在 OpenCL C 中编写快速线性系统求解器
Writing a fast linear system solver in OpenCL C
我正在编写一个 OpenCL 内核，它将涉及求解一个线性系统。目前我的内核太慢了，提高线性系统部分的性能似乎是一个很好的起点。
我还应该指出，我并不是要让我的线性求解器平行，我正在处理的问题在宏观层面上已经令人尴尬地平行了。
以下是我编写的 C 代码，用于使用部分旋转的高斯消去法求解 Ax=b，
#import <stdio.h>
#import <math.h>
#import <time.h>
#define K 50
// Solve the system Ax=b using Gaussian elimination with partial pivoting.
void linear_solve(float A[K * K], float b[K])
{
for (long j=0; j<K; j++)
{
// Begin partial pivoting.
float maxval = fabs(A[K * j + j]);
long maxrow = j;
for (long i=j+1; i<K; i++)
{
if (fabs(A[K * j + i]) > maxval)
{
maxval = fabs(A[K * j + i]);
maxrow = i;
}
}
for (long l=0; l<K; l++)
{
float A_temp = A[K * l + maxrow];
A[K * l + maxrow] = A[K * l + j];
A[K * l + j] = A_temp;
}
float b_temp = b[maxrow];
b[maxrow] = b[j];
b[j] = b_temp;
// End partial pivoting.
// Begin putting [A; b] into row echelon form.
for (long i=j; i<K-1; i++)
{
float c = -A[K * j + (i + 1)] / A[K * j + j];
for (long l=j+1; l<K; l++)
A[K * l + (i + 1)] += c * A[K * l + j];
b[i + 1] += c * b[j];
}
// End putting [A; b] into row echelon form.
}
// Begin backsolving for x (by overwritting b).
for (long j=K-1; j>0; j--)
for (long i=j-1; i>=0; i--)
b[i] -= b[j] * A[K * j + i] / A[K * j + j];
for (long j=0; j<K; j++)
b[j] *= 1 / A[K * j + j];
// End backsolving for x.
}
int main()
{
int i, j;
float A[K * K] = {38, 49, 38, 73, 70, 71, 33, 24, 14, 82, 46, 99, 82, 36, 21, 32, 48, 40, 27, 60, 31, 15, 38, 88, 95, 57, 36, 86, 42, 56, 1, 37, 73, 7, 92, 93, 16, 95, 59, 76, 18, 42, 57, 9, 14, 40, 68, 61, 8, 26, 90, 33, 95, 8, 5, 87, 66, 84, 45, 78, 27, 16, 9, 83, 46, 61, 74, 44, 17, 21, 21, 53, 96, 49, 58, 67, 73, 60, 18, 40, 32, 68, 68, 21, 57, 86, 69, 7, 80, 10, 36, 46, 94, 59, 41, 80, 70, 2, 90, 57, 92, 50, 92, 98, 88, 14, 39, 80, 68, 78, 49, 40, 54, 51, 68, 80, 95, 22, 37, 88, 10, 30, 54, 7, 84, 99, 42, 94, 75, 45, 22, 41, 75, 38, 54, 97, 64, 62, 6, 48, 92, 49, 72, 5, 75, 67, 24, 55, 76, 17, 62, 19, 75, 41, 63, 97, 19, 83, 69, 12, 43, 94, 48, 92, 94, 54, 76, 11, 99, 96, 20, 29, 43, 97, 86, 23, 55, 2, 75, 61, 17, 45, 88, 79, 9, 26, 1, 3, 10, 91, 94, 85, 13, 58, 3, 53, 24, 76, 9, 2, 33, 34, 51, 65, 100, 67, 84, 21, 77, 17, 88, 65, 2, 46, 1, 18, 15, 57, 1, 88, 60, 64, 39, 36, 79, 89, 51, 39, 98, 67, 62, 34, 56, 98, 74, 52, 93, 11, 87, 45, 48, 82, 87, 5, 97, 65, 1, 81, 39, 85, 33, 26, 24, 90, 41, 69, 74, 43, 21, 54, 91, 94, 78, 41, 17, 11, 71, 25, 72, 52, 36, 27, 100, 48, 67, 52, 94, 44, 94, 91, 83, 95, 76, 19, 70, 34, 87, 67, 62, 67, 81, 55, 81, 45, 68, 1, 56, 95, 76, 38, 72, 88, 37, 64, 29, 16, 19, 81, 36, 18, 25, 28, 21, 17, 57, 51, 22, 87, 61, 39, 56, 51, 65, 44, 59, 3, 75, 98, 5, 21, 48, 95, 53, 23, 96, 4, 11, 11, 77, 21, 58, 78, 9, 93, 81, 17, 77, 97, 97, 44, 96, 26, 35, 89, 73, 26, 37, 3, 51, 76, 14, 67, 45, 92, 52, 83, 43, 91, 20, 62, 4, 48, 75, 35, 17, 65, 6, 98, 2, 78, 69, 39, 30, 57, 27, 49, 8, 71, 46, 82, 16, 62, 57, 69, 94, 15, 56, 15, 29, 42, 93, 96, 57, 2, 63, 23, 57, 54, 47, 88, 40, 1, 90, 48, 1, 4, 26, 32, 12, 97, 38, 62, 72, 92, 71, 72, 34, 93, 84, 56, 20, 33, 53, 42, 7, 54, 98, 37, 27, 2, 13, 88, 30, 24, 91, 22, 95, 100, 53, 53, 31, 91, 95, 9, 36, 89, 25, 60, 28, 47, 61, 81, 41, 47, 88, 6, 46, 83, 4, 48, 73, 88, 8, 83, 78, 18, 21, 75, 6, 90, 87, 92, 18, 71, 5, 82, 36, 2, 50, 86, 49, 72, 92, 67, 41, 38, 81, 37, 67, 93, 99, 51, 79, 95, 76, 85, 90, 27, 93, 44, 79, 97, 7, 11, 52, 76, 61, 23, 52, 97, 58, 74, 87, 58, 70, 77, 97, 74, 85, 65, 71, 79, 91, 36, 92, 35, 97, 9, 6, 38, 90, 46, 84, 98, 65, 4, 89, 9, 72, 55, 3, 21, 77, 43, 76, 83, 34, 16, 33, 21, 6, 28, 98, 27, 86, 93, 66, 55, 34, 76, 93, 42, 1, 36, 82, 82, 13, 45, 48, 8, 4, 66, 51, 32, 68, 81, 49, 70, 93, 73, 89, 16, 76, 95, 90, 37, 83, 28, 40, 14, 3, 18, 27, 34, 24, 53, 42, 24, 57, 93, 48, 43, 91, 28, 75, 86, 47, 40, 61, 20, 34, 81, 31, 62, 20, 75, 80, 81, 95, 75, 14, 8, 89, 13, 7, 9, 27, 80, 24, 52, 27, 75, 4, 58, 20, 82, 89, 31, 100, 48, 57, 73, 34, 52, 24, 26, 64, 18, 90, 74, 17, 58, 8, 44, 43, 56, 56, 51, 58, 56, 4, 87, 80, 24, 100, 47, 72, 60, 41, 2, 26, 81, 17, 57, 28, 6, 21, 4, 99, 92, 42, 37, 22, 45, 5, 93, 72, 27, 91, 13, 44, 93, 6, 100, 31, 17, 78, 16, 96, 32, 57, 45, 95, 76, 92, 3, 77, 84, 92, 87, 63, 42, 70, 79, 77, 90, 16, 100, 82, 61, 23, 67, 55, 45, 38, 27, 95, 19, 10, 4, 53, 75, 62, 1, 99, 62, 94, 30, 95, 65, 35, 62, 25, 59, 26, 62, 98, 50, 73, 31, 11, 89, 20, 1, 74, 45, 49, 55, 78, 49, 82, 35, 9, 45, 100, 99, 87, 10, 56, 79, 85, 89, 8, 9, 53, 87, 13, 27, 95, 81, 7, 71, 63, 44, 38, 84, 40, 87, 79, 54, 42, 58, 49, 85, 49, 6, 55, 83, 93, 52, 63, 76, 52, 40, 91, 36, 74, 70, 92, 92, 67, 57, 51, 74, 22, 35, 22, 48, 60, 86, 87, 79, 18, 65, 1, 36, 65, 91, 24, 33, 71, 52, 43, 20, 100, 94, 68, 19, 93, 66, 89, 45, 39, 97, 57, 67, 51, 92, 20, 97, 45, 32, 10, 82, 86, 2, 8, 27, 15, 60, 7, 6, 90, 71, 40, 91, 10, 16, 39, 40, 32, 2, 11, 5, 81, 31, 72, 41, 7, 89, 89, 85, 28, 67, 54, 44, 47, 26, 44, 51, 50, 65, 41, 68, 17, 88, 45, 43, 8, 11, 79, 10, 99, 58, 42, 75, 75, 86, 73, 24, 33, 15, 46, 84, 33, 27, 96, 14, 25, 11, 67, 48, 51, 85, 61, 87, 71, 85, 62, 32, 71, 15, 56, 6, 20, 43, 64, 97, 81, 94, 94, 61, 39, 46, 99, 37, 66, 40, 17, 74, 44, 6, 2, 11, 53, 44, 75, 29, 58, 77, 66, 96, 82, 13, 32, 43, 13, 36, 10, 39, 54, 39, 79, 22, 4, 41, 19, 44, 37, 73, 76, 84, 78, 94, 13, 98, 26, 56, 55, 51, 38, 37, 60, 55, 92, 19, 53, 48, 4, 7, 85, 82, 8, 60, 34, 67, 98, 76, 38, 14, 20, 62, 41, 58, 29, 70, 71, 16, 60, 26, 8, 64, 92, 17, 26, 40, 12, 59, 69, 97, 63, 52, 81, 27, 10, 99, 73, 74, 68, 8, 44, 70, 38, 65, 3, 27, 80, 90, 8, 64, 98, 89, 10, 45, 42, 55, 61, 49, 45, 82, 48, 27, 22, 16, 50, 58, 41, 92, 64, 54, 35, 65, 23, 66, 22, 9, 68, 79, 45, 69, 71, 94, 24, 41, 55, 48, 84, 12, 80, 71, 41, 91, 77, 83, 2, 12, 55, 21, 100, 99, 65, 20, 77, 37, 29, 75, 6, 59, 84, 25, 70, 40, 31, 73, 26, 61, 77, 16, 73, 41, 5, 83, 51, 9, 60, 97, 44, 21, 21, 87, 20, 74, 91, 43, 10, 69, 67, 14, 30, 71, 31, 20, 21, 98, 58, 21, 51, 83, 20, 69, 70, 13, 8, 62, 66, 28, 46, 75, 66, 65, 21, 32, 83, 7, 62, 4, 46, 98, 89, 20, 11, 57, 93, 72, 14, 80, 57, 10, 53, 67, 52, 88, 21, 97, 67, 42, 14, 86, 5, 12, 44, 35, 82, 3, 69, 87, 32, 10, 15, 54, 40, 60, 11, 46, 23, 77, 97, 46, 61, 90, 74, 82, 50, 15, 73, 59, 83, 68, 52, 54, 54, 89, 99, 44, 7, 85, 29, 65, 87, 20, 57, 5, 45, 98, 36, 98, 36, 99, 3, 54, 78, 100, 91, 73, 77, 63, 30, 11, 31, 21, 12, 78, 66, 36, 6, 50, 27, 55, 