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

Math Functions / Matrices and Linear Algebra / Linear System Solvers

`dspsolvers`

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*

Double-precision floating point

Single-precision floating point