rankqr

Scilab Function
Last update : 12/01/2004

rankqr - rank revealing QR factorization

### Calling Sequence

  `[Q,R,JPVT,RANK,SVAL]=rankqr(A, [RCOND,JPVT])`

### Parameters

- `A`: real or complex matrix
- `RCOND`: real number used to determine the effective rank of `A`, which is defined as the order of the largest leading triangular submatrix `R11` in the QR factorization with pivoting of`A`, whose estimated condition number < `1/RCOND`.
- `JPVT`

  : integer vector on entry, if `JPVT(i)` is not 0, the `i`-th column of `A` is permuted to the front of `AP`, otherwise column `i` is a free column. On exit, if `JPVT(i) = k`, then the `i`-th column of `A*P` was the `k`-th column of `A`.

- `RANK`: the effective rank of `A`, i.e., the order of the submatrix `R11`. This is the same as the order of the submatrix `T1` in the complete orthogonal factorization of `A`.
- `SVAL`: real vector with 3 components; The estimates of some of the singular values of the triangular factor `R`.

  `SVAL(1)` is the largest singular value of `R(1:RANK,1:RANK)`;

  `SVAL(2)` is the smallest singular value of `R(1:RANK,1:RANK)`;

  `SVAL(3)` is the smallest singular value of `R(1:RANK+1,1:RANK+1)`, if `RANK` < `MIN(M,N)`, or of `R(1:RANK,1:RANK)`, otherwise.

### Description

To compute (optionally) a rank-revealing QR factorization of a real general M-by-N real or complex matrix `A`, which may be rank-deficient, and estimate its effective rank using incremental condition estimation.

The routine uses a QR factorization with column pivoting:

            A * P = Q * R,  where  R = [ R11 R12 ],
                                       [  0  R22 ]

with `R11` defined as the largest leading submatrix whose estimated condition number is less than `1/RCOND`. The order of `R11`, `RANK`, is the effective rank of `A`.

If the triangular factorization is a rank-revealing one (which will be the case if the leading columns were well- conditioned), then `SVAL(1)` will also be an estimate for the largest singular value of `A`, and `SVAL(2)` and `SVAL(3)` will be estimates for the `RANK`-th and `(RANK+1)`-st singular values of `A`, respectively.

By examining these values, one can confirm that the rank is well defined with respect to the chosen value of `RCOND`. The ratio `SVAL(1)/SVAL(2)` is an estimate of the condition number of `R(1:RANK,1:RANK)`.

### Examples

    A=rand(5,3)*rand(3,7);
    [Q,R,JPVT,RANK,SVAL]=rankqr(A,%eps)

### See Also

`qr`, `rank`,

### Used Function

Slicot library routines MB03OD, ZB03OD.