CholeskydecompositionTeodora Aleksić, 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

 

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𝐴=(1 1 11 2 21 2 3) 𝐿=(1 0 0

1 1 01 1 1)

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]; }}

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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

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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.

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Possible solutions

Square rootThere are more ways to calculate the square root of a number than just the standard sqrt function.

Predicting our loopsWe 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){ ... ...}

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Final Kernel graph

Possible improvements

So far, our stream went through the entire input and output matrix.

We can improve our performancesby only calculating those elements that matter to us.

2 3

6A =

1

9

54

7 8

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Cholesky decompositio

nIs it a good

algorithm for Maxeler?

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“I have not failed. I've just found

10,000 ways that won't work.

Thomas A. Edison

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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].◎ Bishop, J., “Linear Algebra: Cholesky Decomposition Example”, 2013., Available:

https://www.youtube.com/watch?v=NppyUqgQqd0 [Accessed: 26-Jan-2016]◎ Smith, J., “Cordelia”, Available: http://

www.slidescarnival.com/cordelia-free-presentation-template/216 [Accessed: 30-Jan-2016]

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Thanks!Any questions?

You can reach me at:at120391d@student.etf.rsOr:teodora_aleksic@ymail.com

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