Implement Hardware-Efficient Complex Partial-Systolic Matrix Solve Using QR Decomposition with Tikhonov Regularization
This example shows how to use the Complex Partial-Systolic Matrix Solve Using QR Decomposition block to solve the regularized least-squares matrix equation
where A is an m-by-n matrix with m >= n, B is m-by-p, X is n-by-p, eye(n), zeros(n,p), and is a regularization parameter.
The least-squares solution is
but is computed without squares or inverses.
Define Matrix Dimensions
Specify the number of rows in matrices A and B, the number of columns in matrix A, and the number of columns in matrix B.
m = 100; % Number of rows in matrices A and B n = 10; % Number of columns in matrix A p = 1; % Number of columns in matrix B
Define Tikhonov Regularization Parameter
Small, positive values of the regularization parameter can improve the conditioning of the problem and reduce the variance of the estimates. While biased, the reduced variance of the estimate often results in a smaller mean squared error when compared to least-squares estimates.
regularizationParameter = 0.01;
Generate Random Least-Squares Matrices
For this example, use the helper function
complexRandomLeastSquaresMatrices to generate random matrices A and B for the least-squares problem AX=B. The matrices are generated such that the real and imaginary parts of the elements of A and B are between -1 and +1, and A has rank r.
rng('default') r = 3; % Rank of A [A,B] = fixed.example.complexRandomLeastSquaresMatrices(m,n,p,r);
Select Fixed-Point Data Types
Use the helper function
complexQRMatrixSolveFixedpointTypes to select fixed-point data types for input matrices A and B, and output X such that there is a low probability of overflow during the computation.
The real and imaginary parts of the elements of A and B 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(:)) max_abs_B = sqrt(2); % Upper bound on max(abs(B(:)) precisionBits = 32; % Number of bits of precision T = fixed.complexQRMatrixSolveFixedpointTypes(m,n,max_abs_A,max_abs_B,... precisionBits,,,regularizationParameter); A = cast(A,'like',T.A); B = cast(B,'like',T.B); OutputType = fixed.extractNumericType(T.X);
Open the Model
model = 'ComplexPartialSystolicQRMatrixSolveModel'; open_system(model);
The Data Handler subsystem in this model takes complex matrices A and B as inputs. It sends rows of A and B to QR block using the AMBA AXI handshake protocol. The
validIn signal indicates when data is available. The
ready signal indicates that the block can accept the data. Transfer of data occurs only when both the
ready signals are high. You can set a delay between the feeding in rows of A and B in the Data Handler to emulate the processing time of the upstream block.
validIn remains high when
rowDelay is set to 0 becuase this indicates the Data Handler always has data available.
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 Matrix Solve Using QR Decomposition block.
numSamples = 1; % Number of sample matrices rowDelay = 1; % Delay of clock cycles between feeding in rows of A and B fixed.example.setModelWorkspace(model,'A',A,'B',B,'m',m,'n',n,'p',p,... 'regularizationParameter',regularizationParameter,... 'numSamples',numSamples,'rowDelay',rowDelay,'OutputType',OutputType);
Simulate the Model
out = sim(model);
Construct the Solution from the Output Data
The Complex Partial-Systolic Matrix Solve Using QR Decomposition block outputs data one row at a time. When a result row is output, the block sets
validOut to true. The rows of X are output in the order they are computed, last row first, so you must reconstruct the data to interpret the results. To reconstruct the matrix X from the output data, use the helper function
X = fixed.example.matrixSolveModelOutputToArray(out.X,n,p,numSamples);
Verify the Accuracy of the Output
Verify that the relative error between the fixed-point output and builtin MATLAB in double-precision floating-point is small.
A_lambda = double([regularizationParameter*eye(n);A]); B_0 = [zeros(n,p);double(B)]; X_double = A_lambda\B_0; relativeError = norm(X_double - double(X))/norm(X_double)
relativeError = 1.3921e-05
mlint warnings in this file.
%#ok<*NOPTS> %#ok<*NASGU> %#ok<*ASGLU>
Complex Partial-Systolic Matrix Solve Using QR Decomposition