### 8. Numerical Examples In this section, we present numerical examples to illustrate the performance of the QMR method. The examples are based on a variety of test problems, including both small and large systems. The results show that the QMR method is effective in solving non-Hermitian linear systems, and it often outperforms other methods such as BCG, CGS, and Bi-CGSTAB in terms of convergence speed and numerical stability. One of the key advantages of the QMR method is its ability to handle breakdowns and numerical instabilities that are common in other iterative methods. The look-ahead Lanczos algorithm used in the QMR method allows for the generation of basis vectors for the Krylov subspaces induced by A, which helps to avoid the problems associated with the standard Lanczos algorithm. In addition, the QMR method has a good convergence property, which is reflected in the error bounds derived in Section 6. These bounds show that the QMR method can achieve a high degree of accuracy, especially when the matrix A has a favorable spectrum. The numerical examples also demonstrate the effectiveness of the QMR method in practice. For example, in one test problem, the QMR method was able to converge to the exact solution in a number of iterations that was significantly fewer than those required by other methods. This suggests that the QMR method is a promising approach for solving non-Hermitian linear systems. Overall, the numerical examples show that the QMR method is a robust and efficient method for solving non-Hermitian linear systems. It is particularly well-suited for problems where the standard BCG method may fail due to breakdowns or numerical instabilities. The QMR method's ability to handle these issues makes it a valuable tool for solving a wide range of linear systems.### 8. Numerical Examples In this section, we present numerical examples to illustrate the performance of the QMR method. The examples are based on a variety of test problems, including both small and large systems. The results show that the QMR method is effective in solving non-Hermitian linear systems, and it often outperforms other methods such as BCG, CGS, and Bi-CGSTAB in terms of convergence speed and numerical stability. One of the key advantages of the QMR method is its ability to handle breakdowns and numerical instabilities that are common in other iterative methods. The look-ahead Lanczos algorithm used in the QMR method allows for the generation of basis vectors for the Krylov subspaces induced by A, which helps to avoid the problems associated with the standard Lanczos algorithm. In addition, the QMR method has a good convergence property, which is reflected in the error bounds derived in Section 6. These bounds show that the QMR method can achieve a high degree of accuracy, especially when the matrix A has a favorable spectrum. The numerical examples also demonstrate the effectiveness of the QMR method in practice. For example, in one test problem, the QMR method was able to converge to the exact solution in a number of iterations that was significantly fewer than those required by other methods. This suggests that the QMR method is a promising approach for solving non-Hermitian linear systems. Overall, the numerical examples show that the QMR method is a robust and efficient method for solving non-Hermitian linear systems. It is particularly well-suited for problems where the standard BCG method may fail due to breakdowns or numerical instabilities. The QMR method's ability to handle these issues makes it a valuable tool for solving a wide range of linear systems.