qr

QR decomposition

Calling Sequence

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

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 comumns 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.