97, 79, 85, 29, 91, 72, 64, 18, 78, 77, 93, 74, 76, 33, 68, 71, 48, 10, 4, 19, 32, 53, 87, 75, 11, 25, 71, 23, 55, 16, 74, 28, 66, 90, 49, 75, 95, 19, 50, 75, 49, 52, 28, 57, 90, 20, 77, 52, 9, 42, 4, 20, 49, 78, 99, 78, 38, 100, 90, 7, 12, 8, 35, 26, 49, 54, 78, 43, 86, 23, 55, 11, 79, 20, 56, 61, 26, 81, 42, 93, 4, 3, 84, 3, 55, 46, 27, 67, 74, 28, 100, 44, 5, 14, 65, 22, 71, 13, 61, 65, 53, 14, 44, 53, 67, 69, 2, 76, 76, 90, 63, 21, 46, 46, 96, 19, 40, 12, 22, 45, 98, 6, 81, 7, 70, 51, 16, 62, 66, 33, 21, 69, 34, 24, 92, 23, 14, 51, 84, 36, 73, 83, 45, 52, 93, 20, 21, 61, 58, 75, 85, 36, 92, 29, 26, 100, 86, 79, 46, 43, 95, 9, 8, 98, 29, 27, 70, 93, 60, 20, 14, 10, 77, 71, 12, 38, 91, 59, 57, 84, 77, 15, 81, 17, 10, 42, 89, 4, 72, 16, 85, 27, 80, 85, 85, 9, 94, 3, 59, 30, 43, 30, 87, 20, 19, 33, 92, 8, 52, 46, 67, 26, 76, 3, 21, 71, 10, 37, 49, 61, 15, 70, 57, 66, 55, 52, 87, 36, 18, 30, 69, 28, 68, 26, 82, 86, 87, 16, 15, 46, 92, 54, 100, 92, 89, 52, 97, 53, 21, 31, 51, 31, 17, 46, 68, 53, 93, 64, 87, 43, 39, 94, 2, 38, 30, 87, 35, 53, 97, 28, 54, 58, 42, 55, 23, 27, 2, 27, 4, 78, 31, 14, 87, 21, 75, 26, 28, 67, 56, 65, 80, 10, 21, 48, 71, 52, 24, 67, 38, 62, 68, 93, 17, 56, 85, 87, 75, 62, 68, 45, 88, 49, 97, 78, 14, 94, 3, 67, 86, 9, 24, 92, 2, 12, 89, 73, 94, 63, 89, 65, 92, 61, 100, 90, 44, 57, 17, 74, 59, 5, 63, 5, 73, 46, 76, 69, 12, 97, 91, 9, 6, 61, 37, 5, 20, 39, 32, 19, 14, 46, 2, 46, 41, 28, 39, 29, 41, 59, 25, 97, 94, 63, 31, 64, 63, 72, 41, 46, 58, 79, 79, 35, 49, 42, 43, 82, 32, 41, 37, 84, 96, 100, 33, 87, 38, 89, 97, 25, 56, 61, 4, 100, 9, 83, 66, 77, 65, 22, 81, 52, 27, 6, 79, 29, 34, 15, 64, 22, 80, 61, 10, 74, 1, 68, 80, 74, 86, 98, 9, 24, 76, 57, 23, 5, 50, 7, 11, 80, 39, 10, 75, 38, 73, 8, 47, 3, 92, 90, 51, 42, 22, 45, 63, 27, 62, 78, 38, 5, 46, 46, 80, 51, 6, 43, 43, 7, 13, 50, 10, 64, 4, 67, 94, 69, 58, 58, 77, 71, 42, 80, 35, 15, 34, 65, 23, 43, 21, 24, 69, 24, 37, 68, 11, 38, 18, 12, 37, 41, 81, 12, 3, 91, 44, 98, 5, 1, 90, 53, 100, 90, 26, 36, 23, 14, 76, 23, 70, 58, 7, 35, 42, 11, 19, 48, 11, 24, 61, 49, 52, 69, 68, 82, 11, 57, 87, 65, 68, 54, 69, 39, 99, 1, 86, 44, 35, 36, 58, 73, 17, 14, 14, 87, 20, 57, 11, 65, 98, 77, 10, 51, 45, 50, 28, 56, 23, 64, 6, 11, 15, 93, 32, 77, 45, 57, 84, 49, 66, 98, 71, 8, 35, 62, 23, 82, 30, 75, 41, 15, 52, 22, 93, 68, 12, 83, 76, 19, 93, 67, 19, 35, 76, 49, 95, 40, 21, 78, 76, 86, 26, 31, 85, 15, 29, 82, 68, 54, 29, 70, 79, 93, 35, 2, 60, 78, 74, 32, 77, 94, 21, 21, 87, 48, 58, 76, 5, 87, 41, 6, 74, 83, 2, 56, 8, 2, 81, 3, 59, 7, 49, 62, 72, 98, 81, 68, 6, 82, 20, 97, 71, 16, 10, 58, 37, 98, 49, 23, 61, 80, 15, 77, 26, 56, 99, 21, 19, 60, 80, 61, 31, 6, 59, 70, 7, 87, 41, 9, 2, 34, 43, 84, 12, 24, 67, 63, 40, 78, 3, 100, 22, 100, 61, 59, 92, 26, 9, 39, 56, 93, 74, 47, 21, 71, 67, 81, 40, 74, 56, 34, 35, 82, 94, 35, 35, 15, 52, 44, 5, 83, 30, 10, 18, 65, 31, 45, 49, 100, 41, 26, 51, 3, 86, 17, 62, 13, 92, 58, 76, 53, 34, 81, 98, 57, 99, 81, 67, 23, 25, 99, 88, 62, 99, 37, 85, 17, 60, 23, 56, 97, 65, 41, 91, 16, 90, 47, 86, 56, 99, 44, 28, 18, 89, 27, 43, 43, 14, 64, 96, 8, 92, 74, 65, 24, 26, 96, 92, 19, 57, 24, 25, 3, 80, 99, 89, 78, 78, 80, 89, 27, 6, 49, 78, 81, 75, 99, 21, 64, 51, 98, 32, 53, 59, 74, 33, 1, 93, 9, 1, 24, 15, 8, 55, 76, 51, 98, 41, 77, 48, 81, 47, 76, 47, 65, 25, 2, 80, 67, 9, 85, 18, 73, 35, 50, 69, 46, 33, 14, 47, 25, 93, 28, 39, 12, 87, 85, 81, 16, 51, 91, 93, 32, 60, 55, 43, 54, 32, 57, 4, 30, 20, 15, 96, 64, 3, 99, 41, 5, 78, 28, 52, 39, 45, 41, 54, 1, 13, 53, 84, 75, 24, 100, 44, 8, 18, 46, 42, 86, 65, 27, 74, 1, 75, 99, 90, 33, 31, 4, 22, 17, 30, 44, 36, 72, 47, 75, 100, 47, 85, 86, 59, 37, 32, 30, 67, 98, 94, 85, 93, 1, 81, 60, 33, 97, 88, 73, 68, 8, 35, 30, 83, 19, 99, 74, 21, 93, 42, 80, 95, 27, 65, 24, 73, 31, 43, 92, 81, 24, 70, 67, 78, 48, 47, 70, 76, 12, 79, 89, 7, 28, 83, 78, 22, 25, 32, 17, 4, 68, 42, 15, 1, 3, 18, 43, 75, 48, 84, 17, 60, 100, 73, 59, 80, 68, 13, 89, 7, 93, 16, 22, 1, 58, 92, 87, 90, 23, 95, 76, 67, 10, 14, 70, 17, 99, 77, 6, 63, 69, 2, 93, 27, 29, 88, 39, 35, 25, 50, 91, 13, 16, 91, 50, 53, 54, 12, 53, 25, 11, 6, 10, 44, 36, 87, 67, 69, 5, 5, 78, 25, 19, 24, 50, 88, 62, 24, 89, 39, 86, 6, 7, 70, 56, 92, 18, 76, 57, 50, 28, 71, 50, 74, 19, 89, 49, 8, 76, 92, 80, 41, 34, 33, 63, 88, 31, 95, 97, 71, 52, 36, 26, 99, 72, 50, 76, 33, 62, 79, 11, 76, 54, 64, 42, 76, 5, 45, 79, 61, 39, 66, 72, 74, 76, 25, 63, 35, 100, 42, 61, 12, 9, 41, 95, 90, 48, 24, 8, 66, 65, 29, 74, 97, 54, 51, 31, 31, 51, 30, 63, 32, 70, 79, 49, 7, 35, 53, 76, 83, 62, 20, 13, 92, 95, 40, 99, 10, 98, 13, 7, 88, 16, 40, 10, 22, 29, 88, 64, 39, 13, 26, 12, 27, 69, 70, 23, 41, 67, 50, 96, 24, 97, 29, 31, 42, 27, 90, 50, 69, 42, 92, 22, 88, 23, 35, 83, 82, 74, 50, 72, 98, 94, 94, 46, 82, 16, 35, 88, 46, 89, 77, 86, 19, 17, 20, 5, 13, 25, 69, 79, 90, 55, 88, 71, 13, 30};
float b[K] = {66, 97, 50, 69, 24, 42, 23, 82, 25, 79, 66, 26, 76, 25, 75, 25, 43, 40, 55, 8, 20, 53, 66, 94, 57, 10, 39, 70, 5, 57, 22, 36, 45, 94, 24, 44, 89, 41, 14, 87, 9, 46, 74, 23, 72, 62, 52, 74, 36, 13};
clock_t begin = clock();
linear_solve(A, b);
clock_t end = clock();
double time_spent = (double)(end - begin) / CLOCKS_PER_SEC;
printf("seconds: %f\n", time_spent);
printf("Result vector is: ");
for (i=0; i<K; i++)
{
printf("%f,", b[i]);
}
printf("\n");
return 0;
}
以下是求解 Ax=b 的 Julia 代码，通过调用 LAPACK（LU 分解后跟 upper/lower 三角求解器），
using BenchmarkTools
