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Implement Hardware-Efficient Complex Partial-Systolic Q-less QR Decomposition

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This example shows how to implement a hardware-efficient Q-less QR decomposition using the Complex Partial-Systolic Q-less QR Decomposition block.

Economy Size Q-less QR Decomposition

The Complex Partial-Systolic Q-less QR Decomposition block performs the first step of solving the matrix equation A'AX = B which transforms A in-place to upper-triangular R, then solves the transformed system R'RX = B, where R'R = A'A.

Define Matrix Dimensions

Specify the number of rows and columns in matrix A.

m = 5;  % Number of rows in matrix A
n = 3;  % Number of columns in matrix A

Generate Matrix A

Use the helper function complexUniformRandomArray to generate a random matrix A such that the real and imaginary parts of the elements of A are between -1 and +1, and A is full rank.

rng('default')
A = fixed.example.complexUniformRandomArray(-1,1,m,n);

Select Fixed-Point Data Types

Use the helper function qlessqrFixedpointTypes to select fixed-point data types for matrix A that guarantee no overflow will occur in the transformation of A in-place to R.

The real and imaginary parts of the elements of A are between -1 and 1, so the maximum possible absolute value of any element is sqrt(2).

max_abs_A = sqrt(2);  % Upper bound on max(abs(A(:))
precisionBits = 24;   % Number of bits of precision
T = fixed.qlessqrFixedpointTypes(m,max_abs_A,precisionBits);
A = cast(A,'like',T.A);

Open the Model

model = 'ComplexPartialSystolicQlessQRModel';
open_system(model);

The Data Handler subsystem in this model takes complex matrix A as input. The ready port triggers the Data Handler. After sending a true validIn signal, there may be some delay before ready is set to false. When the Data Handler detects the leading edge of the ready signal, the block sets validIn to true and sends the next row of A. This protocol allows data to be sent whenever a leading edge of the ready signal is detected, ensuring that all data is processed.

Set Variables in the Model Workspace

Use the helper function setModelWorkspace to add the variables defined above to the model workspace. These variables correspond to the block parameters for the Complex Partial-Systolic Q-less QR Decomposition block.

numSamples = 1; % Number of sample matrices
fixed.example.setModelWorkspace(model,'A',A,'m',m,'n',n,...
    'numSamples',numSamples);

Simulate the Model

out = sim(model);

Construct the Solution from the Output Data

The Complex Partial-Systolic QR Decomposition block outputs matrix R at each time step. When a valid result matrix is output, the block sets validOut to true.

R = out.R;

R is an upper-triangular matrix.

R

R = 

   2.1863 + 0.0000i   0.6427 - 1.0882i  -0.5771 - 0.3089i
   0.0000 + 0.0000i   1.8126 + 0.0000i   0.2095 + 0.0599i
   0.0000 + 0.0000i   0.0000 + 0.0000i   1.7760 + 0.0000i

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 29
        FractionLength: 24

isequal(R,triu(R))

ans =

  logical

   1

Verify the Accuracy of the Output

To evaluate the accuracy of the Complex Partial-Systolic Q-less QR Decomposition block, compute the relative error.

relative_error = norm(double(R'*R - A'*A))/norm(double(A'*A))

relative_error =

   9.1285e-07

Suppress mlint warnings.

%#ok<*NOPTS>