矩阵相乘的快速算法

矩阵相乘的快速算法

算法介绍

矩阵相乘在进行3D变换的时候是经常用到的。在应用中常用矩阵相乘的定义算法对其进行计算。这个算法用到了大量的循环和相乘运算，这使得算法效率不高。而矩阵相乘的计算效率很大程度上的影响了整个程序的运行速度，所以对矩阵相乘算法进行一些改进是必要的。

这里要介绍的矩阵算法称为斯特拉森方法，它是由v.斯特拉森在1969年提出的一个方法。

我们先讨论二阶矩阵的计算方法。

对于二阶矩阵

a11 a12 b11 b12

A = a21 a22

B = b21 b22

先计算下面7个量(1)

x1 = (a11 + a22) * (b11 + b22);

x2 = (a21 + a22) * b11;

x3 = a11 * (b12 - b22);

x4 = a22 * (b21 - b11);

x5 = (a11 + a12) * b22;

x6 = (a21 - a11) * (b11 + b12);

x7 = (a12 - a22) * (b21 + b22);

再设C = AB。根据矩阵相乘的规则，C的各元素为(2)

c11 = a11 * b11 + a12 * b21

c12 = a11 * b12 + a12 * b22

c21 = a21 * b11 + a22 * b21

c22 = a21 * b12 + a22 * b22

比较(1)(2)，C的各元素可以表示为(3)

c11 = x1 + x4 - x5 + x7

c12 = x3 + x5

c21 = x2 + x4

c22 = x1 + x3 - x2 + x6

根据以上的方法，我们就可以计算4阶矩阵了，先将4阶矩阵A和B划分成四块2阶矩阵，分别利用公式计算它们的乘积，再使用(1)(3)来计算出最后结果。

ma11 ma12 mb11 mb12

A4 = ma21 ma22 B4 = mb21 mb22

其中

a11 a12 a13 a14 b11 b12 b13 b14

ma11 = a21 a22 ma12 = a23 a24 mb11 = b21 b22 mb12 = b23 b24

a31 a32 a33 a34 b31 b32 b33 b34

ma21 = a41 a42 ma22 = a43 a44 mb21 = b41 b42 mb22 = b43 b44

实现

// 计算2X2矩阵

void Multiply2X2(float& fOut_11, float& fOut_12, float& fOut_21, float& fOut_22,

float f1_11, float f1_12, float f1_21, float f1_22,

float f2_11, float f2_12, float f2_21, float f2_22)

{

const float x1((f1_11 + f1_22) * (f2_11 + f2_22));

const float x2((f1_21 + f1_22) * f2_11);

const float x3(f1_11 * (f2_12 - f2_22));

const float x4(f1_22 * (f2_21 - f2_11));

const float x5((f1_11 + f1_12) * f2_22);

const float x6((f1_21 - f1_11) * (f2_11 + f2_12));

const float x7((f1_12 - f1_22) * (f2_21 + f2_22));

fOut_11 = x1 + x4 - x5 + x7;

fOut_12 = x3 + x5;

fOut_21 = x2 + x4;

fOut_22 = x1 - x2 + x3 + x6;

}

// 计算4X4矩阵

void Multiply(CLAYMATRIX& mOut, const CLAYMATRIX& m1, const CLAYMATRIX& m2) {

float fTmp[7][4];

// (ma11 + ma22) * (mb11 + mb22)

Multiply2X2(fTmp[0][0], fTmp[0][1], fTmp[0][2], fTmp[0][3],

m1._11 + m1._33, m1._12 + m1._34, m1._21 + m1._43, m1._22 + m1._44,

m2._11 + m2._33, m2._12 + m2._34, m2._21 + m2._43, m2._22 + m2._44);

// (ma21 + ma22) * mb11

Multiply2X2(fTmp[1][0], fTmp[1][1], fTmp[1][2], fTmp[1][3],

m1._31 + m1._33, m1._32 + m1._34, m1._41 + m1._43, m1._42 + m1._44,

m2._11, m2._12, m2._21, m2._22);

// ma11 * (mb12 - mb22)

Multiply2X2(fTmp[2][0], fTmp[2][1], fTmp[2][2], fTmp[2][3],

m1._11, m1._12, m1._21, m1._22,

m2._13 - m2._33, m2._14 - m2._34, m2._23 - m2._43, m2._24 - m2._44);

// ma22 * (mb21 - mb11)

Multiply2X2(fTmp[3][0], fTmp[3][1], fTmp[3][2], fTmp[3][3],

m1._33, m1._34, m1._43, m1._44,

m2._31 - m2._11, m2._32 - m2._12, m2._41 - m2._21, m2._42 - m2._22);

// (ma11 + ma12) * mb22

Multiply2X2(fTmp[4][0], fTmp[4][1], fTmp[4][2], fTmp[4][3],

m1._11 + m1._13, m1._12 + m1._14, m1._21 + m1._23, m1._22 + m1._24,

m2._33, m2._34, m2._43, m2._44);

// (ma21 - ma11) * (mb11 + mb12)

Multiply2X2(fTmp[5][0], fTmp[5][1], fTmp[5][2], fTmp[5][3],

m1._31 - m1._11, m1._32 - m1._12, m1._41 - m1._21, m1._42 - m1._22,

m2._11 + m2._13, m2._12 + m2._14, m2._21 + m2._23, m2._22 + m2._24);

// (ma12 - ma22) * (mb21 + mb22)

Multiply2X2(fTmp[6][0], fTmp[6][1], fTmp[6][2], fTmp[6][3],

m1._13 - m1._33, m1._14 - m1._34, m1._23 - m1._43, m1._24 - m1._44,

m2._31 + m2._33, m2._32 + m2._34, m2._41 + m2._43, m2._42 + m2._44);

// 第一块

mOut._11 = fTmp[0][0] + fTmp[3][0] - fTmp[4][0] + fTmp[6][0];

mOut._12 = fTmp[0][1] + fTmp[3][1] - fTmp[4][1] + fTmp[6][1];

mOut._21 = fTmp[0][2] + fTmp[3][2] - fTmp[4][2] + fTmp[6][2];