qr

Arguments

X

real or complex matrix

tol

nonnegative real number

Q

square orthogonal or unitary matrix

R

matrix with same dimensions as X

E

permutation matrix

rk

integer (QR-rank of X)

Description

[Q,R] = qr(X)

produces an upper triangular matrix R of the same dimension as X and an orthogonal (unitary in the complex case) matrix Q so that X = Q*R. [Q,R] = qr(X,"e") produces an "economy size": If X is m-by-n with m > n, then only the first n columns of Q are computed as well as the first n rows of R.

From Q*R = X , it follows that the kth column of the matrix X, is expressed as a linear combination of the k first columns of Q (with coefficients R(1,k), ..., R(k,k)). The k first columns of Q make an orthogonal basis of the subspace spanned by the k first columns of X. If column k of X (i.e. X(:,k) ) is a linear combination of the first p columns of X, then the entries R(p+1,k), ..., R(k,k) are zero. It this situation, R is upper trapezoidal. If X has rank rk, rows R(rk+1,:), R(rk+2,:), ... are zeros.

[Q,R,E] = qr(X)

produces a (column) permutation matrix E, an upper triangular R with decreasing diagonal elements and an orthogonal (or unitary) Q so that X*E = Q*R. If rk is the rank of X, the rk first entries along the main diagonal of R, i.e. R(1,1), R(2,2), ..., R(rk,rk) are all different from zero. [Q,R,E] = qr(X,"e") produces an "economy size": If X is m-by-n with m > n, then only the first n columns of Q are computed as well as the first n rows of R.

[Q,R,rk,E] = qr(X ,tol)

returns rk = rank estimate of X i.e. rk is the number of diagonal elements in R which are larger than a given threshold tol.

[Q,R,rk,E] = qr(X)

returns rk = rank estimate of X i.e. rk is the number of diagonal elements in R which are larger than tol=R(1,1)*%eps*max(size(R)). See rankqr for a rank revealing QR factorization, using the condition number of R.

Examples

// QR factorization, generic case
// X is tall (full rank)
X=rand(5,2);[Q,R]=qr(X); [Q'*X R]

//X is fat (full rank)
X=rand(2,3);[Q,R]=qr(X); [Q'*X R]

//Column 4 of X is a linear combination of columns 1 and 2:
X=rand(8,5);X(:,4)=X(:,1)+X(:,2); [Q,R]=qr(X); R, R(:,4)

//X has rank 2, rows 3 to $ of R are zero:
X=rand(8,2)*rand(2,5);[Q,R]=qr(X); R

//Evaluating the rank rk: column pivoting ==> rk first
//diagonal entries of R are non zero :
A=rand(5,2)*rand(2,5);
[Q,R,rk,E] = qr(A,1.d-10);
norm(Q'*A-R)
svd([A,Q(:,1:rk)])    //span(A) =span(Q(:,1:rk))

See also

• rankqr — rank revealing QR factorization
• rank — rank
• svd — singular value decomposition
• rowcomp — row compression, range
• colcomp — column compression, kernel, nullspace

Used Functions

qr decomposition is based the Lapack routines DGEQRF, DGEQPF, DORGQR for the real matrices and ZGEQRF, ZGEQPF, ZORGQR for the complex case.