[英]Cache friendly method to multiply two matrices
I intend to multiply 2 matrices using the cache-friendly method ( that would lead to less number of misses) 我打算使用缓存友好的方法将2个矩阵相乘(这将导致更少的未命中)
I found out that this can be done with a cache friendly transpose function. 我发现这可以通过缓存友好的转置函数来完成。
But I am not able to find this algorithm. 但我无法找到这个算法。 Can I know how to achieve this? 我可以知道如何实现这一目标吗?
The word you are looking for is thrashing . 你正在寻找的是捶打 。 Searching for thrashing matrix multiplication in Google yields more results . 在Google中搜索颠簸矩阵乘法会产生更多结果 。
A standard multiplication algorithm for c = a*b would look like c = a * b的标准乘法算法看起来像
void multiply(double[,] a, double[,] b, double[,] c)
{
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
for (int k = 0; k < n; k++)
C[i, j] += a[i, k] * b[k, j];
}
Basically, navigating the memory fastly in large steps is detrimental to performance. 基本上,以大步骤快速导航存储器对性能是不利的。 The access pattern for k in B[ k , j] is doing exactly that. B [ k ,j]中k的访问模式正是这样做的。 So instead of jumping around in the memory, we may rearrange the operations such that the most inner loops operate only on the second access index of the matrices: 因此,我们可以重新排列操作,使得大多数内部循环仅在矩阵的第二个访问索引上运行,而不是在内存中跳转:
void multiply(double[,] a, double[,] B, double[,] c)
{
for (i = 0; i < n; i++)
{
double t = a[i, 0];
for (int j = 0; j < n; j++)
c[i, j] = t * b[0, j];
for (int k = 1; k < n; k++)
{
double s = 0;
for (int j = 0; j < n; j++ )
s += a[i, k] * b[k, j];
c[i, j] = s;
}
}
}
This was the example given on that page. 这是该页面上给出的示例。 However, another option is to copy the contents of B[k, *] into an array beforehand and use this array in the inner loop calculations. 但是,另一个选择是预先将B [k,*]的内容复制到数组中,并在内部循环计算中使用此数组。 This approach is usually much faster than the alternatives, even if it involves copying data around. 这种方法通常比替代方法快得多 ,即使它涉及复制数据。 Even if this might seem counter-intuitive, please feel free to try for yourself. 即使这看似违反直觉,请随意尝试。
void multiply(double[,] a, double[,] b, double[,] c)
{
double[] Bcolj = new double[n];
for (int j = 0; j < n; j++)
{
for (int k = 0; k < n; k++)
Bcolj[k] = b[k, j];
for (int i = 0; i < n; i++)
{
double s = 0;
for (int k = 0; k < n; k++)
s += a[i,k] * Bcolj[k];
c[j, i] = s;
}
}
}
@Cesar's answer is not correct. @Cesar的回答不正确。 For example, the inner loop 例如,内循环
for (int k = 0; k < n; k++)
s += a[i,k] * Bcolj[k];
goes through the i-th column of a. 通过a的第i列。
The following code should ensure we always visit data row by row. 以下代码应确保我们始终逐行访问数据。
void multiply(const double (&a)[I][K],
const double (&b)[K][J],
double (&c)[I][J])
{
for (int j=0; j<J; ++j) {
// iterates the j-th row of c
for (int i=0; i<I; ++i) {
c[i][j] = 0;
}
// iterates the j-th row of b
for (int k=0; k<K; ++k) {
double t = b[k][j];
// iterates the j-th row of c
// iterates the k-th row of a
for (int i=0; i<I; ++i) {
c[i][j] += a[i][k] * t;
}
}
}
}
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