Cholesky decomposition Teodora Aleksic 3912012 Cholesky decomposition Cholesky
- Slides: 12
Cholesky decomposition Teodora Aleksic, 391/2012
Cholesky decomposition ◎ Cholesky factorization is a decomposition of a symmetric, positive-definite matrix (A) into a lower triangular matrix (L) ◎ It is mainly used for solutions of linear equations and Monte Carlo simulations 2/12
C implementation for (int element i = 0; i < n; i++) Each of the for (int j = 0; j < (i+1); j++) { resulting is doublematrix sum = a[i][j]; for (int k =with 0; k < j; one k++) of calculated sum -= l[i][k] * l[j][k]; these two equations. == j){ we can Usingif(ithem l[i][j] = sqrt(sum); assemble the C code. } } else{ l[i][j] = sum / l[i][i]; } 3/12
Maxeler implementation challenges ◎ Our output matrix also serves as our input matrix ◎An alternate solution to the square root ◎ Overcoming the barrier between regular Java types and DFE 4/12
L_Test matrix as our matrix L Since we are comparing the C and Maxeler implementation, we can use the result of the C decomposition as our input for the Maxeler calculation. 5/12
Possible solutions Square root There are more ways to calculate the square root of a number than just the standard sqrt function. Predicting our loops We can predict the lengths of our loops instead having them depend on our DFE variables. DFEVar res 1 = sum / 2; DFEVar temp = res 1; do{ temp = res 1; res 1 = (temp + (sum / temp)) / 2; }while((temp - res 1) != zero) for(int k = 0; k < (N / 2); ++k){. . . } 6/12
Final Kernel graph 7/12
Possible improvements So far, our stream went through the entire input and output matrix. We can improve our performances by only calculating those elements that matter to us. A= 1 2 3 4 5 6 7 8 9 8/12
Cholesky decomposition Is it a good algorithm for Maxeler? 9/12
“ I have not failed. I've just found 10, 000 ways that won't work. Thomas A. Edison 10/12
References This presentations was made using the following resources: Information about the Cholesky decomposition: ◎ https: //en. wikipedia. org/wiki/Cholesky_decomposition ◎ https: //www. youtube. com/watch? v=Nppy. Uqg. Qqd 0 Information about Maxeler: ◎ http: //home. etf. rs/~vm/os/vlsi/index. html 11/12
Thanks! Any questions? You can reach me at: at 120391 d@student. etf. rs 12/12
