Determine Fixed-Point Types for Q-less QR Decomposition

This example shows how to use fixed.qlessqrFixedpointTypes to analytically determine a fixed-point type for the computation of the Q-less QR decomposition.

Define Matrix Dimensions

Specify the number of rows and columns in matrix $$A$$.

m = 10; % Number of rows in matrix A
n = 3;  % Number of columns in matrix A

Generate Matrix A

Use the helper function realUniformRandomArray to generate a random matrix $$A$$ such that the elements of $$A$$ are between $$-1$$ and $$+1$$.

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

Select Fixed-Point Type

Use the fixed.qlessqrFixedpointTypes function to select the fixed-point data type for matrix $$A$$ that guarantees no overflow will occur in the transformation of $$A$$ in-place to $$R=Q^{\prime }A$$.

max_abs_A = 1;  % Upper bound on max(abs(A(:))
precisionBits = 24;  % Number of bits of precision
T = fixed.qlessqrFixedpointTypes(m,max_abs_A,precisionBits)

T = struct with fields:
    A: [0x0 embedded.fi]

T.A is the type computed for transforming $\mathit{A}$ to $\mathit{R}={\mathit{Q}}^{\prime }\mathit{A}$ in-place so that it does not overflow.

T.A

ans = 

[]

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

Use the Specified Type to Compute the Q-less QR Decomposition

Cast the input to the type determined by fixed.qlessqrFixedpointTypes.

A = cast(A,'like',T.A);

Accelerate fixed.qlessQR by using fiaccel to generate a MATLAB® executable (MEX) function.

fiaccel fixed.qlessQR -args {A} -o qlessQR_mex

Compute the QR decomposition.

R = qlessQR_mex(A);

Verify That R Is Upper-Triangular

$$R$$ is an upper-triangular matrix.

R

R = 
    2.2180    0.8559   -0.5607
         0    2.0578   -0.4017
         0         0    1.7117

          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 fixed.qlessQR function, compute the relative error.

$\mathit{R}={\mathit{Q}}^{\prime }\mathit{A}$, and $$Q$$ is orthogonal, so $$R^{\prime }R=A^{\prime }Q{Q}^{\prime }A=A^{\prime }A$$, within rounding error.

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

relative_error = 
1.0699e-06

Suppress mlint warnings.

%#ok<*NOPTS>