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Systolic 4x4 Matrix QR Decomposition Xiangfeng Wang Mark Chen

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Matrix Triangularization Given matrix A ij To triangularize A, we find a square orthogonal matrix Q and left multiply it with A.

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Matrix Triangularization For example, given Q 23 Left multiplying Q 23 with A will zero the A 32 value.

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Matrix Triangularization Using this principle, by setting up our Q correctly Left multiplying this Q with A will eliminate all value below the main diagonal of A.

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QR Decomposition

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The circular cell simply “reflects” or changes the direction of the data flow The square cell performs two functions. For token values (marked with a *), it will perform the sine and cosine values and store it. For all other values it will apply the sine and cosine values and then pass it along its respective path.

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QR Decomposition

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Generating the Sine and Cosine Sine Cosine x y X’ Y’ y’ = x*c + y*s x ’ = y*c – x*s

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sinecosine X’Y’ x=1, y=2, = actan(1/2) = 0.4636, sin = 0.4472, cos = 0.8944 y’= 2.2361, x’ = 1.0646e-004, time for the calculation ~25 cycles

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Generating and Applying the Rotation

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Simulation

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We finished one computational unit. We will build the whole System and figure out the right timing…

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