle_ps(tmp1, tmp1, 0x4E);
    minor1= _mm_add_ps(_mm_mul_ps(row2, tmp1), minor1);
    minor2= _mm_sub_ps(minor2, _mm_mul_ps(row1, tmp1));

    tmp1= _mm_mul_ps(row0, row2);
    tmp1= _mm_shuffle_ps(tmp1, tmp1, 0xB1);
    minor1= _mm_add_ps(_mm_mul_ps(row3, tmp1), minor1);
    minor3= _mm_sub_ps(minor3, _mm_mul_ps(row1, tmp1));
    tmp1= _mm_shuffle_ps(tmp1, tmp1, 0x4E);
    minor1= _mm_sub_ps(minor1, _mm_mul_ps(row3, tmp1));
    minor3= _mm_add_ps(_mm_mul_ps(row1, tmp1), minor3);
    // -----------------------------------------------
    // -----------------------------------------------
    // -----------------------------------------------
    det= _mm_mul_ps(row0, minor0);
    det= _mm_add_ps(_mm_shuffle_ps(det, det, 0x4E), det);
    det= _mm_add_ss(_mm_shuffle_ps(det, det, 0xB1), det);
    tmp1= _mm_rcp_ss(det);
    det= _mm_sub_ss(_mm_add_ss(tmp1, tmp1), _mm_mul_ss(det, _mm_mul_ss(tmp1, tmp1)));
    det= _mm_shuffle_ps(det, det, 0x00);
    minor0 = _mm_mul_ps(det, minor0);
    _mm_storel_pi((__m64*)(src), minor0);
    _mm_storeh_pi((__m64*)(src+2), minor0);
    minor1 = _mm_mul_ps(det, minor1);
    _mm_storel_pi((__m64*)(src+4), minor1);
    _mm_storeh_pi((__m64*)(src+6), minor1);
    minor2 = _mm_mul_ps(det, minor2);
    _mm_storel_pi((__m64*)(src+ 8), minor2);
    _mm_storeh_pi((__m64*)(src+10), minor2);
    minor3 = _mm_mul_ps(det, minor3);
    _mm_storel_pi((__m64*)(src+12), minor3);
    _mm_storeh_pi((__m64*)(src+14), minor3);
}

我在google上搜索过，但是我没有找到任何有用的东西......我还搜索了反向矩阵的高斯 - 约旦方法，但没有...

Answer 1

对于矩阵求逆，使用Choleksy decomposition比使用Gauss-Jordan消除更有效率参见本文 Matrix Inversion Using Cholesky Decomposition。

I implemented Choleksy decomposition with SSE, AVX, and FMA as well as with multiple threads。我获得了英特尔MKL大约50％的性能。我目前正在重写我的内核代码，所以我希望尽快接近MKL的表现。

请注意，反演通常不需要求解方程组。在我的情况下，我只需要Cholesky分解，然后我使用向后和向前替换来解决。

- SSE - 与Crax 4x4的矩阵逆，如何扩展NxN？

1 个答案: