User`s guide

QR Solver
5-351
5QR Solver
Purpose Find a minimum-norm-residual solution to the equation AX=B.
Library Math Functions / Matrices and Linear Algebra / Linear System Solvers
Description The QR Solver block solves the linear system AX=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-hand-side M-by-L matrix, B. A length-M 1-D vector
input at either port is treated as an M-by-1 matrix.
The output at the
x port is the N-by-L matrix, X. X is always sample based, and
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.
Algorithm QR factorization factors a column-permuted variant (A
e
) of the M-by-N input
matrix A as
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
,
and
A
e
QR=
A
e
XB
e
=
QRX B
e
=