Real Partial-Systolic Q-less QR Decomposition
Q-less QR decomposition for real-valued matrices
- Library:
Fixed-Point Designer HDL Support / Matrices and Linear Algebra / Matrix Factorizations

Description
The Real Partial-Systolic Q-less QR Decomposition block uses QR decomposition to compute the economy size upper-triangular R factor of the QR decomposition A = QR, where A is a real-valued matrix, without computing Q. The solution to A'Ax = B is x = R\R'\b.
When Regularization parameter is nonzero, the Real Partial-Systolic Q-less QR Decomposition block computes the upper-triangular factor R of the economy size QR decomposition of where λ is the regularization parameter.
Ports
Input
A(i,:)
— Rows of real matrix A
vector
Rows of real matrix A, specified as a vector. A is an m-by-n matrix where m ≥ 2 and n ≥ 2. If A is a fixed-point data type, A must be signed and use binary-point scaling. Slope-bias representation is not supported for fixed-point data types.
Data Types: single
| double
| fixed point
validIn
— Whether inputs are valid
Boolean
scalar
Whether inputs are valid, specified as a Boolean scalar. This control signal
indicates when the data from the A(i,:)
input port is valid. When
this value is 1 (true
) and the value of ready
is 1 (true
), the block captures the values at the
A(i,:)
input port. When this value is 0
(false
), the block ignores the input samples.
After sending a true
validIn
signal, there may be some delay before
ready
is set to false
. To ensure all data is
processed, you must wait until ready
is set to
false
before sending another true
validIn
signal.
Data Types: Boolean
restart
— Whether to clear internal states
Boolean
scalar
Whether to clear internal states, specified as a Boolean scalar. When this value
is 1 (true
), the block stops the current calculation and clears all
internal states. When this value is 0 (false
) and the value at
validIn
is 1 (true
), the block begins a new
subframe.
Data Types: Boolean
Output
R
— Upper-triangular matrix R
matrix
Economy size QR decomposition matrix R, returned as a vector or
matrix. R is an upper triangular matrix. The output at
R
has the same data type as the input at
A(i,:)
.
Data Types: single
| double
| fixed point
validOut
— Whether output data is valid
Boolean
scalar
Whether the output data is valid, specified as a Boolean scalar. This control
signal indicates when the data at output port R
is valid. When
this value is 1 (true
), the block has successfully computed the
matrix R. When this value is 0 (false
), the
output data is not valid.
Data Types: Boolean
ready
— Whether block is ready
Boolean
scalar
Whether the block is ready, returned as a Boolean scalar. This control signal
indicates when the block is ready for new input data. When this value is
1
(true
) and validIn
is
1
(true
), the block accepts input data in the
next time step. When this value is 0
(false
),
the block ignores input data in the next time step.
After sending a true
validIn
signal, there may be some delay before
ready
is set to false
. To ensure all data is
processed, you must wait until ready
is set to
false
before sending another true
validIn
signal.
Data Types: Boolean
Parameters
Number of rows in matrix A
— Number of rows in input matrix A
4
(default) | positive integer-valued scalar
Number of rows in input matrix A, specified as a positive integer-valued scalar.
Programmatic Use
Block Parameter:
m |
Type: character vector |
Values: positive integer-valued scalar |
Default:
4 |
Number of columns in matrix A
— Number of columns in input matrix A
4
(default) | positive integer-valued scalar
Number of columns in input matrix A, specified as a positive integer-valued scalar.
Programmatic Use
Block Parameter:
n |
Type: character vector |
Values: positive integer-valued scalar |
Default:
4 |
Regularization parameter
— Regularization parameter
0 (default) | real 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.
Programmatic Use
Block Parameter:
regularizationParameter |
Type: character vector |
Values: real nonnegative scalar |
Default:
0 |
Algorithms
Choosing the Implementation Method
Partial-systolic implementations prioritize speed of computations over space constraints, while burst implementations prioritize space constraints at the expense of speed of the operations. The following table illustrates the tradeoffs between the implementations available for matrix decompositions and solving systems of linear equations.
Implementation | Ready | Latency | Area | Sample block or example |
---|---|---|---|---|
Systolic | C | O(n) | O(mn2) | Implement Hardware-Efficient QR Decomposition Using CORDIC in a Systolic Array |
Partial-Systolic | C | O(m) | O(n2) | |
Partial-Systolic with Forgetting Factor | C | O(n) | O(n2) | Fixed-Point HDL-Optimized Minimum-Variance Distortionless-Response (MVDR) Beamformer |
Burst | O(n) | O(mn2) | O(n) |
Where C is a constant proportional to the word length of the data, m is the number of rows in matrix A, and n is the number of columns in matrix A.
Block Timing
The following table provides details on the timing for the QR decomposition blocks.
Block | validIn to ready (c cycles) | validIn to validOut (v cycles) |
---|---|---|
Real Partial-Systolic QR Decomposition | c = w + 8 | v = c(m + n - 1) |
Complex Partial-Systolic QR Decomposition | c = 2w + 15 | v = c(m + n - 1) |
Real Partial-Systolic Q-less QR Decomposition | c = w + 8 | v = c(m + n - 1) |
Complex Partial-Systolic Q-less QR Decomposition | c = 2w + 15 | v = c(m + n - 1) |
Real Partial-Systolic Q-less QR Decomposition with Forgetting Factor | c = w + 8 | v = c(2n - 1) |
Complex Partial-Systolic Q-less QR Decomposition with Forgetting Factor | c = 2w + 15 | v = c(2n - 1) |
In the table, m represents the number of rows in matrix A, and n is the number of columns in matrix A. w represents the word length of A.
If the data type of A is fixed point, then w is the word length.
If the data type of A is double, then w is 53.
If the data type of A is single, then w is 24.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Slope-bias representation is not supported for fixed-point data types.
HDL Code Generation
Generate Verilog and VHDL code for FPGA and ASIC designs using HDL Coder™.
HDL Coder™ provides additional configuration options that affect HDL implementation and synthesized logic.
This block has a single, default HDL architecture.
General | |
---|---|
ConstrainedOutputPipeline | Number of registers to place at
the outputs by moving existing delays within your design. Distributed
pipelining does not redistribute these registers. The default is
|
InputPipeline | Number of input pipeline stages
to insert in the generated code. Distributed pipelining and constrained
output pipelining can move these registers. The default is
|
OutputPipeline | Number of output pipeline stages
to insert in the generated code. Distributed pipelining and constrained
output pipelining can move these registers. The default is
|
Supports fixed-point data types only.
Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.
Version History
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