A = reshape(Float64[38, 49, 38, 73, 70, 71, 33, 24, 14, 82, 46, 99, 82, 36, 21, 32, 48, 40, 27, 60, 31, 15, 38, 88, 95, 57, 36, 86, 42, 56, 1, 37, 73, 7, 92, 93, 16, 95, 59, 76, 18, 42, 57, 9, 14, 40, 68, 61, 8, 26, 90, 33, 95, 8, 5, 87, 66, 84, 45, 78, 27, 16, 9, 83, 46, 61, 74, 44, 17, 21, 21, 53, 96, 49, 58, 67, 73, 60, 18, 40, 32, 68, 68, 21, 57, 86, 69, 7, 80, 10, 36, 46, 94, 59, 41, 80, 70, 2, 90, 57, 92, 50, 92, 98, 88, 14, 39, 80, 68, 78, 49, 40, 54, 51, 68, 80, 95, 22, 37, 88, 10, 30, 54, 7, 84, 99, 42, 94, 75, 45, 22, 41, 75, 38, 54, 97, 64, 62, 6, 48, 92, 49, 72, 5, 75, 67, 24, 55, 76, 17, 62, 19, 75, 41, 63, 97, 19, 83, 69, 12, 43, 94, 48, 92, 94, 54, 76, 11, 99, 96, 20, 29, 43, 97, 86, 23, 55, 2, 75, 61, 17, 45, 88, 79, 9, 26, 1, 3, 10, 91, 94, 85, 13, 58, 3, 53, 24, 76, 9, 2, 33, 34, 51, 65, 100, 67, 84, 21, 77, 17, 88, 65, 2, 46, 1, 18, 15, 57, 1, 88, 60, 64, 39, 36, 79, 89, 51, 39, 98, 67, 62, 34, 56, 98, 74, 52, 93, 11, 87, 45, 48, 82, 87, 5, 97, 65, 1, 81, 39, 85, 33, 26, 24, 90, 41, 69, 74, 43, 21, 54, 91, 94, 78, 41, 17, 11, 71, 25, 72, 52, 36, 27, 100, 48, 67, 52, 94, 44, 94, 91, 83, 95, 76, 19, 70, 34, 87, 67, 62, 67, 81, 55, 81, 45, 68, 1, 56, 95, 76, 38, 72, 88, 37, 64, 29, 16, 19, 81, 36, 18, 25, 28, 21, 17, 57, 51, 22, 87, 61, 39, 56, 51, 65, 44, 59, 3, 75, 98, 5, 21, 48, 95, 53, 23, 96, 4, 11, 11, 77, 21, 58, 78, 9, 93, 81, 17, 77, 97, 97, 44, 96, 26, 35, 89, 73, 26, 37, 3, 51, 76, 14, 67, 45, 92, 52, 83, 43, 91, 20, 62, 4, 48, 75, 35, 17, 65, 6, 98, 2, 78, 69, 39, 30, 57, 27, 49, 8, 71, 46, 82, 16, 62, 57, 69, 94, 15, 56, 15, 29, 42, 93, 96, 57, 2, 63, 23, 57, 54, 47, 88, 40, 1, 90, 48, 1, 4, 26, 32, 12, 97, 38, 62, 72, 92, 71, 72, 34, 93, 84, 56, 20, 33, 53, 42, 7, 54, 98, 37, 27, 2, 13, 88, 30, 24, 91, 22, 95, 100, 53, 53, 31, 91, 95, 9, 36, 89, 25, 60, 28, 47, 61, 81, 41, 47, 88, 6, 46, 83, 4, 48, 73, 88, 8, 83, 78, 18, 21, 75, 6, 90, 87, 92, 18, 71, 5, 82, 36, 2, 50, 86, 49, 72, 92, 67, 41, 38, 81, 37, 67, 93, 99, 51, 79, 95, 76, 85, 90, 27, 93, 44, 79, 97, 7, 11, 52, 76, 61, 23, 52, 97, 58, 74, 87, 58, 70, 77, 97, 74, 85, 65, 71, 79, 91, 36, 92, 35, 97, 9, 6, 38, 90, 46, 84, 98, 65, 4, 89, 9, 72, 55, 3, 21, 77, 43, 76, 83, 34, 16, 33, 21, 6, 28, 98, 27, 86, 93, 66, 55, 34, 76, 93, 42, 1, 36, 82, 82, 13, 45, 48, 8, 4, 66, 51, 32, 68, 81, 49, 70, 93, 73, 89, 16, 76, 95, 90, 37, 83, 28, 40, 14, 3, 18, 27, 34, 24, 53, 42, 24, 57, 93, 48, 43, 91, 28, 75, 86, 47, 40, 61, 20, 34, 81, 31, 62, 20, 75, 80, 81, 95, 75, 14, 8, 89, 13, 7, 9, 27, 80, 24, 52, 27, 75, 4, 58, 20, 82, 89, 31, 100, 48, 57, 73, 34, 52, 24, 26, 64, 18, 90, 74, 17, 58, 8, 44, 43, 56, 56, 51, 58, 56, 4, 87, 80, 24, 100, 47, 72, 60, 41, 2, 26, 81, 17, 57, 28, 6, 21, 4, 99, 92, 42, 37, 22, 45, 5, 93, 72, 27, 91, 13, 44, 93, 6, 100, 31, 17, 78, 16, 96, 32, 57, 45, 95, 76, 92, 3, 77, 84, 92, 87, 63, 42, 70, 79, 77, 90, 16, 100, 82, 61, 23, 67, 55, 45, 38, 27, 95, 19, 10, 4, 53, 75, 62, 1, 99, 62, 94, 30, 95, 65, 35, 62, 25, 59, 26, 62, 98, 50, 73, 31, 11, 89, 20, 1, 74, 45, 49, 55, 78, 49, 82, 35, 9, 45, 100, 99, 87, 10, 56, 79, 85, 89, 8, 9, 53, 87, 13, 27, 95, 81, 7, 71, 63, 44, 38, 84, 40, 87, 79, 54, 42, 58, 49, 85, 49, 6, 55, 83, 93, 52, 63, 76, 52, 40, 91, 36, 74, 70, 92, 92, 67, 57, 51, 74, 22, 35, 22, 48, 60, 86, 87, 79, 18, 65, 1, 36, 65, 91, 24, 33, 71, 52, 43, 20, 100, 94, 68, 19, 93, 66, 89, 45, 39, 97, 57, 67, 51, 92, 20, 97, 45, 32, 10, 82, 86, 2, 8, 27, 15, 60, 7, 6, 90, 71, 40, 91, 10, 16, 39, 40, 32, 2, 11, 5, 81, 31, 72, 41, 7, 89, 89, 85, 28, 67, 54, 44, 47, 26, 44, 51, 50, 65, 41, 68, 17, 88, 45, 43, 8, 11, 79, 10, 99, 58, 42, 75, 75, 86, 73, 24, 33, 15, 46, 84, 33, 27, 96, 14, 25, 11, 67, 48, 51, 85, 61, 87, 71, 85, 62, 32, 71, 15, 56, 6, 20, 43, 64, 97, 81, 94, 94, 61, 39, 46, 99, 37, 66, 40, 17, 74, 44, 6, 2, 11, 53, 44, 75, 29, 58, 77, 66, 96, 82, 13, 32, 43, 13, 36, 10, 39, 54, 39, 79, 22, 4, 41, 19, 44, 37, 73, 76, 84, 78, 94, 13, 98, 26, 56, 55, 51, 38, 37, 60, 55, 92, 19, 53, 48, 4, 7, 85, 82, 8, 60, 34, 67, 98, 76, 38, 14, 20, 62, 41, 58, 29, 70, 71, 16, 60, 26, 8, 64, 92, 17, 26, 40, 12, 59, 69, 97, 63, 52, 81, 27, 10, 99, 73, 74, 68, 8, 44, 70, 38, 65, 3, 27, 80, 90, 8, 64, 98, 89, 10, 45, 42, 55, 61, 49, 45, 82, 48, 27, 22, 16, 50, 58, 41, 92, 64, 54, 35, 65, 23, 66, 22, 9, 68, 79, 45, 69, 71, 94, 24, 41, 55, 48, 84, 12, 80, 71, 41, 91, 77, 83, 2, 12, 55, 21, 100, 99, 65, 20, 77, 37, 29, 75, 6, 59, 84, 25, 70, 40, 31, 73, 26, 61, 77, 16, 73, 41, 5, 83, 51, 9, 60, 97, 44, 21, 21, 87, 20, 74, 91, 43, 10, 69, 67, 14, 30, 71, 31, 20, 21, 98, 58, 21, 51, 83, 20, 69, 70, 13, 8, 62, 66, 28, 46, 75, 66, 65, 21, 32, 83, 7, 62, 4, 46, 98, 89, 20, 11, 57, 93, 72, 14, 80, 57, 10, 53, 67, 52, 88, 21, 97, 67, 42, 14, 86, 5, 12, 44, 35, 82, 3, 69, 87, 32, 10, 15, 54, 40, 60, 11, 46, 23, 77, 97, 46, 61, 90, 74, 82, 50, 15, 73, 59, 83, 68, 52, 54, 54, 89, 99, 44, 7, 85, 29, 65, 87, 20, 57, 5, 45, 98, 36, 98, 36, 99, 3, 54, 78, 100, 91, 73, 77, 63, 30, 11, 31, 21, 12, 78, 66, 36, 6, 50, 27, 55, 97, 79, 85, 29, 91, 72, 64, 18, 78, 77, 93, 74, 76, 33, 68, 71, 48, 10, 4, 19, 32, 53, 87, 75, 11, 25, 71, 23, 55, 16, 74, 28, 66, 90, 49, 75, 95, 19, 50, 75, 49, 52, 28, 57, 90, 20, 77, 52, 9, 42, 4, 20, 49, 78, 99, 78, 38, 100, 90, 7, 12, 8, 35, 26, 49, 54, 78, 43, 86, 23, 55, 11, 79, 20, 56, 61, 26, 81, 42, 93, 4, 3, 84, 3, 55, 46, 27, 67, 74, 28, 100, 44, 5, 14, 65, 22, 71, 13, 61, 65, 53, 14, 44, 53, 67, 69, 2, 76, 76, 90, 63, 21, 46, 46, 96, 19, 40, 12, 22, 45, 98, 6, 81, 7, 70, 51, 16, 62, 66, 33, 21, 69, 34, 24, 92, 23, 14, 51, 84, 36, 73, 83, 45, 52, 93, 20, 21, 61, 58, 75, 85, 36, 92, 29, 26, 100, 86, 79, 46, 43, 95, 9, 8, 98, 29, 27, 70, 93, 60, 20, 14, 10, 77, 71, 12, 38, 91, 59, 57, 84, 77, 15, 81, 17, 10, 42, 89, 4, 72, 16, 85, 27, 80, 85, 85, 9, 94, 3, 59, 30, 43, 30, 87, 20, 19, 33, 92, 8, 52, 46, 67, 26, 76, 3, 21, 71, 10, 37, 49, 61, 15, 70, 57, 66, 55, 52, 87, 36, 18, 30, 69, 28, 68, 26, 82, 86, 87, 16, 15, 46, 92, 54, 100, 92, 89, 52, 97, 53, 21, 31, 51, 31, 17, 46, 68, 53, 93, 64, 87, 43, 39, 94, 2, 38, 30, 87, 35, 53, 97, 28, 54, 58, 42, 55, 23, 27, 2, 27, 4, 78, 31, 14, 87, 21, 75, 26, 28, 67, 56, 65, 80, 10, 21, 48, 71, 52, 24, 67, 38, 62, 68, 93, 17, 56, 85, 87, 75, 62, 68, 45, 88, 49, 97, 78, 14, 94, 3, 67, 86, 9, 24, 92, 2, 12, 89, 73, 94, 63, 89, 65, 92, 61, 100, 90, 44, 57, 17, 74, 59, 5, 63, 5, 73, 46, 76, 69, 12, 97, 91, 9, 6, 61, 37, 5, 20, 39, 32, 19, 14, 46, 2, 46, 41, 28, 39, 29, 41, 59, 25, 97, 94, 63, 31, 64, 63, 72, 41, 46, 58, 79, 79, 35, 49, 42, 43, 82, 32, 41, 37, 84, 96, 100, 33, 87, 38, 89, 97, 25, 56, 61, 4, 100, 9, 83, 66, 77, 65, 22, 81, 52, 27, 6, 79, 29, 34, 15, 64, 22, 80, 61, 10, 74, 1, 68, 80, 74, 86, 98, 9, 24, 76, 57, 23, 5, 50, 7, 11, 80, 39, 10, 75, 38, 73, 8, 47, 3, 92, 90, 51, 42, 22, 45, 63, 27, 62, 78, 38, 5, 46, 46, 80, 51, 6, 43, 43, 7, 13, 50, 10, 64, 4, 67, 94, 69, 58, 58, 77, 71, 42, 80, 35, 15, 34, 65, 23, 43, 21, 24, 69, 24, 37, 68, 11, 38, 18, 12, 37, 41, 81, 12, 3, 91, 44, 98, 5, 1, 90, 53, 100, 90, 26, 36, 23, 14, 76, 23, 70, 58, 7, 35, 42, 11, 19, 48, 11, 24, 61, 49, 52, 69, 68, 82, 11, 57, 87, 65, 68, 54, 69, 39, 99, 1, 86, 44, 35, 36, 58, 73, 17, 14, 14, 87, 20, 57, 11, 65, 98, 77, 10, 51, 45, 50, 28, 56, 23, 64, 6, 11, 15, 93, 32, 77, 45, 57, 84, 