qtyr#

Purpose#

Computes the orthogonal-triangular (QR) decomposition of a matrix $X$ and returns $Q^{'} Y$ and $R$ .

Format#

{ qty, r } = qtyr(y, X)#

Parameters:

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

Returns:

qty (NxL matrix) – unitary matrix
r (KxP matrix) – upper triangular matrix. $K = m i n (N, P)$ .

Examples#

The QR algorithm is the numerically superior method for the solution of least squares problems:

loadm x, y;
{ qty, r } = qtyr(y, x);
q1ty = qty[1:rows(r), .];
q2ty = qty[rows(r)+1:rows(qty), .];

// LS coefficients
b = qrsol(q1ty, r);

// Residual sums of squares
s2 = sumc(q2ty^2);

Remarks#

Given $X$ , there is an orthogonal matrix $Q$ such that $Q^{'} X$ is zero below its diagonal, i.e.,

\begin{array}{r} Q' X = [\begin{array}{c} R \\ 0 \end{array}] \end{array}

where $R$ is upper triangular. If we partition

Q = [\begin{matrix} Q_{1} & Q_{2} \end{matrix}]

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

X = Q_{1} R

is the QR decomposition of $X$ . If $X$ has linearly independent columns, $R$ is also the Cholesky factorization of the moment matrix of $X$ , i.e., of $X^{'} X$ . For most problems $Q$ or $Q_{1}$ is not what is required. Rather, we require $Q^{'} Y$ or $Q_{1}^{'} Y$ where $Y$ is an NxL matrix (if either $Q Y$ or $Q_{1} Y$ are required, see qyr()). Since $Q$ can be a very large matrix, qtyr() has been provided for the calculation of $Q^{'} Y$ which will be a much smaller matrix. $Q_{1}^{'} Y$ will be a submatrix of $Q^{'} Y$ . In particular,

G = Q_{1}^{'} Y = qty [1 : P, .]

and $Q_{2}^{'} Y$ is the remaining submatrix:

H = Q_{2}^{'} Y = qty [P + 1 : N, .]

Suppose that $X$ is an NxK dataset of independent variables, and $Y$ is an Nx1 vector of dependent variables. Then it can be shown that

b = R^{- 1} G

and

\begin{array}{r} s_{j} = \sum_{i = 1}^{N - P} H_{i, j} \\ j = 1, 2, . . . L \end{array}

where b is a PxL matrix of least squares coefficients and s is a 1xL vector of residual sums of squares. Rather than invert $R$ directly, however, it is better to apply qrsol() to

R b = Q_{1}' Y

For rank deficient least squares problems, see qtyre() and qtyrep().

Source#

qtyr.src