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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=QA and B in-place to C=QB.

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 A to R=QA in-place so that it does not overflow.

T.A
ans = 

[]

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 28
        FractionLength: 24

T.B is the type computed for transforming B to C=QB in-place so that it does not overflow.

T.B
ans = 

[]

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 28
        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 A to R=QA and B to C=QB. In this example, B is the identity matrix, so Q=C 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 QQ 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: 60
        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: 28
        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 = 1.3415e-06

Suppress mlint warnings.

%#ok<*NOPTS>

See Also

Functions

Blocks