49, 66, 98, 71, 8, 35, 62, 23, 82, 30, 75, 41, 15, 52, 22, 93, 68, 12, 83, 76, 19, 93, 67, 19, 35, 76, 49, 95, 40, 21, 78, 76, 86, 26, 31, 85, 15, 29, 82, 68, 54, 29, 70, 79, 93, 35, 2, 60, 78, 74, 32, 77, 94, 21, 21, 87, 48, 58, 76, 5, 87, 41, 6, 74, 83, 2, 56, 8, 2, 81, 3, 59, 7, 49, 62, 72, 98, 81, 68, 6, 82, 20, 97, 71, 16, 10, 58, 37, 98, 49, 23, 61, 80, 15, 77, 26, 56, 99, 21, 19, 60, 80, 61, 31, 6, 59, 70, 7, 87, 41, 9, 2, 34, 43, 84, 12, 24, 67, 63, 40, 78, 3, 100, 22, 100, 61, 59, 92, 26, 9, 39, 56, 93, 74, 47, 21, 71, 67, 81, 40, 74, 56, 34, 35, 82, 94, 35, 35, 15, 52, 44, 5, 83, 30, 10, 18, 65, 31, 45, 49, 100, 41, 26, 51, 3, 86, 17, 62, 13, 92, 58, 76, 53, 34, 81, 98, 57, 99, 81, 67, 23, 25, 99, 88, 62, 99, 37, 85, 17, 60, 23, 56, 97, 65, 41, 91, 16, 90, 47, 86, 56, 99, 44, 28, 18, 89, 27, 43, 43, 14, 64, 96, 8, 92, 74, 65, 24, 26, 96, 92, 19, 57, 24, 25, 3, 80, 99, 89, 78, 78, 80, 89, 27, 6, 49, 78, 81, 75, 99, 21, 64, 51, 98, 32, 53, 59, 74, 33, 1, 93, 9, 1, 24, 15, 8, 55, 76, 51, 98, 41, 77, 48, 81, 47, 76, 47, 65, 25, 2, 80, 67, 9, 85, 18, 73, 35, 50, 69, 46, 33, 14, 47, 25, 93, 28, 39, 12, 87, 85, 81, 16, 51, 91, 93, 32, 60, 55, 43, 54, 32, 57, 4, 30, 20, 15, 96, 64, 3, 99, 41, 5, 78, 28, 52, 39, 45, 41, 54, 1, 13, 53, 84, 75, 24, 100, 44, 8, 18, 46, 42, 86, 65, 27, 74, 1, 75, 99, 90, 33, 31, 4, 22, 17, 30, 44, 36, 72, 47, 75, 100, 47, 85, 86, 59, 37, 32, 30, 67, 98, 94, 85, 93, 1, 81, 60, 33, 97, 88, 73, 68, 8, 35, 30, 83, 19, 99, 74, 21, 93, 42, 80, 95, 27, 65, 24, 73, 31, 43, 92, 81, 24, 70, 67, 78, 48, 47, 70, 76, 12, 79, 89, 7, 28, 83, 78, 22, 25, 32, 17, 4, 68, 42, 15, 1, 3, 18, 43, 75, 48, 84, 17, 60, 100, 73, 59, 80, 68, 13, 89, 7, 93, 16, 22, 1, 58, 92, 87, 90, 23, 95, 76, 67, 10, 14, 70, 17, 99, 77, 6, 63, 69, 2, 93, 27, 29, 88, 39, 35, 25, 50, 91, 13, 16, 91, 50, 53, 54, 12, 53, 25, 11, 6, 10, 44, 36, 87, 67, 69, 5, 5, 78, 25, 19, 24, 50, 88, 62, 24, 89, 39, 86, 6, 7, 70, 56, 92, 18, 76, 57, 50, 28, 71, 50, 74, 19, 89, 49, 8, 76, 92, 80, 41, 34, 33, 63, 88, 31, 95, 97, 71, 52, 36, 26, 99, 72, 50, 76, 33, 62, 79, 11, 76, 54, 64, 42, 76, 5, 45, 79, 61, 39, 66, 72, 74, 76, 25, 63, 35, 100, 42, 61, 12, 9, 41, 95, 90, 48, 24, 8, 66, 65, 29, 74, 97, 54, 51, 31, 31, 51, 30, 63, 32, 70, 79, 49, 7, 35, 53, 76, 83, 62, 20, 13, 92, 95, 40, 99, 10, 98, 13, 7, 88, 16, 40, 10, 22, 29, 88, 64, 39, 13, 26, 12, 27, 69, 70, 23, 41, 67, 50, 96, 24, 97, 29, 31, 42, 27, 90, 50, 69, 42, 92, 22, 88, 23, 35, 83, 82, 74, 50, 72, 98, 94, 94, 46, 82, 16, 35, 88, 46, 89, 77, 86, 19, 17, 20, 5, 13, 25, 69, 79, 90, 55, 88, 71, 13, 30], (50,50))
b = Float64[66, 97, 50, 69, 24, 42, 23, 82, 25, 79, 66, 26, 76, 25, 75, 25, 43, 40, 55, 8, 20, 53, 66, 94, 57, 10, 39, 70, 5, 57, 22, 36, 45, 94, 24, 44, 89, 41, 14, 87, 9, 46, 74, 23, 72, 62, 52, 74, 36, 13]
linear_solve(A, b) = A \ b
@benchmark linear_solve(A, b)
C 代码运行大约 166 微秒，而 LAPACK（通过 Julia）代码平均运行 33 微秒（快 5 倍！）。
我想这是对 LAPACK 和相关 Julia 包装器质量的证明。
不幸的是，由于此 C 代码将成为 OpenCL 内核的一部分，我无法真正利用任何一个，有没有办法让我的 C 代码更高效？从而实现更接近于LAPACK的性能？
TL;DR: 当前的 C 代码 在现代硬件上效率低下 。此外，在专用 GPU 或 CUDA 上使用 OpenCL 只会 对于这里相当大的矩阵 很快（即不是 50x50 的矩阵）。
C 代码中最大的问题来自 A[K * l + (i + 1)] += c * A[K * l + j]; 行。实际上，由于循环迭代器是 l，内存访问模式不是连续的而是跨步的。 跨步内存访问模式 在现代硬件架构上比连续模式效率低得多（由于代码矢量化、高速缓存行、内存预取等）。在 GPU 上尤其如此。
您可以通过 转置 A 矩阵来解决此问题。这是修改后的版本：
// Naive (inefficient) transposition
// Please use the much faster BLAS function to do this (if possible)
void transpose(float A[K * K])
{
for (long j=0; j<K; ++j)
{
for (long i=j+1; i<K; ++i)
{
float tmp = A[K * i + j];
A[K * i + j] = A[K * j + i];
A[K * j + i] = tmp;
}
}
}
// Solve the system Ax=b using Gaussian elimination with partial pivoting.
// Work directly on the transposed version of A rather than transposing A every time should be much faster (especially for small matrices).
void fast_linear_solve(float A[K * K], float b[K])
{
// Not useful if A is already transposed
transpose(A);
for (long j=0; j<K; j++)
{
// Begin partial pivoting.
float maxval = fabs(A[K * j + j]);
long maxrow = j;
for (long i=j+1; i<K; i++)
{
if (fabs(A[K * i + j]) > maxval)
{
maxval = fabs(A[K * i + j]);
maxrow = i;
}
}
for (long l=0; l<K; l++)
{
float A_temp = A[K * maxrow + l];
A[K * maxrow + l] = A[K * j + l];
A[K * j + l] = A_temp;
}
float b_temp = b[maxrow];
b[maxrow] = b[j];
b[j] = b_temp;
// End partial pivoting.
// Begin putting [A; b] into row echelon form.
for (long i=j; i<K-1; i++)
{
float c = -A[K * (i + 1) + j] / A[K * j + j];
for (long l=j+1; l<K; l++)
A[K * (i + 1) + l] += c * A[K * j + l];
b[i + 1] += c * b[j];
}
// End putting [A; b] into row echelon form.
}
// Begin backsolving for x (by overwritting b).
for (long j=K-1; j>0; j--)
for (long i=j-1; i>=0; i--)
b[i] -= b[j] * A[K * i + j] / A[K * j + j];
for (long j=0; j<K; j++)
b[j] *= 1 / A[K * j + j];
// End backsolving for x.
// Not useful if A is already transposed
transpose(A);
}
另一个问题来自执行基准测试的方式。实际上，Julia 运行 多次执行代码，而 C 代码执行一次并使用 clock 函数。为了与 Julia 实现进行更公平的比较，C 实现的 linear_solve 函数必须 多次计算 （通过将其放入循环中并处理可能的聪明可能会增加一些偏差的编译器优化）。 gettimeofday 应该优先于 clock（因为前者计算挂钟时间，后者计算用户时间和系统时间之和）。
这是我机器上 50x50 矩阵的（平均）结果（GCC 9.3 使用 -O3，Clang 9.0 也使用 -O3，以及 Julia 1.4）：
Original C code (GCC): 25 us | Original C code (Clang): 25 us
New C code (GCC): 11 us | New C code (Clang): 12 us
Julia code: 80 us
以下是 500x500 随机矩阵的结果：
Original C code (GCC): 37.9 ms | Original C code (Clang): 38.8 ms
New C code (GCC): 6.7 ms | New C code (Clang): 6.1 ms
Julia code: 2.3 ms
大矩阵仍有改进的空间：C 代码可以使用 loop tiling 来改进（以降低代码的可读性和可维护性为代价）。
应该记住，虽然使用（专用）GPU 应该可以提高大矩阵的性能，但是由于 GPU 的相对 高延迟 [=47]，小矩阵不应该是这种情况=]（例如数据传输、同步、内存延迟）除非批处理用于许多小矩阵。
我正在编写一个 OpenCL 内核，它将涉及求解一个线性系统。目前我的内核太慢了，提高线性系统部分的性能似乎是一个很好的起点。
我还应该指出，我并不是要让我的线性求解器平行，我正在处理的问题在宏观层面上已经令人尴尬地平行了。
以下是我编写的 C 代码，用于使用部分旋转的高斯消去法求解 Ax=b，
#import <stdio.h>
#import <math.h>
#import <time.h>
#define K 50
// Solve the system Ax=b using Gaussian elimination with partial pivoting.
void linear_solve(float A[K * K], float b[K])
{
for (long j=0; j<K; j++)
{
// Begin partial pivoting.
float maxval = fabs(A[K * j + j]);
long maxrow = j;
for (long i=j+1; i<K; i++)
{
if (fabs(A[K * j + i]) > maxval)
{
maxval = fabs(A[K * j + i]);
maxrow = i;
}
}
for (long l=0; l<K; l++)
{
float A_temp = A[K * l + maxrow];
A[K * l + maxrow] = A[K * l + j];
A[K * l + j] = A_temp;
}
float b_temp = b[maxrow];
b[maxrow] = b[j];
b[j] = b_temp;
// End partial pivoting.
