qr

Syntax

qr(obj, [mode=’full’], [pivoting=false])

Arguments

obj is a matrix.

mode is a string indicating what information is to be returned. It can be “full”, “economic” or “r”. The default value is “full”.

pivoting is a Boolean value. The default value is false.

Details

Perform the QR decomposition of a matrix. Decompose a matrix A into an orthogonal matrix Q and an upper triangular matrix R, with A=Q*R.

Given an m-by-n matrix A:

• If mode=”full”, return 2 matrices: Q (m-by-m) and R (m-by-n).

• If mode=”economic”, return 2 matrices: Q (m-by-k) and R (k-by-n) with k=min(m,n).

• If mode=‘“r”, only return matrix R (m-by-n).

If pivoting= true, also return a vector P which has the same length as the number of columns of the matrix. P is the pivoting for rank-revealing QR decomposition indicating the location of 1s in the permutation matrix.

Examples

$ A = matrix([2,5,7,5], [5,2,5,4], [8,2,6,4]);

$ Q,R = qr(A);
$ Q;

#0	#1	#2	#3
-0.197066	0.903357	0.300275	0.234404
-0.492665	-0.418267	0.459245	0.609449
-0.68973	-0.02475	0.170745	-0.703211
-0.492665	0.091573	-0.818398	0.281284

$ R;

#0	#1	#2
-10.148892	-7.38997	-8.670898
0	3.922799	6.608121
0	0	1.071571
0	0	0

$ Q,R=qr(A,mode='economic');
$ Q;

#0	#1	#2
-0.197066	0.903357	0.300275
-0.492665	-0.418267	0.459245
-0.68973	-0.02475	0.170745
-0.492665	0.091573	-0.818398

$ R;

#0	#1	#2
-10.148892	-7.38997	-8.670898
0	3.922799	6.608121
0	0	1.071571

$ Q,T,R=qr(A,mode='raw');
$ R;

#0	#1	#2
-10.148892	-7.38997	-8.670898
0.41156	3.922799	6.608121
0.576184	0.3046	1.071571
0.41156	0.156539	0.900419

$ T;
[1.197066,1.790053,1.104512]

$ R

#0	#1	#2
-10.148892	-7.38997	-8.670898
0	3.922799	6.608121
0	0	1.071571

$ Q,T,R,P = qr(A,mode='raw',pivoting=true);
$ Q;

#0	#1	#2
-10.954451	-8.033264	-8.215838
0.105516	-6.20215	-1.45111
0.316548	0.37699	-0.627918
0.211032	0.284188	0.936372

$ T;
[1.730297,1.635478,1.065648]

$ R

#0	#1	#2
-10.954451	-8.033264	-8.215838
0	-6.20215	-1.45111
0	0	-0.627918

$ P;
[2,0,1]