Block Toeplitz matrices are a special class of matrices that exhibit reduced memory requirements and a reduced complexity of matrix-vector multiplications. We herein present an efficient computational approach to solve a sequence of block Toeplitz systems arising from a block Toeplitz system with multiple right-hand sides. Two different numerical schemes are implemented for the solution of the sequence of block Toeplitz systems based on global and block variants of the generalized minimal residual (GMRES) method. The performance of the schemes is assessed in terms of the wall clock time of the iterative solution process, the number of multiplications with the block Toeplitz system matrix and the peak memory usage. To demonstrate the method, two numerical examples are presented. In the first case study, aeroacoustic prediction of an airfoil in turbulent flow is examined, which requires multiple solutions of the wall pressure field beneath the turbulent boundary layer. The fluctuating pressure on the surface of the airfoil is synthesized in terms of uncorrelated wall plane waves, whereby each realization of the wall pressure field is an input to the acoustic solver based on the boundary element method (BEM). The total acoustic response from the airfoil in turbulent flow is then obtained from an ensemble average for the number of realizations considered. The number of realizations to yield a converged solution for the wall pressure field leads to a sequence of block Toeplitz systems. The second case study examines the nonlinear eigenvalue analysis of a sonic crystal barrier composed of locally resonant C-shaped sound-hard scatterers. The periodicity of the sound barrier leads to a block Toeplitz system matrix whereas the nonlinear eigenvalue problem requires the solution of sequences of linear systems. The combined technique to solve the sequences of block Toeplitz systems using the proposed variants of the GMRES is shown to yield a computationally efficient approach for flow noise prediction and nonlinear eigenvalue analysis.

Block Toeplitz 矩阵是一类特殊的矩阵，可以减少内存需求并降低矩阵向量乘法的复杂度。我们在此提出了一种有效的计算方法来解决由具有多个右侧的块托普利茨系统产生的一系列块托普利茨系统。基于广义最小残差 (GMRES) 方法的全局和块变体，为解决块托普利茨系统的序列实现了两种不同的数值方案。根据迭代求解过程的挂钟时间、与块 Toeplitz 系统矩阵的乘法次数和峰值内存使用情况来评估方案的性能。为了演示该方法，给出了两个数值例子。在第一个案例研究中，研究了湍流中翼型的气动声学预测，这需要湍流边界层下方的壁压力场的多个解。翼型表面上的波动压力是根据不相关的壁面波合成的，因此壁面压力场的每个实现都是基于边界元法 (BEM) 的声学求解器的输入。然后从所考虑的实现次数的整体平均值获得湍流中翼型的总声学响应。产生壁压力场收敛解的实现数量导致了一系列块托普利茨系统。第二个案例研究检查了由局部共振 C 形声硬散射体组成的声波晶体屏障的非线性特征值分析。声屏障的周期性导致分块 Toeplitz 系统矩阵，而非线性特征值问题需要线性系统序列的解。使用所提出的 GMRES 变体求解块 Toeplitz 系统序列的组合技术被证明为流噪声预测和非线性特征值分析提供了一种计算有效的方法。