## QR QR Decomposition of a Matrix

Section: Transforms/Decompositions

### Usage

Computes the QR factorization of a matrix. The `qr` function has multiple forms, with and without pivoting. The non-pivot version has two forms, a compact version and a full-blown decomposition version. The compact version of the decomposition of a matrix of size `M x N` is
```  [q,r] = qr(a,0)
```

where `q` is a matrix of size `M x L` and `r` is a matrix of size `L x N` and `L = min(N,M)`, and `q*r = a`. The QR decomposition is such that the columns of `Q` are orthonormal, and `R` is upper triangular. The decomposition is computed using the LAPACK routine `xgeqrf`, where `x` is the precision of the matrix. FreeMat supports decompositions of `single` and `double` types. The second form of the non-pivot decomposition omits the second `0` argument:

```  [q,r] = qr(a)
```

This second form differs from the previous form only for matrices with more rows than columns (`M > N`). For these matrices, the full decomposition is of a matrix `Q` of size `M x M` and a matrix `R` of size `M x N`. The full decomposition is computed using the same LAPACK routines as the compact decomposition, but on an augmented matrix `[a 0]`, where enough columns are added to form a square matrix. Generally, the QR decomposition will not return a matrix `R` with diagonal elements in any specific order. The remaining two forms of the `qr` command utilize permutations of the columns of `a` so that the diagonal elements of `r` are in decreasing magnitude. To trigger this form of the decomposition, a third argument is required, which records the permutation applied to the argument `a`. The compact version is

```  [q,r,e] = qr(a,0)
```

where `e` is an integer vector that describes the permutation of the columns of `a` necessary to reorder the diagonal elements of `r`. This result is computed using the LAPACK routines `(s,d)geqp3`. In the non-compact version of the QR decomposition with pivoting,

```  [q,r,e] = qr(a)
```

the returned matrix `e` is a permutation matrix, such that `q*r*e' = a`.