When I use the QR factorization I do not get a upper triangular matrix.

According to the documentation if A is an mxn matrix the function R=qr(A) should return a mxn upper triangular matrix. If I use the following example:

A=[1 0 0 0 ; 0 1 0 1; 0 1 1 0];
R=qr(A)

I get the following output, which is not an upper triangular matrix

R=
    1.0000         0         0         0
         0   -1.4142   -0.7071   -0.7071
         0    0.4142    0.7071   -0.7071

Likewise if I try to compute the "economy size" QR-decomposition via R=qr(A, 0) I also get a matrix that is not upper triangular:

ans =

      1.0000         0         0         0
           0   -1.4142   -0.7071   -0.7071
           0    0.4142    0.7071   -0.7071

If on the other hand I use the code:

A=[1 0 0 0 ; 0 1 0 1; 0 1 1 0];
[Q,R,P]=qr(A);
R =    1.0000         0         0         0
       0   -1.4142   -0.7071   -0.7071
       0         0    0.7071   -0.7071

I get that R is an upper triangular matrix. Am I misunderstanding the documentation for qr? Ideally I would like to use qr(A,0) to compute the economy QR decomposition of a matrix, but have the output be upper triangular.

Answer 1

[Steven Lord]

Yes, you are misunderstanding the documentation. On the documentation page, the section of the description that describes the syntax "R = qr(A)" appears right after the statement "If A is sparse:"

Your matrix A is full not sparse, and so what you're actually computing is "X = qr(A) and X = qr(A,0) return a matrix X such that triu(X) is the upper triangular factor R." from the section "If A is full:"