# Fixed-Point Matrix Operations in MATLAB

Optimized fixed-point math operations, matrix solvers, and matrix decomposition functions for efficient code

Use these functions to perform fixed-point math and matrix operations and generate efficient code. These functions solve systems of linear equations and perform matrix decompositions in a way that is efficient for embedded devices.

## Functions

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 `ceilDiv` Round the result of division toward positive infinity `fixed.cordicDivide` Fixed-point divide using CORDIC `fixed.cordicReciprocal` Fixed-point reciprocal using CORDIC `fixDiv` Round the result of division toward zero `floorDiv` Round the result of division toward negative infinity `nearestDiv` Round the result of division toward the nearest integer `modByConstant` Modulus after division by a constant denominator
 `fixed.backwardSubstitute` Solve upper-triangular system of equations through backward substitution `fixed.forwardSubstitute` Solve lower-triangular system of equations through forward substitution `fixed.qlessQR` Q-less QR decomposition `fixed.qlessQRUpdate` Update QR factorization `fixed.qrAB` Compute C = Q'*B and upper-triangular factor R `fixed.qrMatrixSolve` Solve system of linear equations Ax = B for x using QR decomposition `fixed.qlessQRMatrixSolve` Solve system of linear equations (A'A)x = B for x using Q-less QR decomposition
 `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 `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 `fixed.realQRMatrixSolveFixedpointTypes` Determine fixed-point types for matrix solution of real-valued AX=B and matrix solution using diagonal loading using QR decomposition `fixed.complexQRMatrixSolveFixedpointTypes` Determine fixed-point types for matrix solution of complex-valued AX=B and matrix solution using diagonal loading using QR decomposition `fixed.realQlessQRMatrixSolveFixedpointTypes` Determine fixed-point types for matrix solution of real-valued A'AX=B using QR decomposition `fixed.complexQlessQRMatrixSolveFixedpointTypes` Determine fixed-point types for matrix solution of complex-valued A'AX=B using QR decomposition `fixed.realSingularValueLowerBound` Estimate lower bound for smallest singular value of real-valued matrix `fixed.complexSingularValueLowerBound` Estimate lower bound for smallest singular value of complex-valued matrix `fixed.forgettingFactor` Compute forgetting factor required for streaming input data `fixed.forgettingFactorInverse` Compute the inverse of the forgetting factor required for streaming input data `fixed.realQuantizationNoiseStandardDeviation` Estimate standard deviation of quantization noise of real-valued signal `fixed.complexQuantizationNoiseStandardDeviation` Estimate standard deviation of quantization noise of complex-valued signal

## 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.

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.

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.

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.

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.

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