qyre#

Purpose#

Computes the orthogonal-triangular (QR) decomposition of a matrix x and returns \(QY\) 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. \(K = min(N,P)\).
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.,

\[\begin{split}Q′R[ ., E ] = \begin{bmatrix} R \\ 0 \end{bmatrix}\end{split}\]

where \(R\) is upper triangular. If we partition

\[Q = [Q_1 Q_2]\]

where \(Q_1\) has \(P\) columns, then

\[X[ ., E ] = Q_1R\]

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 \(QY\), 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