Systolic 4x4 Matrix QR Decomposition

Xiangfeng Wang

Mark Chen

Matrix Triangularization

Given matrix Aij

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

44434241

34333231

24232221

14131211

aaaa

aaaa

aaaa

aaaa

A

Matrix Triangularization

For example, given Q23

Left multiplying Q23 with A will zero the A32 value.

1000

00

00

0001

23 cs

scQ

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.

))()(( 433242213141 QQQQQQQ

QR Decomposition

*44a

14a 13a 12a *11a

24a 23a *22a 21a

34a *33a 32a31a

43a 42a 41a

*11

12

13

14

'

'

'

'

a

a

a

a

*22

23

24

'

'

'

a

a

a *33

34

'

'

a

a*44'a

QR Decomposition

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.

x

y

1x 2x

1y

2y

QR Decomposition

*44a

14a 13a 12a *11a

24a 23a *22a 21a

34a *33a 32a31a

43a 42a 41a

*11

12

13

14

'

'

'

'

a

a

a

a

*22

23

24

'

'

'

a

a

a *33

34

'

'

a

a*44'a

Generating the Sine and Cosine

Sine

Cosine

x

y

X’

Y’

y’ = x*c + y*s x’ = y*c – x*s

sine cosine

X’Y’

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

Generating and Applying the Rotation

Simulation

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