qtyrep#

Purpose#

Computes the orthogonal-triangular (QR) decomposition of a matrix X using a pivot vector and returns \(Q'Y\) and \(R\).

Format#

{ qty, r, e } = qtyrep(y, x, pvt)#

Parameters:

y (NxL matrix) – data
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 placed at the end. Only the free columns will be moved during the decomposition.

Returns:

qty (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.,

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

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 \(X[.,E]\).

qtyrep() 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.

Examples#

// Create a 3x2 matrix
x = { 1 2,
      3 4,
      5 6 };

// Right-hand side vector
y = { 1,
      0,
      0 };

// Pivot vector: all columns are free
pvt = { 0, 0 };

// Compute Q'*Y, R, and permutation vector with controlled pivoting
{ qty, r, e } = qtyrep(y, x, pvt);

print "Q'*Y:";
print qty;
print "R (upper triangular):";
print r;
print "Permutation vector E:";
print e;

The above code produces the following output:

Q'*Y:

 -0.26726124
  0.87287156
  0.40824829

R (upper triangular):

  -7.4833148       -5.8797473
   0.0000000      -0.65465367

Permutation vector E:

   2.0000000
   1.0000000

Source#

qtyr.src