 qr - Maple Help

Matlab

 qr
 compute the QR orthogonal-triangular decomposition of a MapleMatrix or MatlabMatrix in MATLAB(R), where X*P = Q*R Calling Sequence qr(X, output=R) qr(X, output=QR) qr(X, output=QRP) Parameters

 X - MapleMatrix or MatlabMatrix output - specify the form of the output (optional) R - return the upper triangular matrix R QR - return unitary matrix Q and upper triangular R matrix QRP - return Q, R, and permutation matrix P Description

 • The qr command computes the QR orthogonal-triangular decomposition of a matrix (either a Maple matrix or a MatlabMatrix) in MATLAB®. When output=QRP, the result is computed where $XP=QR$. When output=QR, the result is computed where $X=QR$.
 • The matrix X can be either square or rectangular.
 • The matrix X is expressed as product of an upper triangular matrix and either a real orthonormal matrix or a complex unitary matrix.
 • The default if no output option is specified is to return the matrices Q and R. Examples

Define the Maple matrix

 > $\mathrm{with}\left(\mathrm{Matlab}\right):$
 > $\mathrm{maplematrix_a}≔\mathrm{Matrix}\left(\left[\left[3,1,3\right],\left[1,6,4\right],\left[6,7,8\right],\left[3,3,7\right]\right]\right)$
 ${\mathrm{maplematrix_a}}{≔}\left[\begin{array}{ccc}{3}& {1}& {3}\\ {1}& {6}& {4}\\ {6}& {7}& {8}\\ {3}& {3}& {7}\end{array}\right]$ (1)

The QR decomposition of this MapleMatrix is computed and returns Q and R, as follows:

 > $Q,R≔\mathrm{Matlab}\left[\mathrm{qr}\right]\left(\mathrm{maplematrix_a}\right)$

 Q, R := [-0.404519917477945468 , 0.418121005003545431 ,  -0.120768607347027060 , -0.804334137667873206] [-0.134839972492648424 , -0.903141370807658106 , 0.0315048540905287777 , -0.406400406400609482] [-0.809039834955890602 , -0.0836242010007090253 , -0.399061485146697980 , 0.423333756667301608] [-0.404519917477945301 , 0.0501745206004254873 , 0.908389959610246600 ,  0.0931334264668064182] [-7.41619848709566209    -8.09039834955890668    -11.0568777443971715] [                                                                    ] [         0.             -5.43557306504609006    -2.67597443202269013] [                                                                    ] [         0.                      0.             2.92995143041917760 ] [                                                                    ] [         0.                      0.                      0.         ]

The QR decomposition returning only the R matrix is as follows:

 > $M≔\mathrm{Matlab}\left[\mathrm{qr}\right]\left(\mathrm{maplematrix_a},\mathrm{output}='R'\right)$

 [-7.41619848709566209 ,  -8.09039834955890668 ,  -11.0568777443971715] [                                                                    ] [0.0960043149368622339 ,  -5.43557306504609006 , -2.67597443202269013] [                                                                    ] [0.576025889621173404 ,   0.166971439413840017 ,  2.92995143041917760] [                                                                    ] [0.288012944810586701 , 0.0361500352119390120 , -0.704436674263540508]

To force the lower triangle entries to zero, use

 > $\mathrm{Matrix}\left(M,\mathrm{shape}=\mathrm{triangular}\left[\mathrm{upper}\right]\right)$

 [-7.41619848709566209    -8.09039834955890668    -11.0568777443971715] [                                                                    ] [         0.             -5.43557306504609006    -2.67597443202269013] [                                                                    ] [         0.                      0.             2.92995143041917760 ] [                                                                    ] [         0.                      0.                      0.         ]

Note that the R in $\mathrm{output}='R'$ is surrounded by quotation marks, since the variable R was assigned previously.  QR decomposition returning Q, R, and P matrices is as follows:

 > $Q,R,P≔\mathrm{Matlab}\left[\mathrm{qr}\right]\left(\mathrm{maplematrix_a},\mathrm{output}=\mathrm{QRP}\right)$