qqrep#

Purpose#

Computes the orthogonal-triangular (QR) decomposition of a matrix x, such that: \(X[., E] = Q_1R\)

Format#

{ q1, r, e } = qqrep(x, pvt)#

Parameters:

x (NxP matrix) – data
pvt (Px1 vector) –
controls the selection of the pivot columns:

if \(pvt[i] > 0\), \(x[i]\) is an initial column

if \(pvt[i] = 0\), \(x[i]\) is a free column

if \(pvt[i] < 0\), \(x[i]\) is a final column

The initial columns are placed at the beginning of the matrix and the final columns are placedat the end. Only the free columns will be moved during the decomposition.

Returns:

q1 (NxK unitary matrix) – unitary matrix, K = min(N,P).
r (KxP matrix) – upper triangular matrix
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

\[Q⁢ = \begin{bmatrix} Q_1 & Q_2 \end{bmatrix}\]

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

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

is the QR decomposition of \(X[., E]\).

qqrep() allows you to control the pivoting. For example, suppose that x is a dataset with a column of ones in the first column. If there are linear dependencies among the columns of x, the column of ones for the constant may get pivoted away. This column can be forced to be included among the linearly independent columns using pvt.

If you want only the \(R\) matrix, see qrep(). Not computing \(Q_1\) can produce significant improvements in computing time and memory usage.

Source#

qqr.src