// Begin putting [A; b] into row echelon form.
for (long i=j; i<K-1; i++)
{
float c = -A[K * j + (i + 1)] / A[K * j + j];
for (long l=j+1; l<K; l++)
A[K * l + (i + 1)] += c * A[K * l + j];
b[i + 1] += c * b[j];
}
// End putting [A; b] into row echelon form.
}
// Begin backsolving for x (by overwritting b).
for (long j=K-1; j>0; j--)
for (long i=j-1; i>=0; i--)
b[i] -= b[j] * A[K * j + i] / A[K * j + j];
for (long j=0; j<K; j++)
b[j] *= 1 / A[K * j + j];
// End backsolving for x.
}
int main()
{
int i, j;
float A[K * K] = {38, 49, 38, 73, 70, 71, 33, 24, 14, 82, 46, 99, 82, 36, 21, 32, 48, 40, 27, 60, 31, 15, 38, 88, 95, 57, 36, 86, 42, 56, 1, 37, 73, 7, 92, 93, 16, 95, 59, 76, 18, 42, 57, 9, 14, 40, 68, 61, 8, 26, 90, 33, 95, 8, 5, 87, 66, 84, 45, 78, 27, 16, 9, 83, 46, 61, 74, 44, 17, 21, 21, 53, 96, 49, 58, 67, 73, 60, 18, 40, 32, 68, 68, 21, 57, 86, 69, 7, 80, 10, 36, 46, 94, 59, 41, 80, 70, 2, 90, 57, 92, 50, 92, 98, 88, 14, 39, 80, 68, 78, 49, 40, 54, 51, 68, 80, 95, 22, 37, 88, 10, 30, 54, 7, 84, 99, 42, 94, 75, 45, 22, 41, 75, 38, 54, 97, 64, 62, 6, 48, 92, 49, 72, 5, 75, 67, 24, 55, 76, 17, 62, 19, 75, 41, 63, 97, 19, 83, 69, 12, 43, 94, 48, 92, 94, 54, 76, 11, 99, 96, 20, 29, 43, 97, 86, 23, 55, 2, 75, 61, 17, 45, 88, 79, 9, 26, 1, 3, 10, 91, 94, 85, 13, 58, 3, 53, 24, 76, 9, 2, 33, 34, 51, 65, 100, 67, 84, 21, 77, 17, 88, 65, 2, 46, 1, 18, 15, 57, 1, 88, 60, 64, 39, 36, 79, 89, 51, 39, 98, 67, 62, 34, 56, 98, 74, 52, 93, 11, 87, 45, 48, 82, 87, 5, 97, 65, 1, 81, 39, 85, 33, 26, 24, 90, 41, 69, 74, 43, 21, 54, 91, 94, 78, 41, 17, 11, 71, 25, 72, 52, 36, 27, 100, 48, 67, 52, 94, 44, 94, 91, 83, 95, 76, 19, 70, 34, 87, 67, 62, 67, 81, 55, 81, 45, 68, 1, 56, 95, 76, 38, 72, 88, 37, 64, 29, 16, 19, 81, 36, 18, 25, 28, 21, 17, 57, 51, 22, 87, 61, 39, 56, 51, 65, 44, 59, 3, 75, 98, 5, 21, 48, 95, 53, 23, 96, 4, 11, 11, 77, 21, 58, 78, 9, 93, 81, 17, 77, 97, 97, 44, 96, 26, 35, 89, 73, 26, 37, 3, 51, 76, 14, 67, 45, 92, 52, 83, 43, 91, 20, 62, 4, 48, 75, 35, 17, 65, 6, 98, 2, 78, 69, 39, 30, 57, 27, 49, 8, 71, 46, 82, 16, 62, 57, 69, 94, 15, 56, 15, 29, 42, 93, 96, 57, 2, 63, 23, 57, 54, 47, 88, 40, 1, 90, 48, 1, 4, 26, 32, 12, 97, 38, 62, 72, 92, 71, 72, 34, 93, 84, 56, 20, 33, 53, 42, 7, 54, 98, 37, 27, 2, 13, 88, 30, 24, 91, 22, 95, 100, 53, 53, 31, 91, 95, 9, 36, 89, 25, 60, 28, 47, 61, 81, 41, 47, 88, 6, 46, 83, 4, 48, 73, 88, 8, 83, 78, 18, 21, 75, 6, 90, 87, 92, 18, 71, 5, 82, 36, 2, 50, 86, 49, 72, 92, 67, 41, 38, 81, 37, 67, 93, 99, 51, 79, 95, 76, 85, 90, 27, 93, 44, 79, 97, 7, 11, 52, 76, 61, 23, 52, 97, 58, 74, 87, 58, 70, 77, 97, 74, 85, 65, 71, 79, 91, 36, 92, 35, 97, 9, 6, 38, 90, 46, 84, 98, 65, 4, 89, 9, 72, 55, 3, 21, 77, 43, 76, 83, 34, 16, 33, 21, 6, 28, 98, 27, 86, 93, 66, 55, 34, 76, 93, 42, 1, 36, 82, 82, 13, 45, 48, 8, 4, 66, 51, 32, 68, 81, 49, 70, 93, 73, 89, 16, 76, 95, 90, 37, 83, 28, 40, 14, 3, 18, 27, 34, 24, 53, 42, 24, 57, 93, 48, 43, 91, 28, 75, 86, 47, 40, 61, 20, 34, 81, 31, 62, 20, 75, 80, 81, 95, 75, 14, 8, 89, 13, 7, 9, 27, 80, 24, 52, 27, 75, 4, 58, 20, 82, 89, 31, 100, 48, 57, 73, 34, 52, 24, 26, 64, 18, 90, 74, 17, 58, 8, 44, 43, 56, 56, 51, 58, 56, 4, 87, 80, 24, 100, 47, 72, 60, 41, 2, 26, 81, 17, 57, 28, 6, 21, 4, 99, 92, 42, 37, 22, 45, 5, 93, 72, 27, 91, 13, 44, 93, 6, 100, 31, 17, 78, 16, 96, 32, 57, 45, 95, 76, 92, 3, 77, 84, 92, 87, 63, 42, 70, 79, 77, 90, 16, 100, 82, 61, 23, 67, 55, 45, 38, 27, 95, 19, 10, 4, 53, 75, 62, 1, 99, 62, 94, 30, 95, 65, 35, 62, 25, 59, 26, 62, 98, 50, 73, 31, 11, 89, 20, 1, 74, 45, 49, 55, 78, 49, 82, 35, 9, 45, 100, 99, 87, 10, 56, 79, 85, 89, 8, 9, 53, 87, 13, 27, 95, 81, 7, 71, 63, 44, 38, 84, 40, 87, 79, 54, 42, 58, 49, 85, 49, 6, 55, 83, 93, 52, 63, 76, 52, 40, 91, 36, 74, 70, 92, 92, 67, 57, 51, 74, 22, 35, 22, 48, 60, 86, 87, 79, 18, 65, 1, 36, 65, 91, 24, 33, 71, 52, 43, 20, 100, 94, 68, 19, 93, 66, 89, 45, 39, 97, 57, 67, 51, 92, 20, 97, 45, 32, 10, 82, 86, 2, 8, 27, 15, 60, 7, 6, 90, 71, 40, 91, 10, 16, 39, 40, 32, 2, 11, 5, 81, 31, 72, 41, 7, 89, 89, 85, 28, 67, 54, 44, 47, 26, 44, 51, 50, 65, 41, 68, 17, 88, 45, 43, 8, 11, 79, 10, 99, 58, 42, 75, 75, 86, 73, 24, 33, 15, 46, 84, 33, 27, 96, 14, 25, 11, 67, 48, 51, 85, 61, 87, 71, 85, 62, 32, 71, 15, 56, 6, 20, 43, 64, 97, 81, 94, 94, 61, 39, 46, 99, 37, 66, 40, 17, 74, 44, 6, 2, 11, 53, 44, 75, 29, 58, 77, 66, 96, 82, 13, 32, 43, 13, 36, 10, 39, 54, 39, 79, 22, 4, 41, 19, 44, 37, 73, 76, 84, 78, 94, 13, 98, 26, 56, 55, 51, 38, 37, 60, 55, 92, 19, 53, 48, 4, 7, 85, 82, 8, 60, 34, 67, 98, 76, 38, 14, 20, 62, 41, 58, 29, 70, 71, 16, 60, 26, 8, 64, 92, 17, 26, 40, 12, 59, 69, 97, 63, 52, 81, 27, 10, 99, 73, 74, 68, 8, 44, 70, 38, 65, 3, 27, 80, 90, 8, 64, 98, 89, 10, 45, 42, 55, 61, 49, 45, 82, 48, 27, 22, 16, 50, 58, 41, 92, 64, 54, 35, 65, 23, 66, 22, 9, 68, 79, 45, 69, 71, 94, 24, 41, 55, 48, 84, 12, 80, 71, 41, 91, 77, 83, 2, 12, 55, 21, 100, 99, 65, 20, 77, 37, 29, 75, 6, 59, 84, 25, 70, 40, 31, 73, 26, 61, 77, 16, 73, 41, 5, 83, 51, 9, 60, 97, 44, 21, 21, 87, 20, 74, 91, 43, 10, 69, 67, 14, 30, 71, 31, 20, 21, 98, 58, 21, 51, 83, 20, 69, 70, 13, 8, 62, 66, 28, 46, 75, 66, 65, 21, 32, 83, 7, 62, 4, 46, 98, 89, 20, 11, 57, 93, 72, 14, 80, 57, 10, 53, 67, 52, 88, 21, 97, 67, 42, 14, 86, 5, 12, 44, 35, 82, 3, 69, 87, 32, 10, 15, 54, 40, 60, 11, 46, 23, 77, 97, 46, 61, 90, 74, 82, 50, 15, 73, 59, 83, 68, 52, 54, 54, 89, 99, 44, 7, 85, 29, 65, 87, 20, 57, 5, 45, 98, 36, 98, 36, 99, 3, 54, 78, 100, 91, 73, 77, 63, 30, 11, 31, 21, 12, 78, 66, 36, 6, 50, 27, 55, 97, 79, 85, 29, 91, 72, 64, 18, 78, 77, 93, 74, 76, 33, 68, 71, 48, 10, 4, 19, 32, 53, 87, 75, 11, 25, 71, 23, 55, 16, 74, 28, 66, 90, 49, 75, 95, 19, 50, 75, 49, 52, 28, 57, 90, 20, 77, 52, 9, 42, 4, 20, 49, 78, 99, 78, 38, 100, 90, 7, 12, 8, 35, 26, 49, 54, 78, 43, 86, 23, 55, 11, 79, 20, 56, 61, 26, 81, 42, 93, 4, 3, 84, 3, 55, 46, 27, 67, 74, 28, 100, 44, 5, 14, 65, 22, 71, 13, 61, 65, 53, 14, 44, 53, 67, 69, 2, 76, 76, 