## 5 Numerical Experiments and Conclusions
The test systems were automatically generated by routine \"genA\"; the three graphs of Figure 1 are examples from a large number of cases, corresponding to certain systems of dimensions 50, 100, 200 respectively; other parameters like \(c\) (the condition number), \(\theta\) (the factor in Step 4), \(p(n_{1},...,n_{p})\) (the number of partitions and their dimensions), were chosen as follows:  
(a) \(m=50,\ c=3.569,\ \theta=0.02,\ p(...)=5(11,9,13,11,6),\ E_{0}=I\),  
(b) \(m=100,\ c=71.69,\ \theta=0.02,\ p(...)=5(21,19,23,21,16),\ E_{0}=I\),  
(c) \(m=200,\ c=785,\ \theta=0.02,\ p(...)=5(41,39,43,41,36),\ E_{0}=I\). The following conclusions can be drawn. (1) The proposed parallel [PERSON]'s algorithm works well if the system is relatively well conditioned (the condition number should be around ten), and in this case the behavior of Algorithm 3 is very similar to the parallel [PERSON] method and significantly better than the parallel [PERSON] method. (Figure 1a); (2) For medium conditioned systems, the behavior of Algorithm 3 is unpredictable, sometimes it works better than the [PERSON] method (Figure 1b); (3) If the system is poorly conditioned (the condition number is greater than 300) the proposed parallel [PERSON]'s algorithm fails to converge (the Newton method has the same behavior) (Figure 1c). Figure 2a presents the behavior of the sequence generated by the Algorithm 3 for a linear system of dimension \(m=500\), \(c=4.366,\ p(...)=5(101,51,151,101,96),\ \theta=0.2,\ E_{0}=I\). The condition (2) of Theorem 1 is not satisfied and the sequence generated by this algorithm (in this case) does not converge. However, the behavior is interesting, the generated sequence becomes very close to the solution, the error decreases until a very small value (in our example until \(10^{-12}\)), and then the good behavior is broken. The graph corresponding to the sequence generated by Algorithm 4 for a system of dimension \(m=600\) and \(c=3.942,\ p(...)=6(101,51,151,101,101,95),\ \theta=0.03,\ E_{0}=(I+D)^{-1}\) is presented in Figure 2b. We can observe the similarities between this sequence and the sequence obtained by the parallel Newton method. Figure 2: The behavior of the sequence generated by Algorithm 3, Parallel Newton method and Parallel Cimmino method (the graphs (a)) and by Algorithm 4, Parallel Newton method and Parallel Cimmino method (the graphs (b)); (**a**) system which does not satisfy the condition (2); (**b**) system well conditioned. **Remark 2**. : _The numerical experiments presented here, for the most part, have been performed with the basic Algorithm 3; for Algorithm 4 similar conclusions can be drawn. A special characteristics of Algorithm 4, in comparison with Algorithm 3, is that Algorithm 4 is less sensitive to \(\theta\); this parameter can take its values on a much larger interval, \(\theta\in(0,2)\)._  
**Experiment 2. **  
This experiment is devoted to show the behavior of the sequences generated by Algorithm 5 in the case of nonlinear systems. We considered several nonlinear systems of low dimensions and of certain sparsity (every nonlinear equation depends on a few numbers of unknowns). Figure 3 presents the results for the following three nonlinear systems:  
\[(1)\left\{ \begin{array}{l}4x_{1}^{2}-x_{2}x_{6}-3=0,\\ x_{1}-5x_{2}+x_{3}^{2}+x_{7}=0,\\ -x_{1}+5x_{3}-x_{4}^{2}-3=0,\\ x_{1}x_{2}-4x_{4}+x_{5}^{2}+2=0,\\ x_{4}-5x_{5}+2x_{6}=0,\\ x_{1}-6x_{6}+4=0,\\ x_{2}+x_{3}^{2}-4x_{7}+2=0.\end{array} \right.\quad(2)\left\{ \begin{array}{l} 3x_{1}+x_{2}^{2}x_{3}=0,\\ 2x_{2}+x_{3}-2x_{7}^{2}=0,\\ x_{1}-0.2x_{2}x_{4}+3x_{3}=0,\\ 3x_{4}-x_{6}^{2}=0,\\ x_{2}+2x_{5}=0,\\ x_{3}x_{7}+5x_{6}=0,\\ x_{1}^{2}+x_{2}+x_{5}+4x_{7}=0.\end{array} \right.\quad(3)\left\{ \begin{array}{ l}5\frac{x_{1}^{2}+x_{2}^{2}}{x_{1}^{2}+x_{2}^{2}+1}-\frac{10}{3}=0,\\ 4\frac{x_{3}^{2}}{x_{3}^{2}+1}-2=0,\\ 3\frac{x_{2}^{2}+x_{3}^{2}}{x_{2}^{2}+x_{3}^{2}+1}=0.\\ x_{1}^{2}+x_{2}+x_{5}+4x_{7}=0.\end{array} \right.\]  
The proposed parallel Broyden's method is compared with the parallel Newton method described above.