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Find minimum-norm-residual solution to A*X*=B

The QR Solver block solves the linear system A*X*=B,
which can be overdetermined, underdetermined, or exactly determined.
The system is solved by applying QR factorization to the M-by-N matrix,
A, at the `A` port. The input to the `B` port
is the right side M-by-L matrix, B. The block treats length-M unoriented
vector input as an M-by-1 matrix.

The output at the `x` port is the N-by-L matrix,
X. X is chosen to minimize the sum of the squares of the elements
of B-AX*. *When B is a vector, this solution minimizes
the vector 2-norm of the residual (B-AX is the residual). When B is
a matrix, this solution minimizes the matrix Frobenius norm of the
residual. In this case, the columns of X are the solutions to the
L corresponding systems AX_{k}=B_{k},
where B_{k} is the kth column of B, and X_{k} is
the kth column of *X*.

X is known as the minimum-norm-residual solution to AX=B. The
minimum-norm-residual solution is unique for overdetermined and exactly
determined linear systems, but it is not unique for underdetermined
linear systems. Thus when the QR Solver is applied to an underdetermined
system, the output *X* is chosen such that the
number of nonzero entries in X is minimized.

QR factorization factors a column-permuted variant (A_{e})
of the M-by-N input matrix A as

*A*_{e} = *QR*

where Q is a M-by-min(M,N) unitary matrix, and R is a min(M,N)-by-N upper-triangular matrix.

The factored matrix is substituted for A_{e} in

*A*_{e}*X* = *B*_{e}

and

*QRX* = *B*_{e}

is solved for X by noting that Q^{-1} =
Q^{*} and substituting Y = Q^{*}B_{e}.
This requires computing a matrix multiplication for Y and solving
a triangular system for X.

*RX* = *Y*

Levinson-Durbin | DSP System Toolbox |

LDL Solver | DSP System Toolbox |

LU Solver | DSP System Toolbox |

QR Factorization | DSP System Toolbox |

SVD Solver | DSP System Toolbox |

See Linear System Solvers for related information.

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