90, 63, 21, 46, 46, 96, 19, 40, 12, 22, 45, 98, 6, 81, 7, 70, 51, 16, 62, 66, 33, 21, 69, 34, 24, 92, 23, 14, 51, 84, 36, 73, 83, 45, 52, 93, 20, 21, 61, 58, 75, 85, 36, 92, 29, 26, 100, 86, 79, 46, 43, 95, 9, 8, 98, 29, 27, 70, 93, 60, 20, 14, 10, 77, 71, 12, 38, 91, 59, 57, 84, 77, 15, 81, 17, 10, 42, 89, 4, 72, 16, 85, 27, 80, 85, 85, 9, 94, 3, 59, 30, 43, 30, 87, 20, 19, 33, 92, 8, 52, 46, 67, 26, 76, 3, 21, 71, 10, 37, 49, 61, 15, 70, 57, 66, 55, 52, 87, 36, 18, 30, 69, 28, 68, 26, 82, 86, 87, 16, 15, 46, 92, 54, 100, 92, 89, 52, 97, 53, 21, 31, 51, 31, 17, 46, 68, 53, 93, 64, 87, 43, 39, 94, 2, 38, 30, 87, 35, 53, 97, 28, 54, 58, 42, 55, 23, 27, 2, 27, 4, 78, 31, 14, 87, 21, 75, 26, 28, 67, 56, 65, 80, 10, 21, 48, 71, 52, 24, 67, 38, 62, 68, 93, 17, 56, 85, 87, 75, 62, 68, 45, 88, 49, 97, 78, 14, 94, 3, 67, 86, 9, 24, 92, 2, 12, 89, 73, 94, 63, 89, 65, 92, 61, 100, 90, 44, 57, 17, 74, 59, 5, 63, 5, 73, 46, 76, 69, 12, 97, 91, 9, 6, 61, 37, 5, 20, 39, 32, 19, 14, 46, 2, 46, 41, 28, 39, 29, 41, 59, 25, 97, 94, 63, 31, 64, 63, 72, 41, 46, 58, 79, 79, 35, 49, 42, 43, 82, 32, 41, 37, 84, 96, 100, 33, 87, 38, 89, 97, 25, 56, 61, 4, 100, 9, 83, 66, 77, 65, 22, 81, 52, 27, 6, 79, 29, 34, 15, 64, 22, 80, 61, 10, 74, 1, 68, 80, 74, 86, 98, 9, 24, 76, 57, 23, 5, 50, 7, 11, 80, 39, 10, 75, 38, 73, 8, 47, 3, 92, 90, 51, 42, 22, 45, 63, 27, 62, 78, 38, 5, 46, 46, 80, 51, 6, 43, 43, 7, 13, 50, 10, 64, 4, 67, 94, 69, 58, 58, 77, 71, 42, 80, 35, 15, 34, 65, 23, 43, 21, 24, 69, 24, 37, 68, 11, 38, 18, 12, 37, 41, 81, 12, 3, 91, 44, 98, 5, 1, 90, 53, 100, 90, 26, 36, 23, 14, 76, 23, 70, 58, 7, 35, 42, 11, 19, 48, 11, 24, 61, 49, 52, 69, 68, 82, 11, 57, 87, 65, 68, 54, 69, 39, 99, 1, 86, 44, 35, 36, 58, 73, 17, 14, 14, 87, 20, 57, 11, 65, 98, 77, 10, 51, 45, 50, 28, 56, 23, 64, 6, 11, 15, 93, 32, 77, 45, 57, 84, 49, 66, 98, 71, 8, 35, 62, 23, 82, 30, 75, 41, 15, 52, 22, 93, 68, 12, 83, 76, 19, 93, 67, 19, 35, 76, 49, 95, 40, 21, 78, 76, 86, 26, 31, 85, 15, 29, 82, 68, 54, 29, 70, 79, 93, 35, 2, 60, 78, 74, 32, 77, 94, 21, 21, 87, 48, 58, 76, 5, 87, 41, 6, 74, 83, 2, 56, 8, 2, 81, 3, 59, 7, 49, 62, 72, 98, 81, 68, 6, 82, 20, 97, 71, 16, 10, 58, 37, 98, 49, 23, 61, 80, 15, 77, 26, 56, 99, 21, 19, 60, 80, 61, 31, 6, 59, 70, 7, 87, 41, 9, 2, 34, 43, 84, 12, 24, 67, 63, 40, 78, 3, 100, 22, 100, 61, 59, 92, 26, 9, 39, 56, 93, 74, 47, 21, 71, 67, 81, 40, 74, 56, 34, 35, 82, 94, 35, 35, 15, 52, 44, 5, 83, 30, 10, 18, 65, 31, 45, 49, 100, 41, 26, 51, 3, 86, 17, 62, 13, 92, 58, 76, 53, 34, 81, 98, 57, 99, 81, 67, 23, 25, 99, 88, 62, 99, 37, 85, 17, 60, 23, 56, 97, 65, 41, 91, 16, 90, 47, 86, 56, 99, 44, 28, 18, 89, 27, 43, 43, 14, 64, 96, 8, 92, 74, 65, 24, 26, 96, 92, 19, 57, 24, 25, 3, 80, 99, 89, 78, 78, 80, 89, 27, 6, 49, 78, 81, 75, 99, 21, 64, 51, 98, 32, 53, 59, 74, 33, 1, 93, 9, 1, 24, 15, 8, 55, 76, 51, 98, 41, 77, 48, 81, 47, 76, 47, 65, 25, 2, 80, 67, 9, 85, 18, 73, 35, 50, 69, 46, 33, 14, 47, 25, 93, 28, 39, 12, 87, 85, 81, 16, 51, 91, 93, 32, 60, 55, 43, 54, 32, 57, 4, 30, 20, 15, 96, 64, 3, 99, 41, 5, 78, 28, 52, 39, 45, 41, 54, 1, 13, 53, 84, 75, 24, 100, 44, 8, 18, 46, 42, 86, 65, 27, 74, 1, 75, 99, 90, 33, 31, 4, 22, 17, 30, 44, 36, 72, 47, 75, 100, 47, 85, 86, 59, 37, 32, 30, 67, 98, 94, 85, 93, 1, 81, 60, 33, 97, 88, 73, 68, 8, 35, 30, 83, 19, 99, 74, 21, 93, 42, 80, 95, 27, 65, 24, 73, 31, 43, 92, 81, 24, 70, 67, 78, 48, 47, 70, 76, 12, 79, 89, 7, 28, 83, 78, 22, 25, 32, 17, 4, 68, 42, 15, 1, 3, 18, 43, 75, 48, 84, 17, 60, 100, 73, 59, 80, 68, 13, 89, 7, 93, 16, 22, 1, 58, 92, 87, 90, 23, 95, 76, 67, 10, 14, 70, 17, 99, 77, 6, 63, 69, 2, 93, 27, 29, 88, 39, 35, 25, 50, 91, 13, 16, 91, 50, 53, 54, 12, 53, 25, 11, 6, 10, 44, 36, 87, 67, 69, 5, 5, 78, 25, 19, 24, 50, 88, 62, 24, 89, 39, 86, 6, 7, 70, 56, 92, 18, 76, 57, 50, 28, 71, 50, 74, 19, 89, 49, 8, 76, 92, 80, 41, 34, 33, 63, 88, 31, 95, 97, 71, 52, 36, 26, 99, 72, 50, 76, 33, 62, 79, 11, 76, 54, 64, 42, 76, 5, 45, 79, 61, 39, 66, 72, 74, 76, 25, 63, 35, 100, 42, 61, 12, 9, 41, 95, 90, 48, 24, 8, 66, 65, 29, 74, 97, 54, 51, 31, 31, 51, 30, 63, 32, 70, 79, 49, 7, 35, 53, 76, 83, 62, 20, 13, 92, 95, 40, 99, 10, 98, 13, 7, 88, 16, 40, 10, 22, 29, 88, 64, 39, 13, 26, 12, 27, 69, 70, 23, 41, 67, 50, 96, 24, 97, 29, 31, 42, 27, 90, 50, 69, 42, 92, 22, 88, 23, 35, 83, 82, 74, 50, 72, 98, 94, 94, 46, 82, 16, 35, 88, 46, 89, 77, 86, 19, 17, 20, 5, 13, 25, 69, 79, 90, 55, 88, 71, 13, 30};
float b[K] = {66, 97, 50, 69, 24, 42, 23, 82, 25, 79, 66, 26, 76, 25, 75, 25, 43, 40, 55, 8, 20, 53, 66, 94, 57, 10, 39, 70, 5, 57, 22, 36, 45, 94, 24, 44, 89, 41, 14, 87, 9, 46, 74, 23, 72, 62, 52, 74, 36, 13};
clock_t begin = clock();
linear_solve(A, b);
clock_t end = clock();
double time_spent = (double)(end - begin) / CLOCKS_PER_SEC;
printf("seconds: %f\n", time_spent);
printf("Result vector is: ");
for (i=0; i<K; i++)
{
printf("%f,", b[i]);
}
printf("\n");
return 0;
}
以下是求解 Ax=b 的 Julia 代码，通过调用 LAPACK（LU 分解后跟 upper/lower 三角求解器），
using BenchmarkTools
A = reshape(Float64[38, 49, 38, 73, 70, 71, 33, 24, 14, 82, 46, 99, 82, 36, 21, 32, 48, 40, 27, 60, 31, 15, 38, 88, 95, 57, 36, 86, 42, 56, 1, 37, 73, 7, 92, 93, 16, 95, 59, 76, 18, 42, 57, 9, 14, 40, 68, 61, 8, 26, 90, 33, 95, 8, 5, 87, 66, 84, 45, 78, 27, 16, 9, 83, 46, 61, 74, 44, 17, 21, 21, 53, 96, 49, 58, 67, 73, 60, 18, 40, 32, 68, 68, 21, 57, 86, 69, 7, 80, 10, 36, 46, 94, 59, 41, 80, 70, 2, 90, 57, 92, 50, 92, 98, 88, 14, 39, 80, 68, 78, 49, 40, 54, 51, 68, 80, 95, 22, 37, 88, 10, 30, 54, 7, 84, 99, 42, 94, 75, 45, 22, 41, 75, 38, 54, 97, 64, 62, 6, 48, 92, 49, 72, 5, 75, 67, 24, 55, 76, 17, 62, 19, 75, 41, 63, 97, 19, 83, 69, 12, 43, 94, 48, 92, 94, 54, 76, 11, 99, 96, 20, 29, 43, 97, 86, 23, 55, 2, 75, 61, 17, 45, 88, 79, 9, 26, 1, 3, 10, 91, 94, 85, 13, 58, 3, 53, 24, 76, 9, 2, 33, 34, 51, 65, 100, 67, 84, 21, 77, 17, 88, 65, 2, 46, 1, 18, 15, 57, 1, 88, 60, 64, 39, 36, 79, 89, 51, 39, 98, 67, 62, 34, 56, 98, 74, 52, 93, 11, 87, 45, 