About qr decomposition function : qr

X = qr(A) return a matrix X such that triu(X) is the upper triangualr factor R .

Could tell me how to calculate X ? (what is the algorithm to calculate X ?)

Also, why triu(X) is equal to R ?

[Matt J]

That is only true if A is type sparse. If A is not sparse, then qr(A) will return the orthogonal factor Q.

[Titus Edelhofer]

Why not? Also for full matrices X = qr(A) returns a matrix x, where the upper triangular part coincides with the matrix R that is returned when you ask for two output arguments [Q,R]=qr(A).

Titus

[Matt J]

Whoops, that's right. I forgot how that calling syntax worked. However, like the OP, I find it non-intuitive that qr() would return an output of that form. If triu(X) is R, then what is useful about the lower triangular part of the output? Why not just return R instead of forcing the user to call triu(X)?

[Titus Edelhofer]

Good question. The example at the bottom of the doc makes exactly this distinction between sparse and full:

if issparse(A)
  R = qr(A); 
else 
  R = triu(qr(A)); 
end

The values in the lower triangle describe the elementary reflectors for computing the qr decomposition, although I admit I'm not sure what you can use them for ;-).

Titus

Answer 1

[Titus Edelhofer]

Hi,

regarding the algorithm: the help for qr states

% X = QR(A) and X = QR(A,0) return the output of LAPACK's *GEQRF routine. % TRIU(X) is the upper triangular factor R.

So take a look at the GEQRF documentation about the algorithm. It looks as if the algorithm is based on Householder reflections.

Titus