1. Cholesky decomposition Teodora Aleksi ć, 391/2012

2. 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

3. C implementation Each element of the resulting matrix is calculated with one of these two equations. Using them we can assemble the C code. for (int i = 0; i < n; i++) for (int j = 0; j < (i+1); j++) { double sum = a[i][j]; for (int k = 0; k < j; k++) sum -= l[i][k] * l[j][k]; if(i == j){ l[i][j] = sqrt(sum); } else{ l[i][j] = sum / l[i][i]; } 3/12

4. 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

5. 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

6. 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 res1 = sum / 2; DFEVar temp = res1; do{ temp = res1; res1 = (temp + (sum / temp)) / 2; }while((temp - res1) != zero) DFEVar res1 = sum / 2; DFEVar temp = res1; for(int k = 0; k < N; ++k){... } 6/12

7. 7/12 Final Kernel graph

8. 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. 23 6 A = 1 9 54 78 8/12

9. Cholesky decomposition Is it a good algorithm for Maxeler? 9/12

10. “ I have not failed. I've just found 10,000 ways that won't work. Thomas A. Edison 10/12

11. References This presentations was made using the following resources: ◎ Milutinovic, V., Salom, J., Trifunovic, N., Giorgi, R., “Guide to DataFlow SuperComputing ”, Springer, 2015. ◎ Milutinovic, V., et al., “Paradigm Shift in SuperComputing: DataFlow vs. ControlFlow ”, Journal of Big Data, 2015. ◎ Hurson, A., Milutinovic, V., editors, “DataFlow Processing ”, Elsevier, 2015. ◎ Wikipedia.org, “Cholesky decomposition ”, 2002., Available: https://en.wikipedia.org/wiki/Cholesky_decomposition [Accessed: 26-Jan-2016]. https://en.wikipedia.org/wiki/Cholesky_decomposition ◎ Bishop, J., “Linear Algebra: Cholesky Decomposition Example ”, 2013., Available: https://www.youtube.com/watch?v=NppyUqgQqd0 [Accessed: 26-Jan-2016] https://www.youtube.com/watch?v=NppyUqgQqd0 ◎ Smith, J., “Cordelia ”, Available: http://www.slidescarnival.com/cordelia-free- presentation-template/216 [Accessed: 30-Jan-2016]http://www.slidescarnival.com/cordelia-free- presentation-template/216 11/12

12. Thanks! Any questions? You can reach me at: at120391d@student.etf.rs Or: teodora_aleksic@ymail.com 12/12