48, 82, 87, 5, 97, 65, 1, 81, 39, 85, 33, 26, 24, 90, 41, 69, 74, 43, 21, 54, 91, 94, 78, 41, 17, 11, 71, 25, 72, 52, 36, 27, 100, 48, 67, 52, 94, 44, 94, 91, 83, 95, 76, 19, 70, 34, 87, 67, 62, 67, 81, 55, 81, 45, 68, 1, 56, 95, 76, 38, 72, 88, 37, 64, 29, 16, 19, 81, 36, 18, 25, 28, 21, 17, 57, 51, 22, 87, 61, 39, 56, 51, 65, 44, 59, 3, 75, 98, 5, 21, 48, 95, 53, 23, 96, 4, 11, 11, 77, 21, 58, 78, 9, 93, 81, 17, 77, 97, 97, 44, 96, 26, 35, 89, 73, 26, 37, 3, 51, 76, 14, 67, 45, 92, 52, 83, 43, 91, 20, 62, 4, 48, 75, 35, 17, 65, 6, 98, 2, 78, 69, 39, 30, 57, 27, 49, 8, 71, 46, 82, 16, 62, 57, 69, 94, 15, 56, 15, 29, 42, 93, 96, 57, 2, 63, 23, 57, 54, 47, 88, 40, 1, 90, 48, 1, 4, 26, 32, 12, 97, 38, 62, 72, 92, 71, 72, 34, 93, 84, 56, 20, 33, 53, 42, 7, 54, 98, 37, 27, 2, 13, 88, 30, 24, 91, 22, 95, 100, 53, 53, 31, 91, 95, 9, 36, 89, 25, 60, 28, 47, 61, 81, 41, 47, 88, 6, 46, 83, 4, 48, 73, 88, 8, 83, 78, 18, 21, 75, 6, 90, 87, 92, 18, 71, 5, 82, 36, 2, 50, 86, 49, 72, 92, 67, 41, 38, 81, 37, 67, 93, 99, 51, 79, 95, 76, 85, 90, 27, 93, 44, 79, 97, 7, 11, 52, 76, 61, 23, 52, 97, 58, 74, 87, 58, 70, 77, 97, 74, 85, 65, 71, 79, 91, 36, 92, 35, 97, 9, 6, 38, 90, 46, 84, 98, 65, 4, 89, 9, 72, 55, 3, 21, 77, 43, 76, 83, 34, 16, 33, 21, 6, 28, 98, 27, 86, 93, 66, 55, 34, 76, 93, 42, 1, 36, 82, 82, 13, 45, 48, 8, 4, 66, 51, 32, 68, 81, 49, 70, 93, 73, 89, 16, 76, 95, 90, 37, 83, 28, 40, 14, 3, 18, 27, 34, 24, 53, 42, 24, 57, 93, 48, 43, 91, 28, 75, 86, 47, 40, 61, 20, 34, 81, 31, 62, 20, 75, 80, 81, 95, 75, 14, 8, 89, 13, 7, 9, 27, 80, 24, 52, 27, 75, 4, 58, 20, 82, 89, 31, 100, 48, 57, 73, 34, 52, 24, 26, 64, 18, 90, 74, 17, 58, 8, 44, 43, 56, 56, 51, 58, 56, 4, 87, 80, 24, 100, 47, 72, 60, 41, 2, 26, 81, 17, 57, 28, 6, 21, 4, 99, 92, 42, 37, 22, 45, 5, 93, 72, 27, 91, 13, 44, 93, 6, 100, 31, 17, 78, 16, 96, 32, 57, 45, 95, 76, 92, 3, 77, 84, 92, 87, 63, 42, 70, 79, 77, 90, 16, 100, 82, 61, 23, 67, 55, 45, 38, 27, 95, 19, 10, 4, 53, 75, 62, 1, 99, 62, 94, 30, 95, 65, 35, 62, 25, 59, 26, 62, 98, 50, 73, 31, 11, 89, 20, 1, 74, 45, 49, 55, 78, 49, 82, 35, 9, 45, 100, 99, 87, 10, 56, 79, 85, 89, 8, 9, 53, 87, 13, 27, 95, 81, 7, 71, 63, 44, 38, 84, 40, 87, 79, 54, 42, 58, 49, 85, 49, 6, 55, 83, 93, 52, 63, 76, 52, 40, 91, 36, 74, 70, 92, 92, 67, 57, 51, 74, 22, 35, 22, 48, 60, 86, 87, 79, 18, 65, 1, 36, 65, 91, 24, 33, 71, 52, 43, 20, 100, 94, 68, 19, 93, 66, 89, 45, 39, 97, 57, 67, 51, 92, 20, 97, 45, 32, 10, 82, 86, 2, 8, 27, 15, 60, 7, 6, 90, 71, 40, 91, 10, 16, 39, 40, 32, 2, 11, 5, 81, 31, 72, 41, 7, 89, 89, 85, 28, 67, 54, 44, 47, 26, 44, 51, 50, 65, 41, 68, 17, 88, 45, 43, 8, 11, 79, 10, 99, 58, 42, 75, 75, 86, 73, 24, 33, 15, 46, 84, 33, 27, 96, 14, 25, 11, 67, 48, 51, 85, 61, 87, 71, 85, 62, 32, 71, 15, 56, 6, 20, 43, 64, 97, 81, 94, 94, 61, 39, 46, 99, 37, 66, 40, 17, 74, 44, 6, 2, 11, 53, 44, 75, 29, 58, 77, 66, 96, 82, 13, 32, 43, 13, 36, 10, 39, 54, 39, 79, 22, 4, 41, 19, 44, 37, 73, 76, 84, 78, 94, 13, 98, 26, 56, 55, 51, 38, 37, 60, 55, 92, 19, 53, 48, 4, 7, 85, 82, 8, 60, 34, 67, 98, 76, 38, 14, 20, 62, 41, 58, 29, 70, 71, 16, 60, 26, 8, 64, 92, 17, 26, 40, 12, 59, 69, 97, 63, 52, 81, 27, 10, 99, 73, 74, 68, 8, 44, 70, 38, 65, 3, 27, 80, 90, 8, 64, 98, 89, 10, 45, 42, 55, 61, 49, 45, 82, 48, 27, 22, 16, 50, 58, 41, 92, 64, 54, 35, 65, 23, 66, 22, 9, 68, 79, 45, 69, 71, 94, 24, 41, 55, 48, 84, 12, 80, 71, 41, 91, 77, 83, 2, 12, 55, 21, 100, 99, 65, 20, 77, 37, 29, 75, 6, 59, 84, 25, 70, 40, 31, 73, 26, 61, 77, 16, 73, 41, 5, 83, 51, 9, 60, 97, 44, 21, 21, 87, 20, 74, 91, 43, 10, 69, 67, 14, 30, 71, 31, 20, 21, 98, 58, 21, 51, 83, 20, 69, 70, 13, 8, 62, 66, 28, 46, 75, 66, 65, 21, 32, 83, 7, 62, 4, 46, 98, 89, 20, 11, 57, 93, 72, 14, 80, 57, 10, 53, 67, 52, 88, 21, 97, 67, 42, 14, 86, 5, 12, 44, 35, 82, 3, 69, 87, 32, 10, 15, 54, 40, 60, 11, 46, 23, 77, 97, 46, 61, 90, 74, 82, 50, 15, 73, 59, 83, 68, 52, 54, 54, 89, 99, 44, 7, 85, 29, 65, 87, 20, 57, 5, 45, 98, 36, 98, 36, 99, 3, 54, 78, 100, 91, 73, 77, 63, 30, 11, 31, 21, 12, 78, 66, 36, 6, 50, 27, 55, 97, 79, 85, 29, 91, 72, 64, 18, 78, 77, 93, 74, 76, 33, 68, 71, 48, 10, 4, 19, 32, 53, 87, 75, 11, 25, 71, 23, 55, 16, 74, 28, 66, 90, 49, 75, 95, 19, 50, 75, 49, 52, 28, 57, 90, 20, 77, 52, 9, 42, 4, 20, 49, 78, 99, 78, 38, 100, 90, 7, 12, 8, 35, 26, 49, 54, 78, 43, 86, 23, 55, 11, 79, 20, 56, 61, 26, 81, 42, 93, 4, 3, 84, 3, 55, 46, 27, 67, 74, 28, 100, 44, 5, 14, 65, 22, 71, 13, 61, 65, 53, 14, 44, 53, 67, 69, 2, 76, 76, 90, 63, 21, 46, 46, 96, 19, 40, 12, 22, 45, 98, 6, 81, 7, 70, 51, 16, 62, 66, 33, 21, 69, 34, 24, 92, 23, 14, 51, 84, 36, 73, 83, 45, 52, 93, 20, 21, 61, 58, 75, 85, 36, 92, 29, 26, 100, 86, 79, 46, 43, 95, 9, 8, 98, 29, 27, 70, 93, 60, 20, 14, 10, 77, 71, 12, 38, 91, 59, 57, 84, 77, 15, 81, 17, 10, 42, 89, 4, 72, 16, 85, 27, 80, 85, 85, 9, 94, 3, 59, 30, 43, 30, 87, 20, 19, 33, 92, 8, 52, 46, 67, 26, 76, 3, 21, 71, 10, 37, 49, 61, 15, 70, 57, 66, 55, 52, 87, 36, 18, 30, 69, 28, 68, 26, 82, 86, 87, 16, 15, 46, 92, 54, 100, 92, 89, 52, 97, 53, 21, 31, 51, 31, 17, 46, 68, 53, 93, 64, 87, 43, 39, 94, 2, 38, 30, 87, 35, 53, 97, 28, 54, 58, 42, 55, 23, 27, 2, 27, 4, 78, 31, 14, 87, 21, 75, 26, 28, 67, 56, 65, 80, 10, 21, 48, 71, 52, 24, 67, 38, 62, 68, 93, 17, 56, 85, 87, 75, 62, 68, 45, 88, 49, 97, 78, 14, 94, 3, 67, 86, 9, 24, 92, 2, 12, 89, 73, 94, 63, 89, 65, 92, 61, 100, 90, 44, 57, 17, 74, 59, 5, 63, 5, 73, 46, 76, 69, 12, 97, 91, 9, 6, 61, 37, 5, 20, 39, 32, 19, 14, 46, 2, 46, 41, 28, 39, 29, 41, 59, 25, 97, 94, 63, 31, 64, 63, 72, 41, 46, 58, 79, 79, 35, 49, 42, 43, 82, 32, 41, 37, 84, 96, 100, 33, 87, 38, 89, 97, 25, 56, 61, 4, 100, 9, 83, 66, 77, 65, 22, 81, 52, 27, 6, 79, 29, 34, 15, 64, 22, 80, 61, 10, 74, 1, 68, 80, 74, 86, 98, 9, 24, 76, 57, 23, 5, 50, 7, 11, 80, 39, 10, 75, 38, 73, 8, 47, 3, 92, 90, 51, 42, 22, 45, 63, 27, 62, 78, 38, 5, 46, 46, 80, 51, 6, 43, 43, 7, 13, 50, 10, 64, 4, 67, 94, 69, 58, 58, 77, 71, 42, 80, 35, 15, 34, 