A Sample Jordan Block Problem

We now look for a nearby matrix with a Jordan structure $J_4(\alpha) \oplus J_2(\beta) \oplus \mbox{(anything)}$ to the $12 \times 12$ Frank matrix, a matrix whose Jordan structure is known to be very difficult.

>> !cp examples/jordan/*.m .
>> A = frank(12);
>> evs = sort(eig(A)); eigvs = [mean(evs(1:4)) mean(evs(5:6))];
>> parameters(A,eigvs,[4; 2],'bundle')    
   1 Jordan block  of order 4 of eigenvalue 0.076352
   1 Jordan block  of order 2 of eigenvalue 0.464128
>> Y0 = guess;          
>> [fn,Yn] = sg_min(Y0,'dog','euclidean');
iter    grad            F(Y)              flops         step type
0       1.775900e-02    9.122159e-06        16257       none
  invdgrad: max iterations reached inverting the Hessian by MINRES
1       1.011261e-03    4.121994e-07      4638811       good dog
  invdgrad: max iterations reached inverting the Hessian by MINRES
2       4.250874e-04    3.523843e-07      9274101       good dog
  invdgrad: max iterations reached inverting the Hessian by MINRES
3       2.146629e-03    1.696542e-07     13936305       good dog
  invdgrad: max iterations reached inverting the Hessian by MINRES
4       5.260491e-04    5.326314e-08     18553494       good dog
  invdgrad: max iterations reached inverting the Hessian by MINRES
5       4.861126e-04    7.545415e-09     23231554       good dog
  invdgrad: max iterations reached inverting the Hessian by MINRES
6       4.228777e-05    3.014858e-09     27846197       good dog
  invdgrad: max iterations reached inverting the Hessian by MINRES
7       2.954536e-06    2.924391e-09     32508118       good dog
8       2.868836e-08    2.924238e-09     37207124       good dog
9       2.093780e-09    2.924238e-09     38384200       good dog
10      2.479121e-09    2.924238e-09     39163025       good dog
11      1.713240e-09    2.924238e-09     39921924       bad dog

Figure 9.2 shows the convergence curve for this run.

Figure 9.2: Jordan problem