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# Implement Hardware-Efficient Real Burst QR Decomposition

This example shows how to implement a hardware-efficient QR decomposition using the Real Burst QR Decomposition block.

### Economy Size QR Decomposition

The Real Burst QR Decomposition block performs the first step of solving the least-squares matrix equation AX = B which transforms A in-place to R and B in-place to C = Q'B, then solves the transformed system RX = C, where QR is the orthogonal-triangular decomposition of A.

To compute the stand-alone QR decomposition, this example sets B to be the identity matrix so that the output of the Real Burst QR Decomposition block is the upper-triangular R and C = Q'.

### 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 p = m; % Number of columns in matrix B ```

### 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, and A is full rank. Matrix B is the identity matrix.

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

### Select Fixed-Point Data Types

Use the helper function `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 and B in-place to C = Q'B. For more information about how datatypes are selected, see the document FixedPointMatrixLibraryDatatypesExample.pdf in the current directory.

```max_abs_A = 1; % max(abs(A(:)) max_abs_B = 1; % max(abs(B(:)) f = 24; % Fraction length (bits of precision) T = fixed.example.qrFixedpointTypes(m,max_abs_A,max_abs_B,f); A = cast(A,'like',T.A); B = cast(B,'like',T.B); ```

### Open the Model

```model = 'RealBurstQRModel'; open_system(model); ``` The Data Handler subsystem in this model takes real matrices A and B as inputs. The `ready` port triggers the Data Handler. After sending a true `validIn` signal, there may be some delay before `ready` is set to false. When the Data Handler detects the leading edge of the `ready` signal, the block sets `validIn` to true and sends the next row of A and B. This protocol allows data to be sent whenever a leading edge of the `ready` signal is detected, ensuring that all data is processed.

### Set Variables in the Model Workspace

Use the helper function `setModelWorkspace` to add the variables defined above to the model workspace. These variables correspond to the block parameters for the Real Burst QR Decomposition block.

```numSamples = 1; % Number of sample matrices fixed.example.setModelWorkspace(model,'A',A,'B',B,'m',m,'n',n,'p',p,... 'numSamples',numSamples); ```

### Simulate the Model

```out = sim(model); ```

### Construct the Solution from the Output Data

The Real Burst QR Decomposition block outputs data one row at a time. When a result row is output, the block sets `validOut` to true. The rows of matrices R and C are output in reverse order to accommodate back-substitution, so you must reconstruct the data to interpret the results. To reconstruct the matrices R and C from the output data, use the helper function `qrModelOutputToArray`.

```[C,R] = fixed.example.qrModelOutputToArray(out.C,out.R,m,n,p,numSamples); ```

### Extract the Economy-Size Q

The block computes C = Q'B. 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 orothogonal, so Q'Q is the identity matrix within roundoff.

```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 Real Burst QR Decomposition block, compute the relative error.

```relative_error = norm(double(Q*R - A))/norm(double(A)) ```
```relative_error = 1.3415e-06 ```

Suppress mlint warnings.

```%#ok<*NOPTS> ```

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