65, 23, 43, 21, 24, 69, 24, 37, 68, 11, 38, 18, 12, 37, 41, 81, 12, 3, 91, 44, 98, 5, 1, 90, 53, 100, 90, 26, 36, 23, 14, 76, 23, 70, 58, 7, 35, 42, 11, 19, 48, 11, 24, 61, 49, 52, 69, 68, 82, 11, 57, 87, 65, 68, 54, 69, 39, 99, 1, 86, 44, 35, 36, 58, 73, 17, 14, 14, 87, 20, 57, 11, 65, 98, 77, 10, 51, 45, 50, 28, 56, 23, 64, 6, 11, 15, 93, 32, 77, 45, 57, 84, 49, 66, 98, 71, 8, 35, 62, 23, 82, 30, 75, 41, 15, 52, 22, 93, 68, 12, 83, 76, 19, 93, 67, 19, 35, 76, 49, 95, 40, 21, 78, 76, 86, 26, 31, 85, 15, 29, 82, 68, 54, 29, 70, 79, 93, 35, 2, 60, 78, 74, 32, 77, 94, 21, 21, 87, 48, 58, 76, 5, 87, 41, 6, 74, 83, 2, 56, 8, 2, 81, 3, 59, 7, 49, 62, 72, 98, 81, 68, 6, 82, 20, 97, 71, 16, 10, 58, 37, 98, 49, 23, 61, 80, 15, 77, 26, 56, 99, 21, 19, 60, 80, 61, 31, 6, 59, 70, 7, 87, 41, 9, 2, 34, 43, 84, 12, 24, 67, 63, 40, 78, 3, 100, 22, 100, 61, 59, 92, 26, 9, 39, 56, 93, 74, 47, 21, 71, 67, 81, 40, 74, 56, 34, 35, 82, 94, 35, 35, 15, 52, 44, 5, 83, 30, 10, 18, 65, 31, 45, 49, 100, 41, 26, 51, 3, 86, 17, 62, 13, 92, 58, 76, 53, 34, 81, 98, 57, 99, 81, 67, 23, 25, 99, 88, 62, 99, 37, 85, 17, 60, 23, 56, 97, 65, 41, 91, 16, 90, 47, 86, 56, 99, 44, 28, 18, 89, 27, 43, 43, 14, 64, 96, 8, 92, 74, 65, 24, 26, 96, 92, 19, 57, 24, 25, 3, 80, 99, 89, 78, 78, 80, 89, 27, 6, 49, 78, 81, 75, 99, 21, 64, 51, 98, 32, 53, 59, 74, 33, 1, 93, 9, 1, 24, 15, 8, 55, 76, 51, 98, 41, 77, 48, 81, 47, 76, 47, 65, 25, 2, 80, 67, 9, 85, 18, 73, 35, 50, 69, 46, 33, 14, 47, 25, 93, 28, 39, 12, 87, 85, 81, 16, 51, 91, 93, 32, 60, 55, 43, 54, 32, 57, 4, 30, 20, 15, 96, 64, 3, 99, 41, 5, 78, 28, 52, 39, 45, 41, 54, 1, 13, 53, 84, 75, 24, 100, 44, 8, 18, 46, 42, 86, 65, 27, 74, 1, 75, 99, 90, 33, 31, 4, 22, 17, 30, 44, 36, 72, 47, 75, 100, 47, 85, 86, 59, 37, 32, 30, 67, 98, 94, 85, 93, 1, 81, 60, 33, 97, 88, 73, 68, 8, 35, 30, 83, 19, 99, 74, 21, 93, 42, 80, 95, 27, 65, 24, 73, 31, 43, 92, 81, 24, 70, 67, 78, 48, 47, 70, 76, 12, 79, 89, 7, 28, 83, 78, 22, 25, 32, 17, 4, 68, 42, 15, 1, 3, 18, 43, 75, 48, 84, 17, 60, 100, 73, 59, 80, 68, 13, 89, 7, 93, 16, 22, 1, 58, 92, 87, 90, 23, 95, 76, 67, 10, 14, 70, 17, 99, 77, 6, 63, 69, 2, 93, 27, 29, 88, 39, 35, 25, 50, 91, 13, 16, 91, 50, 53, 54, 12, 53, 25, 11, 6, 10, 44, 36, 87, 67, 69, 5, 5, 78, 25, 19, 24, 50, 88, 62, 24, 89, 39, 86, 6, 7, 70, 56, 92, 18, 76, 57, 50, 28, 71, 50, 74, 19, 89, 49, 8, 76, 92, 80, 41, 34, 33, 63, 88, 31, 95, 97, 71, 52, 36, 26, 99, 72, 50, 76, 33, 62, 79, 11, 76, 54, 64, 42, 76, 5, 45, 79, 61, 39, 66, 72, 74, 76, 25, 63, 35, 100, 42, 61, 12, 9, 41, 95, 90, 48, 24, 8, 66, 65, 29, 74, 97, 54, 51, 31, 31, 51, 30, 63, 32, 70, 79, 49, 7, 35, 53, 76, 83, 62, 20, 13, 92, 95, 40, 99, 10, 98, 13, 7, 88, 16, 40, 10, 22, 29, 88, 64, 39, 13, 26, 12, 27, 69, 70, 23, 41, 67, 50, 96, 24, 97, 29, 31, 42, 27, 90, 50, 69, 42, 92, 22, 88, 23, 35, 83, 82, 74, 50, 72, 98, 94, 94, 46, 82, 16, 35, 88, 46, 89, 77, 86, 19, 17, 20, 5, 13, 25, 69, 79, 90, 55, 88, 71, 13, 30], (50,50))
b = Float64[66, 97, 50, 69, 24, 42, 23, 82, 25, 79, 66, 26, 76, 25, 75, 25, 43, 40, 55, 8, 20, 53, 66, 94, 57, 10, 39, 70, 5, 57, 22, 36, 45, 94, 24, 44, 89, 41, 14, 87, 9, 46, 74, 23, 72, 62, 52, 74, 36, 13]
linear_solve(A, b) = A \ b
@benchmark linear_solve(A, b)
C 代码运行大约 166 微秒，而 LAPACK（通过 Julia）代码平均运行 33 微秒（快 5 倍！）。
我想这是对 LAPACK 和相关 Julia 包装器质量的证明。
不幸的是，由于此 C 代码将成为 OpenCL 内核的一部分，我无法真正利用任何一个，有没有办法让我的 C 代码更高效？从而实现更接近于LAPACK的性能？
TL;DR: 当前的 C 代码 在现代硬件上效率低下 。此外，在专用 GPU 或 CUDA 上使用 OpenCL 只会 对于这里相当大的矩阵 很快（即不是 50x50 的矩阵）。
C 代码中最大的问题来自 A[K * l + (i + 1)] += c * A[K * l + j]; 行。实际上，由于循环迭代器是 l，内存访问模式不是连续的而是跨步的。 跨步内存访问模式 在现代硬件架构上比连续模式效率低得多（由于代码矢量化、高速缓存行、内存预取等）。在 GPU 上尤其如此。
您可以通过 转置 A 矩阵来解决此问题。这是修改后的版本：
// Naive (inefficient) transposition
// Please use the much faster BLAS function to do this (if possible)
void transpose(float A[K * K])
{
for (long j=0; j<K; ++j)
{
for (long i=j+1; i<K; ++i)
{
float tmp = A[K * i + j];
A[K * i + j] = A[K * j + i];
A[K * j + i] = tmp;
}
}
}
// Solve the system Ax=b using Gaussian elimination with partial pivoting.
// Work directly on the transposed version of A rather than transposing A every time should be much faster (especially for small matrices).
void fast_linear_solve(float A[K * K], float b[K])
{
// Not useful if A is already transposed
transpose(A);
for (long j=0; j<K; j++)
{
// Begin partial pivoting.
float maxval = fabs(A[K * j + j]);
long maxrow = j;
for (long i=j+1; i<K; i++)
{
if (fabs(A[K * i + j]) > maxval)
{
maxval = fabs(A[K * i + j]);
maxrow = i;
}
}
for (long l=0; l<K; l++)
{
float A_temp = A[K * maxrow + l];
A[K * maxrow + l] = A[K * j + l];
A[K * j + l] = A_temp;
}
float b_temp = b[maxrow];
b[maxrow] = b[j];
b[j] = b_temp;
// End partial pivoting.
// Begin putting [A; b] into row echelon form.
for (long i=j; i<K-1; i++)
{
float c = -A[K * (i + 1) + j] / A[K * j + j];
for (long l=j+1; l<K; l++)
A[K * (i + 1) + l] += c * A[K * j + l];
b[i + 1] += c * b[j];
}
// End putting [A; b] into row echelon form.
}
// Begin backsolving for x (by overwritting b).
for (long j=K-1; j>0; j--)
for (long i=j-1; i>=0; i--)
b[i] -= b[j] * A[K * i + j] / A[K * j + j];
for (long j=0; j<K; j++)
b[j] *= 1 / A[K * j + j];
// End backsolving for x.
// Not useful if A is already transposed
transpose(A);
}
另一个问题来自执行基准测试的方式。实际上，Julia 运行 多次执行代码，而 C 代码执行一次并使用 clock 函数。为了与 Julia 实现进行更公平的比较，C 实现的 linear_solve 函数必须 多次计算 （通过将其放入循环中并处理可能的聪明可能会增加一些偏差的编译器优化）。 gettimeofday 应该优先于 clock（因为前者计算挂钟时间，后者计算用户时间和系统时间之和）。
这是我机器上 50x50 矩阵的（平均）结果（GCC 9.3 使用 -O3，Clang 9.0 也使用 -O3，以及 Julia 1.4）：
Original C code (GCC): 25 us | Original C code (Clang): 25 us
New C code (GCC): 11 us | New C code (Clang): 12 us
Julia code: 80 us
以下是 500x500 随机矩阵的结果：
Original C code (GCC): 37.9 ms | Original C code (Clang): 38.8 ms
New C code (GCC): 6.7 ms | New C code (Clang): 6.1 ms
Julia code: 2.3 ms
大矩阵仍有改进的空间：C 代码可以使用 loop tiling 来改进（以降低代码的可读性和可维护性为代价）。
应该记住，虽然使用（专用）GPU 应该可以提高大矩阵的性能，但是由于 GPU 的相对 高延迟 [=47]，小矩阵不应该是这种情况=]（例如数据传输、同步、内存延迟）除非批处理用于许多小矩阵。