Important: The linalg package has been deprecated. Use the superseding command LinearAlgebra[QRDecomposition], instead.

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

The routine QRdecomp computes the QR decomposition of the matrix A.

For matrices of floating-point entries, the numerically stable Householder-transformations are used.  For symbolic computation, the Gram-Schmidt process is applied.

The result is an upper triangular matrix R.  The orthonormal (unitary) factor Q is passed back to the Q parameter.

The default factorization is the full QR where R will have the same dimension as A.  Q will be a full rank square matrix whose first n columns span the column space of A and whose last m-n columns span the null space of A.

If the (optional) fullspan arg is set to false, a Q1R1 factorization will be given where the Q1 factor will have the same dimension as A and, assuming A has full column rank, the columns of Q will span the column space of A. The R factor will be square and agree in dimension with Q.  The default for fullspan is true.

If A is an n by n matrix then $\prod _{i\],i=1\mathrm{..}n}R\[i$.

If A contains complex entries, the Q factor will be unitary.

The QR factorization can be used to generate a least squares solution to an overdetermined system of linear equations.  If $\mathrm{Ax}=b$, and $\mathrm{QR}=A$ then $\mathrm{Rx}=\mathrm{transpose}\left(Q\right)b$ can be solved through backsubstitution.

The command with(linalg,QRdecomp) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{linalg}\right)\:$

$A≔\mathrm{matrix}\left(3\,3\,\left[1\,2\,3\,0\,0\,1\,2\,3\,4\right]\right)$

$A≔\left[\begin{array}{ccc}1& 2& 3\\ 0& 0& 1\\ 2& 3& 4\end{array}\right]$

$R≔\mathrm{QRdecomp}\left(A\,Q='q'\,\mathrm{rank}='r'\right)$

$R≔\left[\begin{array}{ccc}\sqrt{5}& \frac{8\sqrt{5}}{5}& \frac{11\sqrt{5}}{5}\\ 0& \frac{\sqrt{5}}{5}& \frac{2\sqrt{5}}{5}\\ 0& 0& 1\end{array}\right]$

$\mathrm{rank}\left(R\right)$

$3$

$\mathrm{evalm}\left(q\right)$

$\left[\begin{array}{ccc}\frac{\sqrt{5}}{5}& \frac{2\sqrt{5}}{5}& 0\\ 0& 0& 1\\ \frac{2\sqrt{5}}{5}& -\frac{\sqrt{5}}{5}& 0\end{array}\right]$

$A≔\mathrm{matrix}\left(4\,2\,\left[1\,2\,3\,0\,0\,1\,2\,3\right]\right)$

$A≔\left[\begin{array}{cc}1& 2\\ 3& 0\\ 0& 1\\ 2& 3\end{array}\right]$

$R≔\mathrm{QRdecomp}\left(A\,Q='q'\,\mathrm{fullspan}=\mathrm{false}\right)$

$R≔\left[\begin{array}{cc}\sqrt{14}& \frac{4\sqrt{14}}{7}\\ 0& \frac{\sqrt{462}}{7}\end{array}\right]$

$\mathrm{rank}\left(R\right)$

$2$

$\mathrm{evalm}\left(q\right)$

$\left[\begin{array}{cc}\frac{\sqrt{14}}{14}& \frac{5\sqrt{462}}{231}\\ \frac{3\sqrt{14}}{14}& -\frac{2\sqrt{462}}{77}\\ 0& \frac{\sqrt{462}}{66}\\ \frac{\sqrt{14}}{7}& \frac{13\sqrt{462}}{462}\end{array}\right]$