Gaussian elimination (with no pivoting) in CUDA - c

Question

I am trying to solve Gaussian elimination with CUDA.
I have a N*N matrix. To get new elements of this matrix, I use the CPU code below, where C.width=N:
for(int z=0; z< C.width-1; z++)
{
for ( int c = z+1 ; c < C.width ; c++ )
{
for (int d = z ; d < C.width ; d++ )
{
C.elements[c*C.width+d]=C.elements[c*C.width+d] - (B.elements[c*C.width+z]*C.elements[z*C.width+d]);
}
}
}
I am trying to implement it with CUDA. For example, for N=512
dim3 dimBlock(16,16,1);
dim3 dimGrid(32,32,1);
MatMulKernel<<<dimGrid, dimBlock>>>(d_A, d_B, d_C);
I think for every iteration I should use N-i*N threads to calculate the elements update, that is
if(idx>511 || idy>510)
return;
for(int i=1; i<512;i++)
{
if(idx>=i-1 && idy>=i-1)
C.elements[(idy+1)*C.width+idx]=C.elements[(idy+1)*C.width+idx]-((C.elements[(idy+1)*C.width+(i-1)]/C.elements[(i-1)*C.width+(i-1)])*C.elements[(i-1)*C.width+idx]);
__syncthreads();
}
}
The results obtained on GPU and CPU are the same, but the processing time is Time(CPU)=2*Time(GPU)
For N=512: Time(CPU) = 1900 ms; Time(GPU) = 980 ms
For N=1024: Time(CPU) = 14000 ms; Time(GPU) = 7766 ms`
.
.
.
I think the speed-up should be larger than what I have now. Is there any mistake in my parallel code? Can you help me how can I rewrite my code?
Thanks for any help!

Answer 1

Gaussian elimination can be seen as a two steps procedure. The first step aims at transforming the linear system to an upper triangular linear system and the second consists of solving the so obtained upper triangular linear system. The second step is trivial in CUDA and can be efficiently performed by cublasStrsm. The first step, which you are addressing in your post, is the tricky part.
There are several optimized approaches to solve the first step. I think you approach is somewhat naive and I recommend studying the literature to achieve decent speedups.
Basically, performing the transformation of the original system to an upper triangular one can be performed by a tiling approach which, from some points of view, resembles the tiling approach which is used to perform the matrix-matrix multiplication in the classical example of the CUDA C Programming Guide.
The tiling approach can be performed either by purposely written kernels or by making massive use of cuBLAS routines.
Last month (November 2013), the following paper
Manuel Carcenac, "From tile algorithm to stripe algorithm: a CUBLAS-based parallel implementation on GPUs of Gauss method for the resolution of extremely large dense linear systems stored on an array of solid state devices", Journal of Supercomputing, DOI 10.1007/s11227-013-1043-3
has proposed a tiling/stripping approach based on the use of cuBLAS.
All the above mentioned approaches are summarized in a presentation available at M. Carcenac's webpage entitled Application: linear system resolution with Gauss method.
Furthermore, a downloadable Visual Studio 2010 project implementing all of them with some performance testing is available at the Gaussian elimination with CUDA post. From the available code, you can make your own tests for your architecture of interest and experience the improvements the approach by M. Carcenac is introducing with respect to the others.