qyre#

Purpose#

Computes the orthogonal-triangular (QR) decomposition of a matrix x and returns $Q Y$ and $R$ .

Format#

{ qy, r, e } = qyre(y, x)#

Parameters:

y (NxL matrix) – data
x (NxP matrix) – data

Returns:

qy (NxL matrix) – unitary matrix
r (KxP matrix) – upper triangular matrix. $K = m i n (N, P)$ .
e (Px1 vector) – permutation vector

Remarks#

Given $X [., E]$ , where $E$ is a permutation vector that permutes the columns of $X$ , there is an orthogonal matrix $Q$ such that $Q^{'} X [., E]$ is zero below its diagonal, i.e.,

\begin{array}{r} Q' R [., E] = [\begin{array}{c} R \\ 0 \end{array}] \end{array}

where $R$ is upper triangular. If we partition

Q = [Q_{1} Q_{2}]

where $Q_{1}$ has $P$ columns, then

X [., E] = Q_{1} R

is the QR decomposition of $X [., E]$ .

For most problems $Q$ or $Q_{1}$ is not what is required. Since $Q$ can be a very large matrix, qyre() has been provided for the calculation of $Q Y$ , where $Y$ is some NxL matrix, which will be a much smaller matrix.

If either $Q^{'} Y$ or $Q_{1}^{'} Y$ are required, see qtyre().

If $N < P$ , the factorization assumes the form:

Q' R [., E] = [R_{1} R_{2}]

where $R_{1}$ is a PxP upper triangular matrix and $R_{2}$ is $P \times (N - P) ‘$ . Thus $Q$ is a PxP matrix and $R$ is a PxN matrix containing $R_{1}$ and $R_{2}$ .

Source#

qyr.src