fixed.qrFixedpointTypes

Determine fixed-point types for transforming A and R and B to C=Q'B in-place, where QR=A is QR decomposition of A

Since R2021b

Syntax

T = fixed.qrFixedpointTypes(m,max_abs_A,max_abs_B,precisionBits)

T = fixed.qrFixedpointTypes(m,max_abs_A,max_abs_B,precisionBits,regularizationParameter)

T = fixed.qrFixedpointTypes(___,maxWordLength)

Description

T = fixed.qrFixedpointTypes(m,max_abs_A,max_abs_B,precisionBits) returns fixed-point types for A and B that guarantee no overflow will occur in the QR algorithm.

The QR algorithm transforms A in-place into upper-triangular R and transforms B in-place into C=Q'B, where QR=A is the QR decomposition of A.

T = fixed.qrFixedpointTypes(m,max_abs_A,max_abs_B,precisionBits,regularizationParameter) returns fixed-point types for transforming $$\left[\begin{array}{c}\lambda I_n\\ A\end{array}\right]$$ in-place to $$R=Q\text{'}\left[\begin{array}{c}\lambda I_n\\ A\end{array}\right]$$ and $$\left[\begin{array}{c}{0}_{n,p}\\ B\end{array}\right]$$ in-place to $$C=Q\text{'}\left[\begin{array}{c}{0}_{n,p}\\ B\end{array}\right]$$ where λ is the regularizationParameter, QR is the economy size QR decomposition of $$\left[\begin{array}{c}\lambda I_n\\ A\end{array}\right]$$, A is an m-by-n matrix, p is the number of columns in B, I_n = eye(n), and 0_n,p = zeros(n,p). regularizationParameter is an optional parameter. If not supplied or empty, then the default value is used.

T = fixed.qrFixedpointTypes(___,maxWordLength) specifies the maximum word length of the fixed-point types. maxWordLength is an optional parameter. If not supplied or empty, then the default value is used.

Examples

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>

Input Arguments

m — Number of rows in A

positive integer-valued scalar

Number of rows in A, specified as a positive integer-valued scalar.

Data Types: double

max_abs_A — Maximum of absolute value of A

scalar

Maximum of the absolute value of A, specified as a scalar.

Example: max(abs(A(:)))

Data Types: double

max_abs_B — Maximum of absolute value of B

scalar

Maximum of the absolute value of B, specified as a scalar.

Example: max(abs(B(:)))

Data Types: double

precisionBits — Required number of bits of precision

positive integer-valued scalar

Required number of bits of precision, specified as a positive integer-valued scalar.

Data Types: double

regularizationParameter — Regularization parameter

0 (default) | nonnegative scalar

Regularization parameter, specified as a nonnegative scalar. 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.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi

maxWordLength — Maximum word length of fixed-point types

65535 (default) | positive integer

Maximum word length of fixed-point types, specified as a positive integer.

If the word length of the fixed-point type exceeds the specified maximum word length, the default of 65535 bits is used.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi

Output Arguments

T — Fixed-point types for A and B

structure

Fixed-point types for A and B, returned as a structure. The structure T has fields T.A and T.B. These fields contain fi objects that specify fixed-point types for A and B that guarantee no overflow will occur in the QR algorithm.

The QR algorithm transforms A in-place into upper-triangular R and transforms B in-place into C=Q'B where QR=A is the QR decomposition of A.

Tips

Use fixed.qrFixedpointTypes to compute fixed-point types for the inputs of these functions and blocks.

• fixed.qrAB

• Complex Burst QR Decomposition

• Complex Partial-Systolic QR Decomposition

• Real Burst QR Decomposition

• Real Partial-Systolic QR Decomposition

Algorithms

The number of integer bits required to prevent overflow is derived from the following bounds on the growth of R and C=Q'B [1]. The required number of integer bits is added to the number of bits of precision, precisionBits, of the input, plus one for the sign bit, plus one bit for intermediate CORDIC gain of approximately 1.6468 [2].

The elements of R are bounded in magnitude by

$$\max \left(\left|R(:)\right|\right)\le \sqrt{m}\max \left(\left|A(:)\right|\right).$$

The elements of C=Q'B are bounded in magnitude by

$$\max \left(\left|C(:)\right|\right)\le \sqrt{m}\max \left(\left|B(:)\right|\right).$$

References

[1] Perform QR Factorization Using CORDIC

[2] Voler, Jack E. "The CORDIC Trigonometric Computing Technique." IRE Transactions on Electronic Computers EC-8 (1959): 330-334.

Version History

Introduced in R2021b

R2022b: Support for maximum word length

You can now use the maxWordLenth parameter to specify the maximum word length of the fixed-point types.

R2022a: Support for Tikhonov regularization parameter

The fixed.qrfixedpointtypes function now supports the Tikhonov regularization parameter, regularizationParameter.