## Group MathWorks

Group Description ```Matrices contributed by The Mathworks, Inc. Some matrices come from other sources. pivtol: A small matrix from the spline toolbox that illustrates the dramatic growth factor that can occur with a pivot tolerance of 0.1 (in UMFPACK v4.0). Needs a tolerance of 0.26, for off-diagonal pivoting. Pd: An unsymmetric matrix (x=P\d) with many sparse right-hand-sides. The Pd.A matrix is 8081-by-8081. Pd.b and Pd_rhs.A are the same 8081-by-12406 matrix (the right-hand-sides). It is highly reducible to block triangular form, with 7877 diagonal blocks, the largest of which is 30-by-30. Only 9 of the blocks are bigger than 1-by-1. This is a good candidate for a solver based on dmperm. Harvard500: Web connectivity matrix from Cleve Moler (Numerical Computing with MATLAB, 2004). A(i,j) is nonzero if there is a link from page j to page i. The URL of page j is in Problem.url{j} in the MATLAB form of this problem. For more details, see Cleve Moler's book at http://www.mathworks.com/moler . The matrix is 500-by-500, with a structural rank of 233 and numerical rank of 170. tomography: An earth science problem from David Yang. Contributed by Bobby Cheng. Much like a medical tomography problem. "UMFPACK 4.0 is consistently slower than MATLAB v5." In MATLAB 7.1, [L,U,P] = lu (A) takes 0.07 seconds (using GPLU), but UMFPACK v4.3 with [L,U,P,Q] = lu (A) takes 0.11 seconds. Kuu, Muu: symmetric positive definite matrices which caused problems with UMFPACK v4.0 (in MATLAB 6.5). chol, or UMFPACK v4.3 or later, work fine. QRpivot is a 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'. TS: a counter-example that triggers a bug in MATLAB R2009a. Fixed in R2009b.```
Displaying all 11 collection matrices
Id Name Group Rows Cols Nonzeros
2256 TS MathWorks 2,142 2,142 45,262
1894 QRpivot MathWorks 660 749 3,808
1404 Kaufhold MathWorks 8,765 8,765 42,471
1378 Sieber MathWorks 2,290 2,290 14,873
1331 Muu MathWorks 7,102 7,102 170,134
1330 Kuu MathWorks 7,102 7,102 340,200
1328 tomography MathWorks 500 500 28,726
1172 Harvard500 MathWorks 500 500 2,636
1171 Pd_rhs MathWorks 8,081 12,406 6,323
1170 Pd MathWorks 8,081 8,081 13,036
861 pivtol MathWorks 102 102 306