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): largest singular value of R(1:RANK,1:RANK);
SVAL(2): smallest singular value of R(1:RANK,1:RANK);
SVAL(3): 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).
   

REFERENCE

Slicot library routines MB03OD, ZB03OD.

Examples

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

See Also

qr  rank