Fixed-Point Matrix Operations in MATLAB

Optimized CORDIC-based fixed-point matrix solvers and matrix decomposition functions for efficient code

Use these functions to perform fixed-point matrix operations and generate efficient C/C++ code. These functions solve systems of linear equations and perform core matrix operations, such as QR decomposition and singular value decomposition, using a CORDIC-based algorithm that is efficient for embedded devices. Use the included Fixed-Point Designer™ functions to analytically determine optimal fixed-point data types for the linear system solver and matrix factorization blocks.

For Simulink^® block implementations of these algorithms, see Fixed-Point Matrix Operations in Simulink. For CORDIC-based and other embedded-efficient implementations of math operations in MATLAB^® and Simulink, see Fixed-Point Math Operations in MATLAB and Simulink.

Functions

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Linear System Solvers and Matrix Factorizations

`fixed.backwardSubstitute`	Solve upper-triangular system of equations through backward substitution (Since R2020b)
`fixed.forwardSubstitute`	Solve lower-triangular system of equations through forward substitution (Since R2020b)
`fixed.jacobiSVD`	Fixed-point Jacobi singular value decomposition (Since R2023a)
`fixed.qlessQR`	Q-less QR decomposition (Since R2020b)
`fixed.qlessQRUpdate`	Update QR factorization (Since R2020b)
`fixed.qrAB`	Compute C = Q'B and upper-triangular factor R (Since R2020b)
`fixed.qrMatrixSolve`	Solve system of linear equations Ax = B for x using QR decomposition (Since R2020b)
`fixed.qlessQRMatrixSolve`	Solve system of linear equations (A'A)X = B for X using Q-less QR decomposition (Since R2020b)
`fixed.svd`	Fixed-point Golub-Kahan-Reinsch singular value decomposition (Since R2022b)
`svd`	Fixed-point Golub-Kahan-Reinsch singular value decomposition (Since R2022b)

Analytically Determine Fixed-Point Data Types for Linear System Solvers and Matrix Factorizations

`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)
`fixed.qlessqrFixedpointTypes`	Determine fixed-point types for transforming A to R in-place, where R is upper-triangular factor of QR decomposition of A, without computing Q (Since R2021b)
`fixed.realQRMatrixSolveFixedpointTypes`	Determine fixed-point types for matrix solution of real-valued AX=B using QR decomposition (Since R2021b)
`fixed.complexQRMatrixSolveFixedpointTypes`	Determine fixed-point types for matrix solution of complex-valued AX=B using QR decomposition (Since R2021b)
`fixed.realQlessQRMatrixSolveFixedpointTypes`	Determine fixed-point types for matrix solution of real-valued A'AX=B using QR decomposition (Since R2021b)
`fixed.complexQlessQRMatrixSolveFixedpointTypes`	Determine fixed-point types for matrix solution of complex-valued A'AX=B using QR decomposition (Since R2021b)
`fixed.realSingularValueLowerBound`	Estimate lower bound for smallest singular value of real-valued matrix (Since R2021b)
`fixed.complexSingularValueLowerBound`	Estimate lower bound for smallest singular value of complex-valued matrix (Since R2021b)
`fixed.singularValueUpperBound`	Upper bound of largest singular value of matrix (Since R2022b)
`fixed.realConditionNumberUpperBound`	Estimate of upper bound for 2-norm condition number of real-valued matrix (Since R2022b)
`fixed.complexConditionNumberUpperBound`	Estimate of upper bound for 2-norm condition number of complex-valued matrix (Since R2022b)
`fixed.forgettingFactor`	Compute forgetting factor required for streaming input data (Since R2021b)
`fixed.forgettingFactorInverse`	Compute the inverse of the forgetting factor required for streaming input data (Since R2021b)
`fixed.realQuantizationNoiseStandardDeviation`	Estimate standard deviation of quantization noise of real-valued signal (Since R2021b)
`fixed.complexQuantizationNoiseStandardDeviation`	Estimate standard deviation of quantization noise of complex-valued signal (Since R2021b)

Topics

Determine Fixed-Point Types for QR Decomposition
Use fixed.qrFixedpointTypes to determine fixed-point types for computation of QR decomposition.
Determine Fixed-Point Types for Q-less QR Decomposition
Use fixed.qlessqrFixedpointTypes to determine fixed-point types for computation of Q-less QR decomposition.
Algorithms to Determine Fixed-Point Types for Complex Q-less QR Matrix Solve A'AX=B
Derivation of algorithms for determining fixed-point types for complex Q-less QR matrix solve.
Determine Fixed-Point Types for Complex Q-less QR Matrix Solve A'AX=B
Use fixed.complexQlessQRFixedpointTypes to determine fixed-point types for computation of the complex least-squares matrix equation.
Determine Fixed-Point Types for Complex Q-less QR Matrix Solve with Tikhonov Regularization
This example shows how to use the fixed.complexQlessQRMatrixSolveFixedpointTypes function to analytically determine fixed-point types for the solution of the complex least-squares matrix equation
Algorithms to Determine Fixed-Point Types for Complex Least-Squares Matrix Solve AX=B
Derivation of algorithms for determining fixed-point types for complex QR matrix solve.
Determine Fixed-Point Types for Complex Least-Squares Matrix Solve AX=B
Use fixed.complexQRFixedpointTypes to determine fixed-point types for computation of the complex least-squares matrix equation.
Determine Fixed-Point Types for Complex Least-Squares Matrix Solve with Tikhonov Regularization
This example shows how to use the fixed.complexQRMatrixSolveFixedpointTypes function to analytically determine fixed-point types for the solution of the complex least-squares matrix equation
Algorithms to Determine Fixed-Point Types for Real Q-less QR Matrix Solve A'AX=B
Derivation of algorithms for determining fixed-point types for real Q-less QR matrix solve.
Determine Fixed-Point Types for Real Q-less QR Matrix Solve A'AX=B
Use fixed.realQlessQRFixedpointTypes to determine fixed-point types for computation of the real least-squares matrix equation.
Determine Fixed-Point Types for Real Q-less QR Matrix Solve with Tikhonov Regularization
This example shows how to use the fixed.realQlessQRMatrixSolveFixedpointTypes function to analytically determine fixed-point types for the solution of the real least-squares matrix equation
Algorithms to Determine Fixed-Point Types for Real Least-Squares Matrix Solve AX=B
Derivation of algorithms for determining fixed-point types for real least-squares matrix solve.
Determine Fixed-Point Types for Real Least-Squares Matrix Solve AX=B
Use fixed.realQRMatrixSolveFixedpointTypes to determine fixed-point types for computation of the real least-squares matrix equation.
Determine Fixed-Point Types for Real Least-Squares Matrix Solve with Tikhonov Regularization
This example shows how to use the fixed.realQRMatrixSolveFixedpointTypes function to analytically determine fixed-point types for the solution of the real least-squares matrix equation
Compute Forgetting Factor Required for Streaming Input Data
Use fixed.forgettingFactor and fixed.forgettingFactorInverse to compute forgetting factor.
Estimate Standard Deviation of Quantization Noise of Complex-Valued Signal
Use fixed.complexQuantizationNoiseStandardDeviation to estimate standard deviation of quantization noise.
Estimate Standard Deviation of Quantization Noise of Real-Valued Signal
Use fixed.realQuantizationNoiseStandardDeviation to estimate standard deviation of quantization noise.

Featured Examples

Perform QR Factorization Using CORDIC

Write MATLAB code that works for both floating-point and fixed-point data types. The algorithm used in this example is the QR factorization implemented via CORDIC.

Open Live Script

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