qr QR decomposition

## Syntax

    [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 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
- svd
- rowcomp
- colcomp

## Used Functions

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