Determine Fixed-Point Types for QR Decomposition

This example shows how to use fixed.qrFixedpointTypes to analytically determine fixed-point types for the computation of the QR decomposition.

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 $$B$$. This example sets $$B$$ to be the identity matrix the same size as the number of rows of $$A$$.

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

Generate Matrices A and B

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

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

Select Fixed-Point Types

Use fixed.qrFixedpointTypes to select fixed-point data types for matrices $$A$$ and $$B$$ that guarantee no overflow will occur in the transformation of $$A$$ in-place to $$R=Q^{\prime }A$$ and $$B$$ in-place to $$C=Q^{\prime }B$$.

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

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

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

T.B

ans = 

[]

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

Use the Specified Types to Compute the QR Decomposition

Cast the inputs to the types determined by fixed.qrFixedpointTypes.

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

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

fiaccel fixed.qrAB -args {A,B} -o qrAB_mex

Compute the QR decomposition.

[C,R] = qrAB_mex(A,B);

Extract the Economy-Size Q

The function fixed.qrAB transforms $\mathit{A}$ to $\mathit{R}={\mathit{Q}}^{\prime }\mathit{A}$ and $$B$$ to $$C=Q^{\prime }B$$. In this example, $$B$$ is the identity matrix, so $$Q=C^{\prime }$$ is the economy-size orthogonal factor of the QR decomposition.

Q = C';

Verify That Q Is Orthogonal and R Is Upper-Triangular

$$Q$$ is orthogonal, so $$Q^{\prime }Q$$ is the identity matrix within rounding error.

I = Q'*Q

I = 
    1.0000   -0.0000   -0.0000
   -0.0000    1.0000   -0.0000
   -0.0000   -0.0000    1.0000

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 62
        FractionLength: 48

$$R$$ 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.qrAB function, compute the relative error.

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

relative_error = 
1.5208e-06

Suppress mlint warnings.

%#ok<*NOPTS>