## MathWorks/QRpivot

Limitation of basic solution to x=A\b using qr(A); needs min 2-norm
Name QRpivot MathWorks 1894 660 749 3,808 3,808 Counter Example Problem No 2008 P. Quillen T. Davis
Structural Rank 660 true 9 1 0 0% 0% no no real
SVD Statistics
Matrix Norm 9.332323e+00
Minimum Singular Value 3.205865e-02
Condition Number 2.911016e+02
Rank 660
sprank(A)-rank(A) 0
Null Space Dimension 0
Full Numerical Rank? yes
Download ```Counter-example problem from The MathWorks, Pat Quillen This matrix was obtained from a MATLAB user. It illustrates the limitations inherent in computing a basic solution to an under- determined system without the use of column pivoting. With column pivoting (which can only be done in MATLAB with full matrices) the problem is solved properly. When finding the min 2-norm solution (ignoring fill-in): [Q,R] = qr (A') ; x = Q*(R'\b) ; a good solution is found. To reduce fill-in: p = colamd (A') ; [Q,R] = qr (A (p,:)') ; x = Q*(R'\b(p)) ; which also finds a good solution. However, x=A\b computes a basic solution, using this algorithm: q = colamd (A) ; [Q,R] = qr (A (:,q)) ; x = R\(Q'*b) ; x (q) = q ; which finds an error with norm(A*x-b) of 1e-9 in MATLAB 7.6. With random permutations, and determining the cond(R1) of the leading trianglar part (R is "squeezed" and the columns can be partitioned into [R1 R2] where R1 is square and upper triangular) leads to the following results. Note that the error is high when condest(R1) is high. Note in particular the last trial. So this clinches the question. MATLAB's QR, and my new sparse QR, both use a rank-detection method (by Heath) that does not do column pivoting, and which is known to fail for some problems - for which Grimes & Lewis' method will likely succeed. The advantage of my QR is that I now always return R as upper trapezoidal, so if the user is concerned, he/she can easily check condest(R(:,1:m)) if m < n. err 7.71e-07 condest R1 2.18e+12 err 1.25e-09 condest R1 9.82e+08 err 2.47e-09 condest R1 2.46e+11 err 4.00e-09 condest R1 4.03e+09 err 9.88e-10 condest R1 4.73e+09 err 2.25e-08 condest R1 5.34e+09 err 2.00e-08 condest R1 1.04e+09 err 1.09e-09 condest R1 6.83e+08 err 6.18e-08 condest R1 8.13e+10 err 3.13e-10 condest R1 4.23e+09 err 6.64e-10 condest R1 2.46e+10 err 5.76e-09 condest R1 4.31e+09 err 7.61e-07 condest R1 5.08e+10 err 2.27e-09 condest R1 4.94e+09 err 3.99e-10 condest R1 2.80e+09 err 1.37e-09 condest R1 3.13e+09 err 6.93e-05 condest R1 1.84e+14 err 1.35e-08 condest R1 7.18e+09 err 1.09e-08 condest R1 1.79e+11 err 1.81e-09 condest R1 2.99e+08 err 1.55e-01 condest R1 2.45e+18 In summary, this is a "feature" not a "bug". If you want a reliable solution to an underdetermined system, find the min 2norm solution via a QR factorization of A'.```