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;
// output
[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;
// output
[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;
// output
[2,0,1]